ANALYSIS IN METRIC SPACES

HELI TUOMINEN

Contents

1. About these notes 22. Classical extension results 32.1. Tietze(-Urysohn) extension theorem 32.2. Extension of Lipschitz functions 52.3. Whitney covering, partition of unity and extension 72.4. About Sobolev extension domains in Rn (1970 - ) 93. Doubling measures, covering theorems and Hardy-Littlewood maximal

function 103.1. Basic properties of doubling measures 113.2. Covering theorems 143.3. Hardy-Littlewood maximal function and the Lebesgue differentiation

theorem 153.4. Noncentered maximal function 183.5. Lebesgue differentiation theorem 184. Mapping properties of the maximal function 194.1. Regularity of the maximal function in the Euclidean setting 194.2. Hardy-Littlewood maximal function of Lipschitz funtion in metric

spaces 204.3. The discrete convolution and the discrete maximal operator 225. Sobolev spaces in metric measure spaces - M1,p(X) 286. Sobolev spaces in metric measure spaces - N1,p(X) 386.1. Modulus of a curve family 416.2. Properties of p-weak upper gradients 446.3. How to find p-weak upper gradients 466.4. Minimal p-weak upper gradient 496.5. Convergence properties of p-weak upper gradients 526.6. Definition and basic properties of N1,p(X) 546.7. Nontriviality of spaces N1,p(X) 567. About Poincare inequalities 597.1. Bi-Lipschitz invariance of the Poincare inequality 627.2. p-Poncare inequality - improvement and dependence of p 648. Removability for Sobolev spaces and Poincare inequality 698.1. Classical results 698.2. How does the removability property of a fixed set depend on p? 70

Versio: April 7, 2014.1

About the metric setting 729. Extension results for Sobolev spaces in the metric setting 749.1. Measure density from extension 759.2. Extension from measure density 79References 84

1. About these notes

You are reading the lecture notes of the course ”Analysis in metric spaces” givenat the University of Jyvaskyla in Spring semester 2014.

We start with the classical extension results of Tietze, McShane and Whitneyand hope to end up to recent extension results for Sobolev spaces in the metricsetting. The main topics of the course are doubling measures, covering theorems,maximal functions, Lipschitz-functions, Poincare inequalities, Sobolev spaces andextension domains. Prerequisites for the course are courses ”topology 1” (metricspaces), ”measure and integration theory”, course ”real analysis” and familiaritywith classical Sobolev spaces are recommended.

Notation and standard assumptions:We assume that X = (X, d, µ) is a metric measure space equipped with a metric d

and a Borel regular, doubling outer measure µ for which the measure of every ball ispositive and finite. The doubling property means that there exists a fixed constantcD > 0, called the doubling constant, such that

µ(B(x, 2r)) ≤ cDµ(B(x, r))

for every ball B(x, r) = y ∈ X : d(y, x) < r.The distance between sets A ⊂ X and B ⊂ X is

d(A,B) = infd(a, b) : a ∈ A, b ∈ B,and for x ∈ X, the distance of x to the set A is

d(x,A) = d(x, A) = infd(x, a) : a ∈ A.The diameter of the set F ⊂ X is

diamF = supd(y, z) : y, z ∈ F.If u : X → R is function and t ∈ R, we sometimes use a short notation u > tfor sets x ∈ X : u(x) > t. The Lebesgue measure of a mesurable set E ⊂ Rn isdenoted by Ln(E) or by |E|.

We sometimes say that a measurable function u : X → R is p-integrable, if∫Xup dµ <∞.

By χE, we denote the characteristic function of a set E ⊂ X. In general, C isa positive constant whose value is not necessarily the same at each occurrence. Bywriting C = C(c1, c2), we mean that the constant C depends only on the constantsc1 and c2. If there is a positive constant C1 such that C−1

1 A ≤ B ≤ C1A, we saythat A and B are comparable.

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2. Classical extension results

This section contains the classical extension results of Tietze, McShane and Whit-ney for continuous and Lipschitz functions. We also discuss Whitney decomposition,a covering of a complement of a closed set in Rn by cubes whose diameters are com-parable to the distance of the cube to the closed set, and a corresponding partitionof unity. We close the section by a brief history about Sobolev extension domains.

2.1. Tietze(-Urysohn) extension theorem. Tietze extension theorem says thatwe can always find a continuous extension for a continuous, real valued functiondefined on a closed set. It is equivalent to the Urysohn lemma, which says thatwhenever E ⊂ X and F ⊂ X are disjoint, nonempty closed sets, then there existsa continuous function g such that g|E = 0, g|F = 1 and 0 < g(x) < 1 for allx ∈ X \ (E ∪ F ).

Theorem 2.1 (Tietze). Let F ⊂ X be a closed set and let u : F → R be a continuousfunction. Then there is a continuous function U : X → R such that U |F = u.Moreover, if there is a constant M > 0 such that |u(x)| ≤ M for all x ∈ F , then|U(x)| < M for all x ∈ X \ F .

There are several different proofs for the Tietze extension theorem. Below, wegive two different proofs. In the first one, we use the following lemma.

Lemma 2.2. Let F ⊂ X be a closed set. If u : F → R is a continuous function andM > 0 is a constant such that |u(x)| ≤M for all x ∈ F , then there is a continuousfunction v : X → R such that

(i) |v(x)| ≤ 13M for all x ∈ F ,

(ii) |v(x)| < 13M for all x ∈ X \ F ,

(iii) |u(x)− v(x)| ≤ 23M for all x ∈ F .

Proof. Let

A =x ∈ F : u(x) ≤ −1

3M

and B =x ∈ F : u(x) ≥ 1

3M.

The sets A and B are disjoint and, by the continuity of u and the assumption|u| ≤M , also closed. If A 6= ∅ and B 6= ∅, then, using the continuity of the distancefunction and the definition of the sets A and B, it is easy to see that the functionv : X → R,

v(x) =M

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d(x,A)− d(x,B)

d(x,A) + d(x,B)

has the required properties.If one of the sets A and B is empty and the other, denoted by C, is not, then

v(x) = M3

(1−min1, d(x,C)

)satisfies (i)− (iii). If A = ∅ and B = ∅, then we can take v = 1

6M .

Proof 1 of Theorem 2.1. Assume first that |u(x)| ≤M for all x ∈ F .We will use Lemma 2.2 and induction to find continuous functions vi such that

the function series∑∞

i=0 vi converges and gives us the extension U .3

Let v0 : X → R, v0 = 0. Assume that k ∈ N and that there exists continuousfunctions v0, v1, . . . , vk such that

(2.1)∣∣∣u(x)−

k∑i=0

vi(x)∣∣∣ ≤ (2

3

)kM for all x ∈ F.

Lemma 2.2 applied to continuous function u−∑k

i=0 vi and constant (23)kM gives us

a continuous function vk+1 for which

(i) |vk+1(x)| ≤ 13(2

3)kM for all x ∈ F ,

(ii) |vk+1(x)| < 13(2

3)kM for all x ∈ X \ F ,

(iii) |u(x)−∑k+1

i=0 vi(x)| ≤ (23)k+1M for all x ∈ F .

Since the geometric series∑∞

k=0(23)k converges, the Weierstrass theorem implies that

the function series∑∞

k=0 vk+1, and hence also∑∞

k=0 vk, converges uniformly on Xto a continuous function U . By (2.1), U = u in F . Moreover, by (ii), and the factthat v0 = 0, we have, for each x ∈ X \ F ,

|U(x)| =∣∣∣ ∞∑k=0

vk

∣∣∣ =∣∣∣ ∞∑k=0

vk+1

∣∣∣ ≤ ∞∑k=0

|vk+1| <∞∑k=0

1

3

(2

3

)kM = M.

Hence the theorem follows for continuous and bounded functions.

Assume then that u is unbounded. Let h : R → (−1, 1) be a strictly increasing,continuous function that is onto and let f : F → (−1, 1), f = h u. Since fis bounded and continuous,the first part of the proof gives a continuous functiong : X → (−1, 1) such that g|F = f . Now the function U : X → R, U = h−1 g, isthe required extension of u.

The second proof of the Tietze extension theorem shows also that

(2.2) supx∈F

u(x) = supx∈X

U(x) and infx∈F

u(x) = infx∈X

U(x).

Proof 2 of Theorem 2.1. We can assume that there are no isolated points in X \ F .Assume first that u is bounded and that u ≥ 0.

We will show that the function U : X → R,

U(x) =

u(x), if x ∈ F,

1d(x)

∫ 2d(x)

d(x)Mx(r) dr if x ∈ X \ F,

where d(x) = d(x, F ) and for r > 0,

Mx(r) = supy∈F∩B(x,r)

u(y),

is the desired extension of u. Clearly 0 ≤ U(x) ≤ supF u for all x ∈ X.Note that for each x ∈ X, Mx is integrable on bounded intervals of [0,∞) as a

bounded and increasing function, and that d(x) > 0 for each x ∈ X \ F because Fis closed.

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To show the continuity of U , let x ∈ X. If x is an interior point of F , then thecontinuity follows from the continuity of u. If x ∈ ∂F , then by the definition of U ,for each y ∈ X \ F , we have

infF∩B(x,3d(x,y))

u ≤ U(y) ≤ supF∩B(x,3d(x,y))

u.

This together with the continuity of u in F shows that U is continuous at x.Let then x ∈ X \ F and let y ∈ X be such that

d(x, y) < 14d(x, F ).

By the choice of y and the definition of the function M , we have that |d(x)−d(y)| ≤d(x, y) and Mx(r) ≥ My(r − d(x, y)) for all r > d(x) and d(y) > 3d(x, y). Hence,with h = d(x, y),

U(y)− U(x) =1

d(y)

∫ 2d(y)

d(y)

My(r) dr −1

d(x)

∫ 2d(x)

d(x)

Mx(r) dr

≤ 1

d(y)

∫ 2d(y)

d(y)

My(r) dr −1

d(y) + h

∫ 2d(y)−2h

d(y)+h

My(r − h) dr

=1

d(y)

∫ 2d(y)−3h

d(y)

My(r) dr +1

d(y)

∫ 2d(y)

2d(y)−3h

My(r) dr

− 1

d(y) + h

∫ 2d(y)−3h

d(y)

My(s) ds

=h

d(y)(d(y) + h)

∫ 2d(y)−3h

d(y)

My(r) dr +1

d(y)

∫ 2d(y)

2d(y)−3h

My(r) dr

≤ 4h supF u

d(y),

which implies that U is continuous at x.To show that (2.2) holds, let m = infF u and M = supF u. Now the function

v : F → R,

v(x) = u(x)−m,satisfies 0 ≤ v(x) ≤M−m for all x ∈ F . Hence, by the first part of the proof, thereis an extension V of v for which 0 ≤ V (x) ≤ M − m. The function U : X → R,U(x) = V (x) +m, is the desired extension of u.

Remark 1. Note that the Tietze theorem does not hold for open sets: considerX = R \ 0 with the euclidean metric and the function u : X → [0, 1], u(x) = 0, ifx < 0 and u(x) = 1, if x > 0.

2.2. Extension of Lipschitz functions. Lipschitz functions are smooth functionsof metric spaces. Extension problems for Lipschitz mappings between metric spacesare under active study. The main question is, for which metric spaces X and Y ,each Lipschitz mapping u : A → Y , A ⊂ X, can be extended to a Lipschitz map-ping U : X → Y . These problems require tools from several areas of mathematics

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including geometric measure theory, topology and geometry, see for example books[7], [9], [10] and the references therein.

We will recall here the classical results of McShane and Kirszbraun. The formersays that every Lipschitz function u : A → R defined on a subset of a metric spaceX can be extended to a Lipschitz function in the whole space. Thus, real-valuedLipschitz functions can be assumed to be defined in X.

Definition 2.3. Let (X, dX) and (Y, dY ) be metric spaces. A mapping u : X → Yis Lipschitz continuous (or L-Lipschitz), if there is a constant L ≥ 0 such that

(2.3) dY (u(a), u(b)) ≤ LdX(a, b) for all a, b ∈ X.

The infimum of constants L satisfying (2.3) is called the Lipschitz constant of u anddenoted by LIP(u).

Example 2.4. There are many Lipschitz mappings in metric spaces.

(1) If x0 ∈ X, then the distance function dx0 : X → R,

dx0(x) = d(x, x0),

is an 1-Lipschitz function.(2) If X,Y and Z are metric spaces and u : X → Y and v : Y → Z are Lipschitz

mappings, then v u : X → Z is Lipschitz and

LIP(v u) ≤ LIP(u) LIP(v).

Theorem 2.5 (McShane). Let A ⊂ X and let u : A→ R be an L-Lipschitz function.Then there exists an L-Lipschitz function U : X → R such that U |A = u.

Proof. Define functions U, U : X → R,

(2.4) U(x) = infa∈Au(a) + Ld(a, x) and U(x) = sup

a∈Au(a)− Ld(a, x).

Using the fact that the distance function is 1-Lipschitz and the definitions of U andU , it is easy to see that U and U are L-Lipschitz and that U |A = U |A = u.

The function U in (2.4) is largest possible extension of u in the sense that ifV : X → R is an L-Lipschitz function and V |A = u, then V ≤ U . Similarly, U isthe smallest one.20.1 =============================

Remark 2. By applying Theorem 2.5 to the coordinate functions of an L-Lipschitzfunction u : A → Rn, A ⊂ X, we obtain an L

√n-Lipschitz extension U : X → Rn.

If X = Rm, then there is actually an L-Lipschitz extension. This is the Kirszbrauntheorem from [51], for a proof, see also [38]. By Valentine [78], an extension withthe same Lipschitz constant, exists also if X and Y are Hilbert spaces, A ⊂ X andu : A → Y is Lipschitz. A generalization to metric spaces with curvature boundswas given by Lang and Schroeder in [60].

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2.3. Whitney covering, partition of unity and extension. In [80], Whitneyconstructed an extension for Ck-functions defined on a closed set of Rn (differentia-bility defined using Taylor polynomials and uniformity condition for the remainder).The main tools used in the proof are a decomposition of an open set Ω ⊂ Rn intodisjoint cubes whose diameter is comparable to the distance of the cube to Rn \ Ωand a partition of unity. Starting from the work of Jones [42], Whitney type coveringhas been an essential tool when showing that domains satisfying given conditionshave an extension property for example for Sobolev functions. We will return toWhitney type coverings in the metric space setting later in the course.

Below, by a cube, we mean a closed cube in Rn whose sides are paraller to thecoordinate axes. If the center of the cube is xi and the side length ri, then we writeQ = Qi = Q(xi, ri). Then diamQi =

√nri. Two cubes Qi and Qj are said to be

disjoint if their interiors are disjoint, that is, intQi ∩ intQj = ∅. The collection ofcubes given by Lemma 2.6 is called Whitney decomposition, Whitney covering orWhitney cubes.

Lemma 2.6. [Whitney covering] Let Ω = Rn \ F , where F ⊂ Rn is a nonemptyclosed set. Let 0 < ε < 1/4. There is a collection W = Q1, Q2, . . . of disjointcubes such that

(1) Ω = ∪iQi,(2) diamQi ≤ d(Qi, F ) ≤ 4 diamQi for each i,(3) if Qi ∩Qj 6= ∅ (boundaries have a common point), then 1

4ri ≤ rj ≤ 4ri,

(4) for each Qi, there are at most 12n cubes Qj ∈ W such that Qi ∩Qj 6= ∅.(5)

∑iχQ∗i (x) ≤ 12n for each x ∈ Ω, where Q∗i = Q(xi, (1 + ε)ri).

Proof. We will give an idea of the proof only, for details, see for example [75, ChapterVI], [61, Thm C.26].

We begin by dividing Rn into disjoint closed cubes having side length 2−k, k ∈ Z.The first generation G0 consists of cubes whose vertices belong to Zn and whose sidelength is 1. From G0, we obtain a family Gkk∈Z of collections of cubes: the sidelenght of each cube Q ∈ Gk is 2−k and from each such Q we obtain 2n cubes ofside length 2−k−1 by bisecting the sides of Q. The collection Gk+1 consists of thosesmaller cubes.

In addition to cubes of different sizes, we construct layers ”around the set F”.For each k ∈ Z, let

Ωk =x ∈ Ω :

√n2−k+1 < d(x, F ) ≤

√n2−k+2

.

Then Ω = ∪k∈ZΩk.For the first selection of cubes, we take from each Gk cubes that intersect the layer

Ωk and define

W0 =⋃k∈Z

Q ∈ Gk : Q ∩ Ωk 6= ∅

.

The collection W0 satisfies properties (1) and (2) but it contains too many cubes.To obtain a covering that satisfies also (3)-(5), we have to throw away unnecessary

cubes. Note first that if Q1, Q2 ∈ W0 are two cubes whose interiors intersect,7

then one of them is contained in the other: Assume that Q1 ∈ Gk, Q2 ∈ Gl, andintQ1 ∩ intQ2 6= ∅. If k = l, then Q1 = Q2. If k > l, then Q1 ⊂ Q2.

Let Q ∈ W0 and let Q′ ∈ W0 be a maximal cube that contains Q. Such a cubeQ′ exists by property (2) and the inclusion property discussed above. Now W , thecollection of maximal cubes inW0, consist of disjoint cubes and satisfies (3)-(5). For(5), note that for two cubes Q1, Q2 ∈ W , the interiors of Q∗2 and Q1 intersect onlyif Q1 ∩Q2 6= ∅. Hence (5) follows from (4).

Partition of unity connected to the Whitney covering. A partition of unity connectedto a covering of a set consists of compactly supported, nonnegative smooth (in themetric setting Lipschitz) functions, whose sum in each point is 1. It is used to trans-fer local information to the global level for example in extension and interpolationproblems.

To construct a partition of unity connected to the Whitney coveringW = Qii∈N,where Qi = Qi(xi, ri), of an open set Ω ⊂ Rn, let Q0 = Q(0, 1) be the unit cubecentered at 0 and having side length 1. Let 0 < ε < 1 and let ϕ : Rn → [0, 1]be a smooth function such that ϕ|Q0 = 1 and suppϕ ⊂ Q(0, 1 + ε). By definingϕ∗i : Rn → [0, 1],

ϕ∗i (x) = ϕ(x− xi

ri

),

we obtain a function having similar properties in Qi as the function ϕ has in Q0.Using these adjusted functions, we define functions ϕi : Rn → [0, 1],

ϕi(x) =ϕ∗i (x)∑i ϕ∗i (x)

,

for which ∑i

ϕi(x) = 1 for all x ∈ Ω.

The family ϕii∈N is called a partition of unity connected (subordinated) to theWhitney covering W .

Whitney extension. Let F ⊂ Rn be a closed set. Below, we will construct a simpleextension operator using Whitney covering and partition of unity. This operatorextends functions with zero or one order of differentiability to functions which aresmooth in the complement of F . It is ”the mother” of all extension operators usedfor Sobolev-type functions (that can be defined via pointwise definition).

Let u : F → R be a function. Let W be a Whitney covering of Ω = Rn \ F givenby Lemma 2.6 and let ϕii∈N be the corresponding partition of unity. For eachQi ∈ W , let yi ∈ F such that d(Qi, yi)=d(Qi, F ). Such a point exists for each ibecause F is closed. Define a function Eu : Rn → R,

(2.5) Eu(x) =

u(x), if x ∈ F,∑

i u(yi)ϕi(x) if x ∈ Ω.

Note that, by the bounded overlap of dilated cubes Q∗i , Lemma 2.6 (5), and the factthat the support of ϕi is in Q∗i , the sum in (2.5) is finite for each x ∈ Ω and thenumber of nonzero terms does not depend on x.

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Theorem 2.7. Let F ⊂ Rn be a closed set. Let u : F → R be a continuous function.Then Eu : Rn → R, defined by (2.5), is continuous in Rn and C∞ in Ω.

For the proof of Theorem 2.7, see [75, Section VI.2.2]. Moreover, this simpleextension operator E, u 7→ Eu, is linear and it maps Lip(α, F ), α-Holder continuousfunctions on F continuously to Lip(α,Rn). The norm of the extension operator isindependent of F .

2.4. About Sobolev extension domains in Rn (1970 - ). Recall that, for adomain (open and connected set) Ω ⊂ Rn the Sobolev space W 1,p(Ω), 1 ≤ p < ∞,consists of the functions u ∈ Lp(Ω) whose all first order weak derivatives Dju belongto Lp(Ω). We then write

‖u‖W 1,p(Ω) = ‖u‖Lp(Ω) +n∑j=1

‖Dju‖Lp(Ω).

We say that Ω is a W 1,p-extension domain if there exists a bounded linear extensionoperator E : W 1,p(Ω)→ W 1,p(Rn). By the boundedness, there is a constant cE > 0such that

‖Eu‖W 1,p(Rn) ≤ cE‖Eu‖W 1,p(Ω)

for all u ∈ W 1,p(Ω). (Extension domains for spaces W k,p and for other functionspaces are defined similarly.)

Below are some important extension results for Sobolev spaces. For exact def-initions of the geometric conditions and proofs, see the corresponding articles orbooks.

(1) (Calderon 1960 [12], Stein 1970 [75]) Every Lipschitz domain (”locally agraph of a Lipschitz function”) is a W k,p-extension domain. In [12], 1 <p <∞, the extension operator E depends on k, and Eu|Rn\Ω = 0 wheneversuppu ⊂ Ω is compact. In [75], 1 ≤ p ≤ ∞ and the same operator E worksfor all p and k.

(2) (Jones 1982 [42]) Each (ε, δ)-domain (”locally connected in a quantitativesense”) is a W k,p-extension domain for all 1 ≤ p ≤ ∞. The extensionoperator E depends on k.

If Ω ⊂ R2 is a finitely connected W 1,2-extension domain, then Ω is an(ε, δ)-domain. Hence extension for p = 2 implies extension for all p.

(3) (Koskela [53]) If Ω ⊂ Rn is a W 1,p-extension domain and p > n − 1, thenthere exists constants c, δ > 0 such that

(2.6) |Ω ∩B(x, r)| ≥ c|B(x, r)| = crn

for all x ∈ Ω, 0 < r < δ.

Measure density condition (2.6) together with the Lebesgue differentiation theoremimplies that no point of ∂Ω is a density point and hence |∂Ω = 0|. Recent results in[31], [32] (see also the papers of Shvartsman) show the regularity of the boundaryis not necessary condition for the extension property but a kind measure densitycondition is. Our goal is, at the end of the course, to study a connection of a metric

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version of (2.6) and the extension property for Sobolev spaces defined in metricmeasure spaces.21.1 =============================

3. Doubling measures, covering theorems and Hardy-Littlewoodmaximal function

To do the first order calculus in the metric setting, we need a measure and somekind of substitute for derivatives. We begin with the measure. We assume through-out the course that X = (X, d, µ) is a metric measure space equipped with a metricd and a Borel regular outer measure µ. Recall that a set function µ : P(X)→ [0,∞]is called an outer measure if µ(∅) = 0 and µ is countably subadditive, that is,

(3.1) µ(A) ≤∞∑i=1

µ(Ai) whenever A ⊂∞⋃i=1

µ(Ai).

We call such a µ a measure. The Borel regularity of the measure µ means that allBorel sets are µ-measurable and that for every set A ⊂ X there is a Borel set D ⊂ Xsuch that A ⊂ D and µ(A) = µ(D).

We denote open balls in X with a center x ∈ X and a radius 0 < r <∞ by

B(x, r) = y ∈ X : d(y, x) < r.

The corresponding closed ball is

B(x, r) = y ∈ X : d(y, x) ≤ r.

Note that a ball in a metric space as a set does not necessarily have a unique centerand radius. By writing B(x, r), we fix the center and the radius. Note also that

B(x, r) may be a larger set than B(x, r), the topological closure of the open ballB(x, r).

If B = B(x, r) is a ball and 0 < t <∞, then tB is the ball with the same centeras B and radius tr.

It is possible to develop the basic theory of Sobolev spaces in metric measurespaces without any other special assumption on the measure. However, to get aricher theory, some further assumptions on the measure on the geometry of thespace are needed. The standard assumptions are the doubling property of the mea-sure and the validity of a (weak) Poincare inequality that implies that the space isquasiconvex.

Definition 3.1. Measure µ is doubling, if there is a constant cd > 0 such that

(3.2) µ(B(x, 2r)) ≤ cdµ(B(x, r))

for each x ∈ X and all r > 0. We call cd a doubling constant of µ.

We assume that µ is doubling and that it is nondegenerate in the sense thatthere exist a ball B ⊂ X such that 0 < µ(B) < ∞. The doubling conditionguarantees that the measure of every open set is then positive and the measure ofeach bounded set is finite. Such a metric measure space X can be written as a union

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of countable family of open sets with finite measure, for example, if x0 ∈ X, thenX = ∪i∈NB(x0, i).

By iterating doubling condition (3.2), we see that

(3.3) µ(B(x, tr)) ≤ cdtsµ(B(x, r)),

where s = log2 cd, for all balls B(x, r) and all t ≥ 1.

3.1. Basic properties of doubling measures. The pioneering work in analysisin metric spaces with a doubling measure is a book [14] by Coifman and Weiss. Theycall the (quasi)metric spaces (where a constant is allowed in the triangle inequality)with a doubling measure spaces of homogenous type.

Lemma 3.2. Let B(x,R) be a ball. If y ∈ B(x,R), 0 < r ≤ R, and δ ≥ log2 cd,then

(3.4)µ(B(y, r))

µ(B(x,R))≥ 4−δ

( rR

)δ.

Proof. Let k ∈ N be such that 2kr < R ≤ 2k+1r. Then B(x,R) ⊂ 2k+2B(y, r) andB(y, r) ⊂ B(x, 2R) and the doubling property of µ gives

µ(B(x,R)) ≤ ck+2d µ(B(y, r)) ≤ ck+3

d µ(B(x,R)).

Since δ ≥ log2 cd, we obtain

µ(B(y, r)) ≥ c−(k+2)d µ(B(x,R)) = (2log2 cd)−(k+2)µ(B(x,R))

≥ 4−δ(2k)−δµ(B(x,R)) ≥ 4−δ( rR

)δµ(B(x,R)).

Hence the doubling condition gives an upper bound for the dimension of the metricspace. The number s = log2 cd is sometimes called the doubling dimension of µ.

Next we recall some definitions for metric spaces. A metric space is proper if allclosed and bounded sets are compact. A set F ⊂ X is totally bounded, if for anyε > 0, there exists a finite set a1, a2, . . . , an ⊂ A such that A ⊂ ∪ni=1B(ai, ε).(This is equivalent to the existence of a finite ε-net.)

In a doubling metric space, bounded sets are totally bounded (exercise). Thistogether with standard results from metric spaces implies the following importantresult.

Lemma 3.3. A doubling metric measure space is proper if and only if it is complete.

Proof. Assume first that X is proper. (The doubling property is not needed to provethis direction.) Let (xi)i∈N be a Cauchy sequence in X. As a Cauchy sequence,

(xi)i∈N is bounded, and hence there is R > 0 such that xi ∈ B(x1, R) for all i ∈ N.

Since X is proper, B(x1, R) is compact. Hence the sequence (xi)i∈N has a limit inX.

Assume then that X is complete. Let F ⊂ X be closed and bounded. By thedoubling property of µ, F is totally bounded. Since a closed subset of a completemetric space is compact if and only if it is totally bounded, we see that F is compact.Hence X is proper.

11

Example 3.4 (Metric spaces with a doubling measure, all examples but (2) beloware from [5]).

(1) (X, d, µ) = (Rn, | · |,Ln), Euclidean space with the Lebesgue measure is adoubling metric measure space with cd = 2n.

(2) Let (X, d, µ) = (Ω, | · |,Ln|Ω), where Ω satisfies measure density condition

Ln(Ω ∩B(x, r)) ≥ cLn(B(x, r)) = crn

for all x ∈ Ω, r > 0. Then, for each ball B = B(x, r) in X, we have

µ(2B) = Ln(Ω ∩ 2B) ≤ Ln(2B) = 2nLn(B)

≤ 2n

cLn(Ω ∩B) = 2n

cµ(B),

and hence µ is doubling with cd = 2n/c.(3) If X = (M, g) is a complete Riemannian manifold of dimension n and with

nonnegative Ricci curvature, and µ is the canonical measure associated tothe metric tensor g, then µ is doubling with cd = 2n.

(4) Let X = [−1, 0]× [−1, 1]∪ [0, 1]×0 be equipped with the Euclidean metricand with the measure µ = L2|X +H1|[0,1]×0, where H1 is the 1-dimensionalHausdorff measure. Then µ is doubling with cd = 4 and doubling dimension2.

(5) Cantor sets constructed as follows are metric measure spaces: Let F be afinite set having k points, k ≥ 2, and

F∞ = x = (xi)i∈N : xi ∈ F.Let a ∈ (0, 1). Then da : F∞ × F∞ → [0,∞),

da(x, y) =

0, if x = y,

aj, if xi = yi for i < j and xj 6= yj,

is a metric in F∞. Let then ν be a uniformly distributed probability measureon F . Define measure µ on F∞ as the product measure of ν infinitely manytimes. Then one can show that

µ(B(x, aj)) = k−j

and that (F∞, da, µ) is doubling metric measure space with dimension s givenby

as = k−1.

If k = 2 and a = 1/3, then F∞ is bi-Lipschitz equivalent to the standard1/3-Cantor set.

27.1 =============================

Sometimes we would like to estimate the measure of B(x, tr), 0 < t < 1, bycµ(B(x, r)), 0 < c < 1, and have an upper bound for the ratio of measures of theballs corresponding to (3.4). This is possible if X is complete, or more generally, ifall annuli in X are nonempty. (This condition is sometimes called a RD-condition,RD coming from ”reverse doubling”, see for example papers of Dachun Yang andYuan Zhou.)

12

Lemma 3.5. Let X be connected. If 0 < t < 1, then there is a constant c1 = c1(t, cd)such that 0 < c1 < 1 and

(3.5) µ(B(x, tr)) ≤ c1µ(B(x, r))

for all balls B(x, r) with 0 < r < 1/2 diamX. Moreover, there exist constantsc2, α > 0 such that

(3.6)µ(B(y, r))

µ(B(x,R))≤ c2

( rR

)αwhenever B(x,R) is a ball, y ∈ B(x,R) and 0 < r ≤ R.

Proof. We begin with (3.5). Let 0 < t < 1 and let B(x, r) be a ball. Let y be apoint such that

d(y, x) = 12(1 + t)r < r

and let By = B(y, 12(1− t)r). Such a point exists by the connectedness of X. Now

By ⊂ B(x, r) \ B(x, tr) and B(x, r) ⊂ aBy, where a = 4/(1 − t), and hence, using(3.3), we have

µ(B(x, r)) ≤ µ(aBy) ≤ cdalog2 cdµ(By).

Thus

µ(B(x, tr)) ≤ µ(B(x, r))− µ(By) ≤ (1− (cdalog2 cd)−1)µ(B(x, r)),

from which (3.5) follows because cdalog2 cd > 1.

To prove (3.6), let B(x,R) be a ball, y ∈ B(x,R) and 0 < r ≤ R. Then B(y,R) ⊂B(x, 2R) and the doubling condition together with an application of (3.5) gives that

µ(B(y, r))

µ(B(x,R))≤ cd

µ(B(y, r))

µ(B(y,R))≤ c2

( rR

)α.

Hence, by Lemmas 3.2 and 3.5, in a connected space X with diamX =∞, thereare constants c > 0 and 0 < α ≤ s <∞, such that

1

c

( rR

)s≤ µ(B(y, r))

µ(B(x,R))≤ c( rR

)αfor all balls B(x,R) and B(y, r) with y ∈ B(x,R) and 0 < r ≤ R. Note also thatby Lemma 3.5, µ(x) = 0 for all x ∈ X.

Exercise 1. A metric space is doubling if there is a constant cD such that for eachx ∈ X and all r > 0, the ball B(x, r) can be covered by at most cD balls of radiusr/2. Show that if X is a metric space with a doubling measure µ, then X is doublingas a metric space. (In the other direction, if X is a complete, doubling metric space,then is a doubling measure in X, see [62].)

13

3.2. Covering theorems. In many situations, we would like to have an oppositeinequality to (3.1), a kind of quasiadditivity,

∞∑i=1

µ(Ai) ≤ cµ( ∞⋃i=1

Ai

),

where sets Ai usually come from a covering of a set A ⊂ X. This is obtained bystandard covering theorems. Using covering theorems, we can extract from a givenfamily of balls, a better (disjoint) subfamily such that enlarged balls of the subfamilystill cover our set and, moreover, have a bounded overlap.

The most general covering theorem, the so-called 5B- or 5R-covering theorem,deals with covers of metric spaces by balls having finite radius and is usually provedusing the Zorn lemma. The second one is a version of the first with balls of equalradii.

Lemma 3.6. Let B be a family of balls with supdiamB : B ∈ B <∞. Then thereis a countable, disjoint subfamily B′ = Bii∈N of B such that⋃

B∈B

B ⊂⋃i∈N

5Bi.

The countability of the covering follows from the doubling condition since ballshave finite and nonzero measure. The claim is true in any separable metric space.Moreover, it holds in any metric space except for the countability of B′. See forexample [6, Thm 2.2.3] or [37, Thm 1.2], for a proof.

Lemma 3.7. Let B be a family of balls of radius r > 0. Then there is a countable,pairwise disjoint subfamily B′ = Bii∈N of B and a constant N = N(cd) > 0 suchthat ⋃

B∈B

B ⊂⋃i∈N

5Bi

and ∑Bi∈B′

χ5Bi(x) ≤ N for each x ∈ X.

Proof. According to Lemma 3.6, it is enough to show the bounded overlap propertyof the balls 5Bi. Let x ∈ X, and let

Ix = i : Bi ∈ B′, x ∈ 5Bi.If i ∈ Ix, then B(x, r) ⊂ 6Bi. This together with the doubling property of µ impliesthat

µ(Bi) ≥ c−3d µ(6Bi) ≥ c−3

d µ(B(x, r)) ≥ c−6d µ(B(x, 6r)).

On the other hand, Bi ⊂ B(x, 6r) for each i ∈ Ix. As the balls Bi are disjoint, wehave

µ(B(x, 6r)) ≥∑i∈Ix

µ(Bi) ≥ c−6d

∑i∈Ix

µ(B(x, 6r)).

Because the measure of each ball is positive and finite, there is only a finite numberof terms in the above sum. Hence µ(B(x, 6r)) ≥ c−6

d #Ixµ(B(x, 6r)), and we cantake N = c6

d. 14

The proof of Lemma 3.7 shows that if we enlarge the balls in the covering bya fixed constant, then the enlarged balls have overlap bounded with a constantdepending only on the doubling constant of µ and the fixed constant.

3.3. Hardy-Littlewood maximal function and the Lebesgue differentiationtheorem. Maximal functions are important tools for example in geometric analy-sis, harmonic analysis, in the theory of singular integrals, and in the PDE theory.Usually, they are used as tools but the behaviour of maximal operators is an inter-esting question as its own. One can study mapping properties, such as boundednessand continuity, of maximal operators between different functions spaces.

By the classical Lebesgue differentiation theorem, almost every point is a Lebesguepoint for a locally integrable function,

limr→0

∫B(x,r)

u(y) dy = u(x)

for Ln-almost x ∈ Rn. One way to prove this result is to use weak type inequality forthe Hardy-Littlewood maximal function and density of continuous functions in L1,see [75, Chapter 1] in the case of the Lebesgue measure in Rn. This proof generalizesto the metric setting.

By saying that a measurable function u : X → [−∞,∞] is locally integrable, wemean that is integrable on balls. Similarly, the class of functions that belong toLp(B), p > 0, in all balls B, is denoted by Lploc(X). The integral average of a locallyintegrable function u over a ball B is

uB =

∫B

u dµ =1

µ(B)

∫B

u dµ.

Definition 3.8. The Hardy–Littlewood maximal function of a locally integrablefunction u is

(3.7) Mu(x) = sup0<r<∞

∫B(x,r)

|u| dµ,

and its restricted version for R > 0,

(3.8) MRu(x) = sup0<r<R

∫B(x,r)

|u| dµ.

It is easy to see that the corresponding Hardy–Littlewood maximal operatorsM : u 7→ Mu and MR : u 7→ MRu are sublinear,

M(u+ v) ≤Mu+Mv and M(λu) ≤ λMu

for all locally integrable u and v and all λ ≥ 0.One of the mostly used properties of the maximal operator is the boundedness

result in Lp-spaces, originally proved by Hardy and Littlewood [35] and Wiener [81].The maximal operatorM maps L1(X) to weak-L1(X) and is bounded in Lp(X) forp > 1.

Theorem 3.9 (Maximal function theorem).15

(1) There is constant C1 = C1(cd) > 0 such that

(3.9) µ(x ∈ X :Mu(x) > t

)≤ C1

t

∫X

|u| dµ

for all t > 0 and for all u ∈ L1(X).(2) If p > 1, then there is a constant Cp = Cp(p, cd) > 0 such that

(3.10) ‖Mu‖Lp(X) ≤ Cp‖u‖Lp(X)

for all u ∈ Lp(X).

28.1 =============================

Proof. Because the 5B-covering lemma (Lemma 3.7) requires the balls to have uni-formly bounded diameter, we prove the claim first for the restricted maximal func-tion (3.8) with constants independent of R and then pass to the limit R→∞. Forthat, let R > 0.

Claim (1): Let u ∈ L1(X), t > 0 and let

Et = x ∈ X :MRu(x) > t.

For each x ∈ Et, let B(x, rx) be a ball such that 0 < rx < R and

(3.11)

∫B(x,rx)

|u| dµ > tµ(B(x, rx)).

By Lemma 3.7, there are disjoint balls Bi = B(xi, ri), i = 1, 2, . . . , satisfying (3.11)such that Et ⊂ ∪i5Bi. Hence, using the doubling property of µ and (3.11), we obtain

µ(Et) ≤∑i

µ(5Bi) ≤ C∑i

µ(Bi) ≤C

t

∑i

∫Bi

|u| dµ ≤ C

t

∫X

|u| dµ,

from which the first claim follows by letting R→∞.

Claim (2): Let u ∈ Lp(X). By the definition of Mu, if p =∞, then C∞ = 1.Assume that 1 < p <∞. Let t > 0. We divide u into a small and large part,

u = uχ|u|≤t/2 + uχ|u|>t/2 = g + h.

Then, by the sublinearity of the maximal operator MR,

MRu(x) ≤MRg(x) +MRh(x) ≤ t

2+MRh(x)

for each x ∈ X, and hence

MRu > t ⊂ MRh > t/2.

Using the assumption u ∈ Lp(X) with p > 1 and the definition of h, we see that∫X

|h| dµ =

∫|u|>t/2

|u|p|u|1−p dµ ≤( t

2

)1−p∫X

|u|p dµ,

and hence h ∈ L1(X).16

Using integration in terms of the distribution function (3.12) three times, the firstpart of the proof for h, and change of variables, we obtain∫

X

|MRu|p dµ = p

∫ ∞0

tp−1µ(MRu > t

)dt ≤ p

∫ ∞0

tp−1µ(MRh > t/2

)dt

≤ C

∫ ∞0

tp−1t−1

∫|u|>t/2

|u| dµ dt

≤ C

∫ ∞0

tp−2

(t

2µ(|u| > t/2

)+

∫ ∞t/2

µ(|u| > s

)ds

)dt

≤ C

∫X

|u|p dµ+ C

∫ ∞0

∫ 2s

0

µ(|u| > s

)tp−2 dt ds

≤ C

∫X

|u|p dµ.

The second claim follows by letting R→∞.

By keeping track on the constants in the Maximal function theorem, we see thatin (2), we can take Cp = C2p

p−1, where C depends only on cd. Note that Cp → ∞ as

p→ 1. In fact, the maximal function of an L1-function is hardly ever integrable. Itis easy to show that in the Euclidean case with Lebesgue measure, if u ∈ L1(Rn)and Mu ∈ L1(Rn), then u = 0. See also [74].

Remark 3. As an application of the Fubini theorem, we obtain the following use-ful formula, sometimes called Cavalieri principle, for Lp-integrals in terms of thedistribution function. If µ is a Borel measure, u ≥ 0 a measurable function and0 < p <∞, then

(3.12)

∫X

up dµ = p

∫ ∞0

tp−1µ(x ∈ X : u(x) > t

)dt.

To see that, we write Et = u > t for t ≥ 0, and define f : [0,∞)×X → R,

f(t, x) = χEt(x) =

1, if u(x) > t,

0, if u(x) ≤ t.

Then

p

∫ ∞0

tp−1µ(Et) dt = p

∫ ∞0

tp−1

∫X

χEt(x) dµ(x) dt

=

∫X

∫ ∞0

ptp−1f(t, x) dt dµ(x)

=

∫X

∫ u(x)

0

ptp−1 dt dµ(x) =

∫X

up dµ.

Remark 4. Theorem 3.9 implies that for a function u ∈ Lp(X), 1 ≤ p ≤ ∞,Mu isfinite almost everywhere. Note also that ”Maximal function sees things that happenfar away...” - a local version of the Lp-boundedness

‖Mu‖Lp(Ω) ≤ Cp‖u‖Lp(Ω)

17

does not necessarily hold for all u ∈ Lp(Ω). It holds if the restriction of µ to Ω, µ|Ω,is doubling or if suppu ⊂ Ω.

3.4. Noncentered maximal function. The Hardy–Littlewood maximal functionhas a variant, where the supremum of integral averages in (3.7) is taken over allballs containing the point x. In some situations, this version is smoother and moreuseful thanMu. A noncentered maximal function of a locally integrable function uis

(3.13) M∗u(x) = supx∈B

∫B

|u| dµ.

It is easy to see that for each locally integrable function u and all x ∈ X;

(3.14) c−2d M∗u(x) ≤Mu(x) ≤M∗u(x);

the first inequality follows from the doubling property of µ and the second from thedefinitions.

Definition (3.13) implies that sets

x ∈ X :M∗u(x) > tare open for each t > 0 and hence M∗u is lower semicontinuous.

A similar proof as for the maximal function Mu in Theorem 3.9 (1) shows that

µ(M∗u > t

)≤ C

t

∫M∗u>t

|u| dµ

for all t > 0 and for all u ∈ L1(X). This together with (3.14) and (3.9) implies that

limt→∞

tµ(M∗u > t

)= lim

t→∞tµ(Mu > t

)= 0

for all u ∈ L1(X).

3.5. Lebesgue differentiation theorem.

Definition 3.10. A point x ∈ X is a Lebesgue point of a locally integrable functionu : X → R, if

(3.15) limr→0

∫B(x,r)

u dµ = u(x).

Theorem 3.11 (Lebesgue differentiation theorem). Let u : X → R be a locallyintegrable function. Then µ-almost every point x ∈ X is a Lebesgue point of u.Moreover,

(3.16) limr→0

∫B(x,r)

|u(y)− u(x)| dµ(y) = 0

for µ-almost every point x ∈ X.

Remark 5. In the literature, the existence of limits (3.15) and (3.16) almost ev-erywhere are both used as a definition of a Lebesgue point. Clearly, (3.16) implies(3.15).

18

About the proof of Theorem 3.11. The argument in [75, Chapter 1] for the first claimgeneralizes to the metric setting. It uses weak type inequality (3.9) for the maximalfunction and the density of continuous functions in L1(X) (which is true in everymetric space with a Borel measure), see [6, Thm 5.2.6] for the proof in the metricsetting. Another way to prove the theorem is to use Vitali covering theorem, see[37, Thm 1.8], [41, 2.4.13].

The second claim follows by using the first claim for locally integrable functionsui,

ui(y) = |u(y)− qi|,with Q = (qi)i∈N.

3.2 =============================

4. Mapping properties of the maximal function

In this section, we discuss the mapping properties of the maximal function. Weare interested in the following question: Does the maximal operator preserve the reg-ularity properties of functions? In definition (3.7), there are two competing things:the integral average is smoothing whereas supremum usually reduces smoothness.

We begin with the boundedness results in the classical Euclidean setting.

4.1. Regularity of the maximal function in the Euclidean setting. In theEuclidean space, many boundedness properties of the maximal operator follow fromthe fact that it commutes with translations: if u : Rn → R is locally integrable, then

(4.1) (Mu)h(x) =Muh(x)

for all x, h ∈ Rn. Here, for a function v : Rn → R,

vh(x) = v(x+ h) for all x, h ∈ Rn.

In the classical case, not only the noncentered maximal function M∗u of eachlocally integrable u but also Mu is lower semicontinuous (Exercise). Moreover, themaximal function of a continuous function u : Rn → R is continuous. If u : Rn → Ris L-Lipschitz, then (4.1) together with the sublinearity of M implies that for allx, h ∈ Rn,

|Mu(x+ h)−Mu(x)| = |(Mu)h(x)−Mu(x)|= |Muh(x)−Mu(x)| ≤ M(uh − u)(x)

= supr>0

∫B(x,r)

|u(y + h)− u(y)| dy ≤ L|h|,

and hence M is also L-Lipschitz (if not identically infinite). Moreover, by theRademacher theorem,Mu is differentiable almost everywhere. The same continuityargument holds also for Holder continuous functions.

How about the differentiability properties? Since |u| is not differentiable and thesupremum of differentiable functions is not differentiable in general, the maximalfunction of a differentiable function is not differentiable in general. The study ofthe properties of the maximal function of a Sobolev function started by Kinnunen

19

in [46], see also [34] and [48] for the local case. A simple argument using thecharacterization of Sobolev spaces W 1,p(Rn), 1 < p < ∞, by integrated differencequotients shows that the maximal operator is bounded in Sobolev spaces. Moreover,there is a pointwise estimate for the weak derivatives of the maximal function interms of the maximal functions of the weak derivatives of the function itself. Recallthat u ∈ Lp(Rn) belongs to W 1,p(Rn) if and only if

lim suph→0

‖uh − u‖Lp(Rn)

|h|<∞,

for a proof, see for example [23, Chapter 7.11]

Theorem 4.1. Let u ∈ W 1,p(Rn), 1 < p <∞. Then Mu ∈ W 1,p(Rn) and

|DiMu(x)| ≤ MDiu(x) for all i = 1, 2, . . . , n

for almost every x ∈ Rn.

Proof. We will prove the boundedness of M in W 1,p(Rn). By the fact that Mcommutes with translations, the sublinearity of M and the boundedness of M inLp(Rn) for p > 1, we obtain

‖(Mu)h −Mu‖Lp(Rn) = ‖Muh −Mu‖Lp(Rn) ≤ C‖M(uh − u)‖Lp(Rn)

≤ C‖uh − u‖Lp(Rn).

The claim follows from the characterization of W 1,p(Rn) mentioned above.

Theorem can be used to prove that for u ∈ W 1,p(Rn), 1 < p < ∞, Mu isp-quasicontinuous (in the Sobolev-capacity sense) and to study the pointwise prop-erties u, see [46].

Remark 6. Since the Hardy-Littlewood maximal operator is not bounded in L1(Rn),we cannot expect it to be bounded in W 1,1(Rn) (consider a smooth bump functionu that belongs to W 1,1(Rn) but Mu is not integrable). Hence the more interest-ing question is: Is the operator u 7→ |DMu| from W 1,1(Rn) to L1(Rn) bounded?In R, the answer is yes, proven for the noncentered maximal operator by Tanakain [76] and for M in the recent preprint [58] by Kurka. The case n > 1 is openand the techniques used in the one-dimensional case does not generalize to higherdimensions. For partial results in Rn, also for the local case, see [34] and [33].

4.2. Hardy-Littlewood maximal function of Lipschitz funtion in metricspaces. The proofs using the fact that M commutes with translations cannot begeneralized to the metric setting. It turns out that the doubling property of themeasure does not guarantee that the maximal function of a Lipschitz function isLipschitz. The next example of Buckley in [11] shows that it can be discontinuous.

Example 4.2. Let X be the subset of the complex plane consisting of the real lineand the points x on the unit circle whose argument θ lies in the interval [0, π

2]. Equip

X with the Euclidean metric and the 1-dimensional Hausdorff measure.Let u : X → [0, 1] be a Lipschitz function such that u(x) = 0, if x ∈ R or

Arg(x) ≤ π/5, and u(x) = 1, if Arg(x) ≥ π/4.20

We will show that Mu has a jump discontinuity at the origin. Since

Mu(0) = limr→1+

∫B(0,r)

u dµ =1

2 + π2

∫B(0,1)

u dµ,

we have that

Mu(0) ≤π2− π

5

2 + π2

=3π

20 + 5π.

If x < 0, then B(x, r(x)), where r(x) = d(x, eiπ/4), includes points on the arc if andonly if their argument exceeds π

4. It follows that

limx→0−

Mu(x) ≥ limx→0−

∫B(x,r(x))

u dµ =π4

2 + π4

=π

8 + π>Mu(0).

Annular decay and maximal operator. If the measures of annuli in X behave nicely,then the maximal operatorM maps Lipschitz functions to Holder continuous func-tions. The space of bounded β-Holder continuous functions is equipped with thenorm

‖u‖C0,β(X) = ‖u‖L∞ + supx 6=y

|u(x)− u(y)|d(x, y)β

,

and β = 1 gives bounded Lipschitz functions on X.

Definition 4.3. Let 0 < δ ≤ 1. The metric measure space X satisfies the δ-annulardecay property, if there exists a constant C > 0 such that for all x ∈ X, R > 0, and0 < h < R, we have

(4.2) µ(B(x,R) \B(x,R− h)

)≤ C

( hR

)δµ(B(x,R)).

Geodesic spaces, and more generally, length spaces (where the distance betweenany pair of points equals the infimum of the lengths of the rectifiable paths joiningthem), satisfy the annular decay for some δ > 0, see [11]. The strong annular decaycondition, (4.2) with δ = 1, holds only for few spaces, for example for Rn and theHeisenberg group Hn. Recall that X is geodesic, if every two points x, y ∈ X canbe joined by a curve γ for which `(γ) = d(x, y).

Theorem 4.4. Let 0 < δ ≤ 1 and 0 < β ≤ 1. If X satisfies the δ-annular decayproperty, then

M : C0,β(X)→ C0,α(X),

where α = minβ, δ, is bounded. In particular, if X satisfies the strong annulardecay property, then M maps Lipschitz functions boundedly to Lipschitz functions.

Proof. Let u ∈ C0,β(X) with ‖u‖C0,β(X) = 1. Since ‖Mu‖L∞(X) = ‖u‖L∞(X), wehave to estimate only the second part of the norm of ‖Mu‖C0,β(X).

Let x, y ∈ X. It suffices to show that there is a constant C > 0, independent ofu, x, and y, such that

(4.3) Mu(x)−Mu(y) ≤ Cd(x, y)α.21

We may also assume that d(x, y) ≤ 1. Let r > 0 be such that∫B(x,r)

|u| dµ ≥Mu(x)− d(x, y)α.

Assume first that r ≤ d(x, y). (This is usually the easy case in this type of estimates).Then, by the β-Holder continuity of u,

|u(z)− u(w)| ≤ 3βd(x, y)β,

for all z, w ∈ B(x, r) ∪B(y, r). Hence∣∣∣∫B(x,r)

|u| dµ−∫B(y,r)

|u| dµ∣∣∣ =

∣∣∣∫B(x,r)

|u| − |u|B(y,r) dµ∣∣∣

=∣∣∣∫B(x,r)

∫B(y,r)

|u(z)| − |u(w)| dµ(w) dµ(z)∣∣∣

≤∫B(x,r)

∫B(y,r)

|u(z)− u(w)|| dµ(w) dµ(z)

≤ 3βd(x, y)β ≤ 3βd(x, y)α.

This together with the selection of r > 0 implies that

Mu(y) ≥∫B(y,r)

|u| dµ ≥∫B(x,r)

|u| dµ− 3βd(x, y)α

≥Mu(x)− (3β + 1)d(x, y)α,

from which (4.3) follows.Assume then that r > d(x, y). A calculation using δ-annular decay (4.2) and the

assumption u ∈ C0,β(X), shows that∫B(x,r)

|u| dµ−∫B(y,r+d(x,y))

|u| dµ ≤ Cd(x, y)α,

from which the claim follows. For details, see [11, Thm 1.1].

4.2 =============================

4.3. The discrete convolution and the discrete maximal operator. By Ex-ample 4.2, the standard Hardy–Littlewood maximal operator does not preserve thesmoothness of the functions in a doubling metric measure space without an ad-ditional assumption on the space. In this section, we will construct a maximaloperator which is based on a discrete convolution. It turns out that this operatorhas better regularity properties in doubling metric measure spaces than the stan-dard Hardy–Littlewood maximal operator. The discrete convolution itself has beenan important tool in harmonic analysis in doubling metric spaces since the works[14] and [63]. Nowadays, it is a standard tool in geometric analysis in metric spaces.Since Lipschitz functions are ”the smooth functions” in metric spaces, partition ofunity used in the discrete convolution consists of Lipschitz functions.

22

The idea behind the discrete maximal function is the fact that in the Euclideansetting, the Hardy–Littlewood maximal function is a supremum of convolutions.Namely, for each locally integrable u and each ball B(x, r) ⊂ Rn,∫

B(x,r)

|u(y)| dy =1

|B(x, r)|

∫B(x,r)

|u(y)| dy = |u| ∗ χr(x),

where ∗ is convolution and χr =χB(0,r)

|B(0,r)| . Hence

Mu(x) = supr>0

(|u| ∗ χr)(x).

The discrete maximal function was introduced and used to study pointwise proper-ties of Sobolev functions in the metric setting by Kinnunen and Latvala in [47]. Itsproperties are also studied, for example, in [1], [2], [50] and [36].

We begin the construction of the discrete maximal function with a covering of thespace and a partition of unity subordinate to this covering.

Covering. Let r > 0. By the 5B-covering lemma, Lemma 3.7, there are balls Bi =B(xi, r), i = 1, 2, . . . , and a constant N = N(cd), such that

X =⋃i

Bi and∑i

χ6Bi ≤ N.

Partition of unity. As in Section 2.3, connected to the covering Bii, there is apartition of unity ϕii, consisting of functions ϕi : X → [0, 1], i = 1, 2 . . . with thefollowing properties: There are positive constants ν and L depending only on thedoubling constant of µ, such that for each i, ϕi = 0 in X \ 6Bi, ϕi ≥ ν in 3Bi, ϕi isLipschitz with constant L/r, and∑

i

ϕi(x) = 1 for all x ∈ X.

To construct such functions, we can first define 1/(3r)-Lipschitz functions ϕ∗i : X →[0, 1], i = 1, 2 . . . ,

ϕ∗i (x) =

1, if x ∈ 3Bi,

2− d(x,xi)3

, if x ∈ 6Bi \ 3Bi,

0, if x ∈ X \ 6Bi.

Using these cut-off functions, we define functions ϕi : X → [0, 1], i = 1, 2 . . . , for thepartition of unity,

ϕi(x) =ϕ∗i (x)∑i ϕ∗i (x)

for all x ∈ X.

It is easy to see that functions ϕi satisfy the desired properties (exercise).23

Discrete convolution. The discrete convolution of a locally integrable function u atthe scale r is ur : X → R,

(4.4) ur(x) =∑i

ϕi(x)u3Bi

for every x ∈ X.Note that by the properties of the covering and the partition of unity, the sum

in (4.4) is finite at each point and upper bound for number of the nonzero termsdepends only on the doubling constant of µ.

Below, when we prove results for the discrete convolution ur, covering of X byballs Bi and the partition of unity ϕi subordinated to the covering Bii are asabove.

Lemma 4.5. Let u : X → R be a a locally integrable function and let ur : X → Rbe a discrete convolution of u as above. Then ur is continuous. Moreover, it isLipschitz on each compact set K ⊂ X.

Proof. Exercise.

Discrete maximal function. Let Q+ = (rj)j∈N, be an enumeration of positive ratio-nals. For each j ∈ N, we take a covering of X by balls B(xi, rj), i = 1, 2, . . . , as

above. The discrete maximal function of u in X is M∗u : X → R,

M∗u(x) = supj|u|rj(x)

for every x ∈ X.The construction of the discrete maximal function depends on the choice of the

coverings, but estimates proved using it that are independent of the chosen coverings.

Lemma 4.6. Let u : X → R be a locally integrable function. Then the discretemaximal function M∗u is lower semicontinuous and the discrete maximal operatorM∗ is sublinear. Moreover, the discrete maximal function is comparable to theHardy-Littlewood maximal function: there is a constant C = C(cd) ≥ 1 such that

(4.5) C−1Mu(x) ≤M∗u(x) ≤ CMu(x)

for all x ∈ X.

Proof. As a supremum of continuous functions, the discrete maximal function islower semicontinuous and hence measurable. The sublinearity,

M∗(u+ v) ≤M∗(u) +M∗(v),

for all u, v ∈ L1loc(X), follows directly from the definitions of the discrete convolution

and the discrete maximal function.To prove the comparability of Mu and M∗u, let x ∈ X. We begin with the

second inequality of (4.5). Let rj ∈ Q+ and let

Ix = i : x ∈ B(xi, 6rj).24

Then B(xi, 3rj) ⊂ B(x, 9rj) ⊂ B(xi, 15rj) whenever i ∈ Ix. This together with thedoubling property of µ, the fact that ϕi|X\B(xi,6rj) = 0 for all i and the boundedoverlap of the balls B(xi, 6rj) imply that

|u|rj(x) =∑i

ϕi(x)|u|B(xi,3rj)

≤∑i∈Ix

ϕi(x)µ(B(xi, 3rj))

µ(B(x, 9rj))

∫B(x,9rj)

|u| dµ

≤ CMu(x)∑i∈Ix

ϕi(x) ≤ CMu(x).

The second inequality of (4.5) follows by taking supremum over rj.To prove the first inequality of (4.5), note that by Lemma 3.2, by taking the

supremum over radii in Q+ instead of all over all r > 0 in the Hardy-Littlewoodmaximal function, we obtain a comparable maximal function. Thus it suffices toshow that

suprj∈Q+

∫B(x,rj)

|u| dµ ≤ CM∗u(x).

Let rj ∈ Q+. Since balls B(xi, rj), i = 1, 2 . . . , form a covering of X, there is i ∈ Nsuch that x ∈ B(xi, rj). Then B(x, rj) ⊂ B(xi, 2rj) and, by the doubling propertyof µ and the fact that ϕi|B(xi,3rj) ≥ ν, we have∫

B(x,rj)

|u| dµ ≤ C

∫B(xi,3rj)

|u| dµ ≤ Cϕ(x)

∫B(xi,3rj)

|u| dµ ≤ CM∗u(x).

The first inequality of (4.5) follows now by taking supremum over rj.

Remark 7. Lemma 4.6 together with Theorem 3.9 shows that the maximal operatorM∗ maps L1(X) to weak-L1(X) and is bounded in Lp(X) for p > 1.

Using the Lebesgue differentiation theorem, it is easy to show that the discreteconvolution approximates locally integrable functions pointwise almost everywhere,ur(x) → u(x) as r → 0 for almost all x ∈ X. When p > 1, this together with thefacts that ur ≤ M∗u and M∗u ∈ Lp(X) and the dominated convergence theorem,imply that in ur → u in Lp(X) as r → 0, (exercise or see [2, Lemma 4.5]).

Using the density of continuous functions in Lp(X), p ≥ 1, we can show that theconvergence is true also in L1(X). Recall the Jensen inequality for sums: if k ∈ N,a1, . . . , ak ∈ R, n1, . . . , nk ≥ 0, and Ψ: R→ R is a convex function, then

Ψ

(n1a1 + · · ·nkakn1 + · · ·nk

)≤ n1Ψ(a1) + · · ·nkΨ(ak)

n1 + · · ·nk.

Below (and later), we also use the following easy application of the Holder inequalityto estimate integral averages: for p ≥ 1∫

B

|u| dµ ≤(∫

B

|u|p dµ)1/p

for all locally integrable functions u and all balls B.25

10.2 =============================

Lemma 4.7. Let u ∈ Lp(X), p ≥ 1. Then ‖ur − u‖Lp(X) as r → 0. Hence (asubsequence) ur(x)→ u(x) as r → 0 for almost every x ∈ X.

Proof. Let r > 0 and let u ∈ Lp(X). First we prove an estimate for ‖ur‖Lp(X).By the Jensen inequality with Ψ(t) = tp, the fact that

∑i ϕi = 1 and the Holder

inequality, we have that |ur|p ≤ (|u|p)r. Hence, by the properties of the functions ϕiand the doubling property of µ, we obtain

(4.6)

∫X

|ur|p dµ ≤∫X

(|u|p)r dµ ≤∑i

∫X

(|u|p)3Biϕi dµ

≤ C∑i

∫6Bi

|u|p dµ ≤ C

∫X

|u|p dµ,

where the constant C depends only on cd.Let ε > 0 and let v be a bounded continuous function with bounded support such

that ‖u− v‖Lp(X) < ε. Estimating as in (4.6), we obtain

‖ur − vr‖Lp(X) = ‖(u− v)r‖Lp(X) ≤ C‖u− v‖Lp(X) < Cε,

where C = C(cd, p), and hence

‖ur − u‖Lp(X) ≤ ‖ur − vr‖Lp(X) + ‖vr − v‖Lp(X) + ‖v − u‖Lp(X)

< ‖vr − v‖Lp(X) + Cε.

Therefore, it suffices to show that ‖vr − v‖Lp(X) → 0 as r → 0. Let x ∈ X andlet Ix = i : x ∈ 6Bi. Then 3Bi ⊂ B(x, 9r) ⊂ 15Bi whenever i ∈ Ix. Using thedoubling property of µ and the properties of the functions ϕi, we have

|vr(x)− v(x)| ≤∑i∈Ix

|v3Bi − v(x)| dµ(y)

≤ C

∫B(x,9r)

|v(y)− v(x)| dµ(y),

which converges to 0 as r → 0 by the continuity of v. Since |vr − v| ≤ 2 sup |v|, theclaim for vr and v follows from the dominated convergence theorem.

Since the maximal operators M and M∗ map L1(X) to weak-L1(X) and arebounded in Lp(X) for p > 1, the maximal function and the discrete maximal func-tion of an Lp-function are finite everywhere. For locally integrable functions, thefiniteness at one point implies finiteness at almost every point.

Lemma 4.8. Let u : X → R be a locally integrable function. If there exists a pointx0 ∈ X such that Mu(x0) <∞, then Mu(x) <∞ for almost all x ∈ X.

Note that, by the comparability ofMu andM∗u, the previous lemma holds alsofor M∗u.

Proof. For each k ∈ N, we divide u in two parts,

u = uχB(x0,2k) + uχX\B(x0,2k) = vk + wk.26

By the sublinearity of the maximal operator,

Mu(x) ≤Mvk(x) +Mwk(x) for all x ∈ X.Since u is locally integrable, vk ∈ L1(X). Hence weak-type estimate (3.9) impliesthat Mvk is finite almost everywhere.

To estimate Mwk, let x ∈ B(x0, k) and let r ≥ k. Then B(x, r) ⊂ B(x0, 2r) ⊂B(x, 3r) and, by the doubling property of µ and the definition of wk,∫

B(x,r)

|wk| dµ ≤ C

∫B(x0,2r)

|wk| dµ ≤ CMwk(x0) ≤ CMu(x0).

The radii r < k does not affect toMwk(x) because wk|B(x0,2k) = 0 and x ∈ B(x0, k).Hence, by taking supremum over r > k, we have that Mwk(x) ≤ CMu(x0) for allx ∈ B(x0, k).

Hence, for each k ∈ N, Mu is finite almost everywhere in B(x0, k). The claimfollows because X = ∪k∈NB(x0, k).

The next results show that the fractional maximal function of a β-Holder contin-uous function is β-Holder continuous.

Theorem 4.9. If u ∈ C0,β(X) with 0 < β ≤ 1, then M∗u ∈ C0,β(X) In particular,the discrete maximal function M∗u of a Lipschitz function of u is Lipschitz (notusually with the same Lipschitz constant) if it is finite almost everywhere.

Note that the proof below shows that if the β-Holder seminorm

(4.7) supx 6=y

|u(x)− u(y)|d(x, y)β

is finite, then it is finite for M∗u also, provided M∗u is finite almost everywhere.

Proof. Since ‖M∗u‖L∞(X) = ‖u‖L∞(X), it suffices to estimate (4.7) for M∗u. Letr > 0. We begin by proving the claim for |u|r. Let x, y ∈ X and let

Ix = i : x ∈ 6Bi and Iy = i : y ∈ 6Bi.Assume first that d(x, y) > r. Using the fact that

∑i ϕi(z) = 1 for all z ∈ X, we

have ∣∣|u|r(x)− |u|r(y)∣∣ ≤ (|u(x)− u(y)|+

∑i∈Ix

ϕi(x)∣∣|u|3Bi − |u(x)|

∣∣+∑i∈Iy

ϕi(y)∣∣|u|3Bi − |u(y)|

∣∣).In the first sum, for i ∈ Ix, by the Holder continuity of u, we have∣∣|u|3Bi − |u(x)|

∣∣ ≤∫3Bi

|u(z)− u(x)| dµ ≤ Crβ.

A similar estimate holds for terms of the second sum. The bounded overlap of theballs 6Bi, i = 1, 2, . . . , and the Holder continuity of u now imply that∣∣|u|r(x)− |u|r(y)

∣∣ ≤ C(d(x, y)β + rβ

)≤ Cd(x, y)β.

27

Assume then that d(x, y) ≤ r. Now∣∣|u|r(x)− |u|r(y)∣∣ ≤∑

i

|ϕi(x)− ϕi(y)|∣∣|u|3Bi − |u(x)|

∣∣,where ϕi(x)−ϕi(y) 6= 0 only if i ∈ Ix∪Iy. If i ∈ Iy, then the assumption d(x, y) ≤ rimplies that x ∈ 7Bi. Hence for each i ∈ Ix ∪ Iy, as above, we have∣∣|u|3Bi − |u(x)|

∣∣ ≤ Crβ.

Using the L/r-Lipschitz-continuity of the functions ϕi and the bounded overlap ofthe balls 6Bi, we obtain∣∣|u|r(x)− |u|r(y)

∣∣ ≤ Cd(x, y)rβ−1 ≤ Cd(x, y)β.

The claim for |u|r follows from these estimates.Then we prove the claim forM∗u. We may assume thatM∗u(x) ≥M∗u(y). Let

ε > 0 and let rε ∈ Q+ be such that

|u|rε(x) >M∗u(x)− ε.

Then, by the first part of the proof,

M∗u(x)−M∗u(y) ≤ |u|rε(x)− |u|rε(y) + ε ≤ Cd(x, y)β + ε.

By letting ε→ 0, we obtain

|M∗u(x)−M∗u(y)| ≤ Cd(x, y)β.

5. Sobolev spaces in metric measure spaces - M1,p(X)

There are several ways to generalize the classical theory of Sobolev spaces to thesetting of metric spaces. In a general metric space, we cannot speak about weakderivatives, but the definition of a Sobolev space can be given, for example, by usingupper gradients or pointwise inequalities. We will not prove all the properties of theSobolev spaces, good references for those are books [6], [8], [37] and [41] and papers[13], [24], [26], [28], [30], [39], [49], [69], [70], for example.

We begin with the Sobolev space M1,p(X) based on a poitwise inequality, definedby Haj lasz in [24].

Definition 5.1. Let u : X → R be a measurable function and let 0 < p < ∞. Ameasurable function g ≥ 0 is a Haj lasz gradient/generalized gradient of u if thereexists E ⊂ X with µ(E) = 0 such that

(5.1) |u(x)− u(y)| ≤ d(x, y)(g(x) + g(y))

for all x, y ∈ X \ E. The collection of Haj lasz gradients of u is denoted by D(u).The space M1,p(X) consists of functions u ∈ Lp(X) for which D(u)∩Lp(X) 6= ∅. Itis equipped with the norm (quasinorm when 0 < p < 1)

‖u‖M1,p(X) = ‖u‖Lp(X) + infg∈D(u)

‖g‖Lp(X).

28

The space M1,p(Ω) for subsets Ω ⊂ X and the local space M1,ploc (X) are defined

in the natural way. Note that the members of M1,p(X) are equivalence classes ofLp(X)-functions and hence defined only almost everywhere. As usual, we call themfunctions.11.2 =============================

Where the pointwise definition comes from? It comes from the following character-ization of the classical Sobolev spaces.

Theorem 5.2. Let u ∈ Lp(Rn), 1 < p < ∞. Then u ∈ W 1,p(Rn), if and only ifthere exists a nonnegative function g ∈ Lp(Rn) such that

(5.2) |u(x)− u(y)| ≤ |x− y|(g(x) + g(y))

for almost all x, y ∈ Rn. Moreover, ‖∇u‖Lp(Rn) is comparable to inf ‖g‖Lp(Rn), wherethe infimum is taken over functions g satisfying (5.2).

We will give a metric space style proof of the necessity direction of the charac-terization. The chaining argument used in the proof is often useful in pointwiseestimates.

Proof. Assume first that u ∈ W 1,p(Rn). Let x, y ∈ Rn be Lebesgue points of u andlet r = |x− y|. For each i = 0, 1, . . . , define balls

Bi(x) = B(x, 2−ir) and Bi(y) = B(y, 2−ir).

Then uBi(x) → u(x) and uBi(y) → u(y) as i→∞, and

|u(x)− u(y)| ≤ |u(x)− uB0(x)|+ |uB0(x) − uB0(y)|+ |u(y)− uB0(y)|.

For the first term |u(x)−uB0(x)|, we obtain, using the Poincare inequality for Sobolevfunctions

|u(x)− uB0(x)| ≤∞∑i=0

|uBi(x) − uBi+1(x)| ≤∞∑i=0

∫Bi+1(x)

|u(z)− uBi(x)| dz

≤∞∑i=0

|Bi(x)||Bi+1(x)|

∫Bi(x)

|u(z)− uBi(x)| dz

≤ C

∞∑i=0

2−ir

∫Bi(x)

|∇u(z)| dz ≤ C|x− y|M|x−y||∇u|(x).

29

Similar estimate holds for the last term |u(y) − uB0(y)|. For the second term, wehave, using the fact that B0(y) ⊂ 2B0(x) and the Poincare inequality, as above,

|uB0(x) − uB0(y)| ≤ |uB0(x) − u2B0(x)|+ |u2B0(x) − uB0(y)|

≤∫B0(x)

|u(z)− u2B0(x)| dz +

∫B0(y)

|u(z)− u2B0(x)| dz

≤ C

∫2B0(x)

|u(z)− u2B0(x)| dz

≤ Cr

∫2B0(x)

|∇u(z)| dz ≤ C|x− y|M2|x−y||∇u|(x).

The calculation above shows that,

|u(x)− u(y)| ≤ C|x− y|(M2|x−y||∇u|(x) +M2|x−y||∇u|(y)

)for Lebesgue points x, y ∈ Rn of u. The claim follows from the boundedness of theHardy-Littlewood maximal function in Lp because |∇u| ∈ Lp(Rn) and p > 1. Thatis, we can take g = CM|∇u|.

For the other direction, see Theorem 8 and Remark 8. One possible way to showthat the validity of (5.2) implies that u ∈ W 1,p(Rn) is first notice that the pair u, gsatisfy a Poincare type inequality and then prove the existence of the weak gradientsusing this inequality.

For a Poincare type inequality for u and g, let B = B(x0, r) be a ball. Integrating(5.2) twice over the ball B, we obtain∫

B

|u− u(x)| dx ≤ 4r

∫B

g(x) dx.

Using this inequality, the Riesz representation theorem and convolutions, one canshow that u has a weak gradient and that the norm estimate holds, see [24, Thm1], [26].

Remark 8. By Theorem 5.2, M1,p(Rn) = W 1,p(Rn) as sets and the norms areequivalent if p > 1. For subsets Ω ⊂ Rn, M1,p(Ω) = W 1,p(Ω), if a bounded domainΩ is nice enough, an extension domain, for example. Note that there are domainswithout the extensions property but for which M1,p(Ω) = W 1,p(Ω), see [67].

A continuous imbedding M1,p(Ω) ⊂ W 1,p(Ω), 1 ≤ p < ∞, holds for all opensets Ω ⊂ Rn. This result, the existence of weak derivatives with the norm bound‖∇u‖Lp(Ω) ≤ C‖gu‖Lp(Ω) can be proved for p > 1 using the Riesz representation the-orem, reflexivity of Lp, convolution approximation and boundedness of the maximaloperator in Lp, see [24, Thm 1], [37, Prop 5.9]. The proof below for all p ≥ 1 is from[26, Prop 1], [37, Remark 5.13].

Theorem 5.3. Let Ω ⊂ Rn be an open set and let 1 ≤ p <∞. If u ∈M1,p(Ω) withg ∈ D(u) ∩ Lp(Ω), then u ∈ W 1,p(Ω), and there is a constant C = C(n) such that

|∇u(x)| ≤ Cg(x)

for almost all x ∈ Ω.30

Proof. Recall that, according to the ACL-characterization of the Sobolev spaceW 1,p(Ω), [84, Thm 2.1.4], a function u ∈ Lp(Ω) belongs to W 1,p(Ω), if and onlyif it has a representative that is absolutely continuous on almost all line segmentsin Ω parallel to the coordinate axes and whose (classical) partial derivatives belongto Lp(Ω).

Thus we may assume that Ω is a finite line segment I ⊂ R. Let u ∈M1,p(I) andlet g ∈ D(u) ∩ Lp(I). We may assume that (5.1) holds in all points of I. We willshow that

(5.3) |u(a)− u(b)| ≤ 4

∫ b

a

g(x) dx

for almost all a, b ∈ I, a < b. By the absolute continuity of the integral, this thenimplies that u is absolute continuous. As an absolutely continuous function, u′ (theclassical derivative) exists almost everywhere, u′ is integrable and the fundamentaltheorem of calculus holds for u. Hence we have that |u′| ≤ 4g.

To show that (5.3) holds, let a, b ∈ I, a < b, be such that g(a) and g(b) are finite(this holds for almost all point by the integrability of g). Let N ∈ N and divide[a, b] into N intervals Ii of length (b− a)/N . For each i = 1, . . . , N , there is a pointxi ∈ Ii such that

g(xi) ≤∫Ii

g(x) dx.

Hence, with the notation x0 = a, xN+1 = b,

|u(a)− u(b)| ≤N∑i=0

|u(xi)− u(xi+1)| ≤N∑i=0

d(xi, xi+1)(g(xi) + g(xi+1))

≤ 4(b− a)

N

N∑i=0

∫Ii

g(x) dx+(b− a)

N(g(a) + g(b))

→ 4

∫ b

a

g(x) dx as N →∞,

from which (5.3), and hence the theorem follows.

Remark 9. The proof of Theorem 5.2 shows that

(5.4) |u(x)− u(y)| ≤ C|x− y|(M2|x−y||∇u|(x) +M2|x−y||∇u|(y)

)whenever u ∈ W 1,p(Rn), 1 ≤ p < ∞, and x, y ∈ Rn are Lebesgue points of u. Ingeneral, as the next example shows, M1,1(Ω) is a smaller space than W 1,1(Ω), evenfor balls and for Ω = Rn.

Indeed, by [27, Thm 4], the above inequality gives a characterization to W 1,1(Rn):A function u ∈ L1(Rn) belongs to W 1,1(Rn), if and only if there exists a functiong ≥ 0 and a constant σ ≥ 1 such that g ∈ L1(Rn) and

|u(x)− u(y)| ≤ |x− y|(Mσ|x−y||g(x) +Mσ|x−y|g(y)

)for almost all x, y ∈ Rn. Instead of charactering Sobolev spaces, inequality (5.1) forn/(n+ 1) < p ≤ 1 characterizes Hardy–Sobolev spaces, see [56].

31

17.2 =============================

Example 5.4 (M1,1 6= W 1,1). Let Ω = (−1/2, 1/2) ⊂ R and let u : Ω → R,u(x) = −x/(|x| log |x|).

-0.4 -0.2 0.2 0.4

-1.5

-1.0

-0.5

0.5

1.0

1.5

Then u ∈ W 1,1(Ω) with the weak derivative u′(x) = |x|−1(log |x|)−2. If thereexists a function g : Ω → [0,∞], g ∈ L1(Ω), such that (5.1) holds, then, for each0 < x < 1/2,

− 2

log x= |u(x)− u(−x)| ≤ 2x(g(x) + g(−x)),

and hence ∫ 1/2

−1/2

g(x) dx =

∫ 1/2

0

g(x) + g(−x) dx ≥∫ 1/2

0

−1

x log xdx =∞,

which is a contradiction with the integrability of g. Hence u /∈M1,1(Ω).

Remark 10. Note that it follows directly from the definition that M1,p(X) =M1,p(X \ E) whenever µ(E) = 0. Such a result does not hold for Sobolev spacesW 1,p, where the removability question is connected to the validity of a Poincareinequality, see [54].

5.0.1. On the properties of M1,p(X). Next we discuss some standard properties ofthe Sobolev spaces M1,p(X) in the metric setting. Many of them hold even withoutthe assumption that µ is doubling.

Theorem 5.5. The space M1,p(X) is a Banach space for all 1 ≤ p <∞.

Proof. Let (ui) be a Cauchy sequence in M1,p(X). Since Lp(X) is a Banach space,there exists a function u ∈ Lp(X) such that ui → u in Lp(X) as i → ∞. We willshow that ui → u in M1,p(X).

We may assume (by taking a subsequence) that

‖ui+1 − ui‖M1,p(X) ≤ 2−i

for all i ∈ N and that ui(x)→ u(x) as i→∞ for almost all x ∈ X. Hence, for eachi ∈ N, there exists gi ∈ Lp(X) such that

|(ui+1 − ui)(x)− (ui+1 − ui)(y)| ≤ d(x, y)(gi(x) + gi(y))32

for almost all x, y ∈ X and that ‖gi‖Lp(X) ≤ 2−i. This implies that

|(ui+k − ui)(x)− (ui+k − ui)(y)| ≤ d(x, y)( ∞∑

j=i

gj(x) +∞∑j=i

gj(y))

for all i, k ≥ 1 for almost all x, y ∈ X. This together with the pointwise convergenceui → u, shows that, letting k →∞, we have

|(u− ui)(x)− (u− ui)(y)| ≤ d(x, y)( ∞∑

j=i

gj(x) +∞∑j=i

gj(y)).

Hence u − ui has a generalized gradient∑∞

j=i gj. As ‖∑∞

j=i gj‖Lp(X) ≤ 2−i+1, we

conclude that u − ui ∈ M1,p(X), that ui → u in M1,p(X). Thus u ∈ M1,p(X) andthe claim follows.

The following generalization of the Leibniz rule holds for functions in M1,p. Itwas proved in [28, Lemma 5.20].

Lemma 5.6. Let Ω ⊂ X. Let u ∈ M1,p(Ω) with gu ∈ D(u) ∩ Lp(Ω). Let ϕ be abounded L-Lipschitz function whose support is in Ω. Then uϕ ∈M1,p(X), and

g =(gu‖ϕ‖∞ + L|u|

)χsuppϕ ∈ D(uϕ) ∩ Lp(X).

Proof. By the triangle inequality,

|u(x)ϕ(x)− u(y)ϕ(y)| ≤ Ld(x, y)|u(x)|+ |ϕ(y)||u(x)− u(y)|and

|u(x)ϕ(x)− u(y)ϕ(y)| ≤ Ld(x, y)|u(y)|+ |ϕ(x)||u(x)− u(y)|.Now it suffices to consider four easy cases depending on whether x or y belongs tosuppϕ or not, and the claim easily follows.

When using the other definitions, we often assume that the space supports a weakPoincare inequality (more about that later). For functions in M1,p(X), a Poincareinequality follows from the definition.

Theorem 5.7 (Poincare for M1,p). Let u ∈ M1,p(X), 1 ≤ p < ∞ with g ∈ D(u).Then ∫

B

|u− uB| dµ ≤ 4r

∫B

g dµ

for all balls B = B(x, r) in X.

Proof. Let B = B(x, r) be a ball. Using (5.1), we obtain∫B

|u(y)− uB| dµ(y) ≤∫B

∫B

|u(y)− u(z)| dµ(z) dµ(y)

≤∫B

∫B

d(y, z)(g(y) + g(z)

)dµ(z) dµ(y)

≤ 2r

∫B

(g(y) + gB

)dµ(y) = 4r

∫B

g dµ.

33

A similar calculation as above using the Holder inequality shows that∫B

|u− uB|p dµ ≤ Crp∫B

gp dµ,

where the constant C ≥ 1 depends only on p. Moreover, balls can be replaced byany bounded measurable sets with positive measure, in that case the diameter ofthe set is used instead of radius.

The classical Lusin(/Luzin) theorem says that each measurable function is con-tinuous outside a set of arbitrary small measure. For Sobolev spaces, even in themetric setting, the approximation can be done by smoother functions that approx-imate the given function in norm. For Sobolev spaces W 1,p, see for example [20,Chapter 6.6.3], [84, Thm 3.10.5, 3.11.6].

The following approximation result by Lipschitz functions, is also a version of the”H=W” theorem, [66], in the metric setting.

Theorem 5.8. Let u ∈ M1,p(X), 1 ≤ p < ∞. For each ε > 0, there is a Lipschitzfunction v : X → R such that

(1) µ(x : u(x) 6= v(x)) < ε and(2) ‖u− v‖M1,p(X) < ε.

Proof. Let gu ∈ D(u) be such that ‖gu‖Lp(X) ≤ 2 infg∈D(u) ‖g‖Lp(X). We may assumethat (5.1) holds for u and gu everywhere.

For each λ > 0, define

Eλ =x ∈ X : |u(x)| ≤ λ and gu(x) ≤ λ

.

Since u, gu ∈ Lp(X), λpµ(X \ Eλ)→ 0 as λ→∞.For all x, y ∈ Eλ,

|u(x)− u(y)| ≤ d(x, y)(g(x) + g(y)) ≤ 2λd(x, y),

and hence u|Eλ is 2λ-Lipschitz. By the McShane extension theorem (Theorem 2.5),there exists a 2λ-Lipschitz function uλ : X → R such that uλ = u in Eλ. We cut thefunction uλ to level λ by defining a function vλ : X → [−λ, λ],

vλ = signuλ min|uλ|, λ.

Then vλ is 2λ-Lipschitz and vλ = uλ = u on Eλ. Thus

µ(x : u(x) 6= vλ

)≤ µ(X \ Eλ)→ 0 as λ→∞,

which proves the first claim.For the second claim, it suffices to show that vλ → u in M1,p(X). Since now∫

X

|u− vλ|p dµ =

∫X\Eλ

|u− vλ|p dµ

≤ 2p(∫

X\Eλ|u|p dµ+ λpµ(X \ Eλ)

)→ 0

as λ→∞, we see that vλ → u in Lp(X).34

As a final step of the proof, we have to find a function gλ ∈ D(u− vλ) such that‖gλ‖Lp(X) → 0 as λ→∞. Using the inequality

|(u(x)− vλ(x))− (u(y)− vλ(y))| ≤ |u(x)− u(y)|+ |vλ(x)− vλ(y)|,

the definition of Eλ and the facts that vλ = u on Eλ and vλ is 2λ-Lipschitz, it iseasy to see that gλ : X → [0,∞],

gλ(x) =

0, if x ∈ Eλ,g(x) + 3λ, if x ∈ X \ Eλ,

is the desired function.

Remark 11. The definition of M1,p(X), of course, provides, for each ε > 0, ageneralized gradient gu of u, such that ‖gu‖Lp(X) < infg∈D(u) ‖g‖Lp(X) + ε. Whenp > 1, there exists a unique minimal generalized gradient, gu ∈ D(u) ∩ Lp(X) forwhich

‖gu‖Lp(X) = infg∈D(u)

‖g‖Lp(X).

Indeed, if (gi) is a sequence of generalized gradients of u such that ‖gi‖Lp(X) →infg∈D(u) ‖g‖Lp(X) as i → ∞, then, as a bounded sequence in in a reflexive Banachspace Lp(X), it contains a weakly convergent subsequence, denoted also by (gi). Bythe Mazur lemma, a sequence (gj) of convex combinations of the functions gi,

gj =

j∑k=1

aj,kgk,

j∑k=i

aj,k = 1, aj,k ≥ 0

converges in Lp(X), to the same limit function g0 ∈ Lp(X) as (gi). It is easy tosee that then gj ∈ D(u) for each j ∈ N, and hence also g0 ∈ D(u). The normminimizing function g0 ∈ Lp(X) is unique because Lp(X) is uniformly convex whenp > 1.

For the functional analytic results used in the proof, see [83, Thm 1, p.126 andThm 2, p.120].

18.1 =============================

Although each function g ∈ D(u) ∩ Lp(X) is a substitute of a derivative of afunction u ∈ M1,p(X), it is behaves more like a maximal function of the gradient,as (5.4) suggests. The definition (5.1) is global, and by this global behaviour, it isnot necessarily true that g ∈ D(u) (even the minimal one), is zero in the sets whereu is constant.

Example 5.9. Let X = [0, 1] ∪ [2, 3] ⊂ R equipped with the Euclidean metric andthe restriction of the Lebesgue measure to X. Then u = χ[0,1] belongs to M1,p(X)for all p ≥ 1, for example, the function g = 1 belongs to D(u) ∩ Lp(X). However,the zero function is not a generalized gradient of u.

An advantage of the pointwise definition is the validity of a Sobolev–Poincareembedding theorem with minimal assumptions on X. Below, s = log2 cd is the

35

doubling dimension of X from Lemma 3.2 and

p∗ =sp

s− pis the Sobolev exponent for s/(s+ 1) ≤ p < s.

The embedding theorem holds also for 0 < p < s/(s+ 1), but in that case p∗ < 1and the function u is not necessarily locally integrable. Hence, in that case, the leftside of (5.5) is replaced with infc∈R(

∫B|u − c|p∗ dµ)1/p∗ . Moreover, it is enough to

assume that µ satisfies a lower bound property, weaker than doubling, see [26, Thm8.7].

Exercise 2. Show that for locally integrable functions, the two different versions ofthe left side of (5.5) are comparable: If B is a ball, u ∈ L1(B) and q ≥ 1, then thereis a constant C = C(q) such that

infc∈R

(∫B

|u− c|q dµ)1/q

≤(∫

B

|u− uB|q dµ)1/q

≤ C infc∈R

(∫B

|u− c|q dµ)1/q

.

Theorem 5.10 (Sobolev-Poincare for M1,p). Let B = B(x, r) be a ball and letu ∈ M1,p(2B), s/(s + 1) ≤ p < ∞. Then there exists constants C, C1 and C2,depending only on p and s such that

(1) If s/(s+ 1) ≤ p < s, then u ∈ Lp∗(B) and

(5.5)

(∫B

|u− uB|p∗dµ

)1/p∗

≤ Cr

(∫2B

gp dµ

)1/p

.

(2) If p = s, then

(5.6)

∫B

exp

(C1µ(2B)1/s

r

|u− uB|‖g‖Ls(2B)

)dµ ≤ C2.

(3) If p > s, then

(5.7) |u(x)− u(y)| ≤ Crs/pd(x, y)1−s/p(∫

2B

gp dµ

)1/p

for all x, y ∈ B.

Proof. The proof is quite long and technical. In the beginning of the proof, it ischecked what we can assume from u and g.

Concerning u; since the left sides of the embedding inequalities don’t change if aconstant is added to u, we may assume that essinfE |u| = 0 for a set E ⊂ 2B withµ(E) > 0. With a clever choice of E, the claim (5.5) is proved with (

∫B|u|p∗ dµ)1/p∗

on the left side.For g; we may assume that g > 0 an a set of positive measure in 2B because

otherwise u is constant and the claims follow. Moreover, we may assume that g(x) ≥2−(1+1/p)(

∫2Bgp dµ)1/p for all x ∈ 2B (otherwise replace g with g = g+(

∫2Bgp dµ)1/p).

The main idea of the proof of (5.5) is to estimate the integrals∫B|u|p∗ dµ and∫

2Bgp dµ using sets

Ek = x ∈ 2B : g(x) ≤ 2k, k ∈ Z,36

and estimates for ak = supEk |u|. Since

(5.8)

∫2B

gp dµ ≈∑k∈Z

2kpµ(Ek \ Ek−1),

and for s/(s+ 1) ≤ p < s,

(5.9)

∫B

|u|p∗ dµ ≤∑k∈Z

ap∗

k µ(B ∩ (Ek \ Ek−1)),

we try to estimate the right side of the latter inequality in terms of the formerestimate for the integral of |g|p.

By the definition of M1,p and the sets Ek, the function u is 2k+1-Lipschitz in Ekand hence, for each x ∈ Ek, y ∈ Ek−1,

|u(x)| ≤ |u(x)− u(y)|+ |u(y)| ≤ 2k+1d(x, y) + ak−1.

Moreover, for each k ∈ Z,

(5.10) µ(2B \ Ek) = µ(x ∈ 2B : g(x) > 2k) ≤ 2−kp∫

2B

gp dµ.

Using the assumptions on g made in the beginning of the proof and weak typeestimate (5.10), one can show that there is an index k0 ∈ Z such that µ(Ek0) > 0and

2k0 ≈ C1/p1

(∫2B

gp dµ)1/p

where C1 = cd/C0 and C0 is the constant from the doubling dimension inequality(3.4).

One of the main steps in the proof is to find for each x ∈ Ek, k > k0, a pointy ∈ Ek−1 such that d(x, y) is sufficiently small,

d(x, y) ≤ 2C−1/s1 r2−(k−1)p/s

(∫2B

gp dµ)1/s

,

Iteration of that together with the Lipschitz continuity of u in sets Ek gives anestimate

(5.11) ak ≤ 8C1/s1

(∫2B

gp dµ)1/s

k−1∑j=k0

2j(1−p/s) + supEk0∩2B

|u|.

Then, then set E with the property essinfE |u| = 0, discussed in the beginning ofthe proof is selected to be Ek0 ∩ 2B. Hence there is a sequence (yi) ⊂ Ek0 such thatu(yi)→ 0 as i→∞. This and the Lipschitz continuity is used to show that

supEk0∩2B

|u| ≤ 8r2k0 ,

and hence ak ≤ ak0 ≤ 8r2k0 for all k > k0. Now a calculation using using (5.9),(5.8), the selection of k0 and the estimates for ak gives the desired upper bound forintegral

∫B|u|p∗ dµ. For more details and inequalities (5.6) and (5.7), see [26, Them

8.7]. 37

In addition to references already mentioned in this section, spaces M1,p(X) arestudied in for example in papers [49] (Sobolev capacity), [28] (Holder quasicontinu-ity), [47] and [50] (Lebesgue points and discrete maximal function).

We close this section by briefly discussing why we need other definitions of Sobolevspaces in the metric setting. A good and bad property of spaces M1,p(X) is thatthey don’t care about the geometry of the underlying space. An advantage of thepointwise definition of spaces M1,p(X) is that the definition makes sense in everymetric measure space even without existence of nonconstant rectifiable curves. Thisis the case for example if X is a Cantor type set or the von Koch snowflake curveequipped with the Euclidean metric and the natural Hausdorff measure on X.

By Theorems 5.7 and 5.10, the functions u ∈M1,p(X) together with the general-ized gradients always satisfy a Poincare type inequality (even without the doublingproperty of µ) and the Sobolev–Poincare embedding theorem holds (a version of itholds also for more general sets than balls, see [6, Thm 5.4.2], [24, Thm 6]). Recallthat, in the classical case, the validity of the classical isoperimetric inequality forsubsets of the Euclidean space is equivalent to the Sobolev embedding theorem forp = 1, see [84, Chapter 2.7]. In the case of M1,p(X), we loose some information viathe global definition of the generalized gradient.

6. Sobolev spaces in metric measure spaces - N1,p(X)

In this section, we will discuss another definition of Sobolev spaces in metric mea-sure spaces, based on upper gradients. It is really useful in spaces with sufficientlymany rectifiable curves and can be used to build the nonlinear potential theory inthe metric setting, see the recent book [8] by A. Bjorn and J. Bjorn. The spaceN1,p(X) was defined by Shanmugalingam in [69] and studied in several papers af-ter that. For the properties of the spaces N1,p(X), see the books [8] and [41] andreferences therein.

In some sense, the existence of large families of rectifiable curves is essential fora reasonable substitute of the first class calculus in the metric setting. We couldhope that a generalization of gradient exist for Lipschitz functions and version of ”afunction is constant in a domain if and only if the gradient is zero” holds. As wesaw in Example 5.9, a generalized gradient of an M1,p(X)-function is not necessarilyzero in the set where the function is constant.

Example 6.1 (From [39]). Let X = ([0, 1], |x−y|1/2,H2) be a ”snowflaked” versionof [0, 1] with the 2-dimensional Hausdorff measure H2. The function u : [0, 1] →[0, 1], u(x) = x, is Lipschitz on X and

|u(x)− u(y)||x− y|1/2

= |x− y|1/2 → 0 as y → x for each x ∈ [0, 1].

Hence, a reasonable ”gradient” u should vanish on [0, 1]. On the other hand, thisinfinitesimal information should imply that u is constant.

In the Euclidean case, a refinement of the ACL-characterization of Sobolev func-tions tells that Sobolev functions are not absolutely continuous only on almost all

38

line segments parallel with coordinate axes but also on almost all compact curves,in the sense of p-modulus, see the discussion in [41, Chapter 5] and [39, Chapter 7].Note that the representative provided by this characterization is even ”better” thanthe ACL-representative, namely the quasicontinuous representative of the Sobolevfunction. This coordinate free characterization of Sobolev spaces W 1,p(Rn) is theidea behind the spaces N1,p(X).

Theorem 6.2 ([39], Thm 7.6.). Let u ∈ Lp(Rn), 1 ≤ p < ∞. Then u has arepresentative in W 1,p(Rn) if and only if there exists a nonnegative Borel functiong ∈ Lp(Rn) such that the inequality

u(γ(a))− u(γ(b)) ≤∫γ

g ds

holds for the representative, for p-almost every curve γ : [a, b]→ Rn.

This result gives a motivation for the definition of upper gradients, originallydefined by name very weak gradients by Heinonen and Koskela in [40] and for p-weak upper gradients which behave better in limiting processes, defined by Koskelaand MacManus in [55]. We begin with the definition of upper gradients and weakupper gradients and give some examples of upper gradients, and after that, definep-modulus, a tool to measure curve families. For properties of curves in metricspaces, we refer to [41, Chapter 4].

Recall that a function v : X → R (a more generally between two topologicalspaces), is a Borel function if the preimage of every open set is a Borel set. Hence,under a Borel function the preimage of every Borel set is a Borel set, see [41, Chapter2.3].

Curves in metric spaces. Before going to the next definition of Sobolev spaces, werecall the definition and the basic facts about curves in metric spaces. By a curvewe mean a continuous map γ : I → X, where I ⊂ R is an interval. A subcurve of γis the restriction of γ to a subinterval J ⊂ I. We say that a curve γ is compact if Iis compact. If nothing else is said, we assume that curves are compact. The lengthof a compact curve γ : [a, b]→ X is

`(γ) = supn∑i=1

d(γ(ti), γ(ti+1)),

where the supremum is taken over finite subdivisions a = t1 ≤ t2 ≤ · · · ≤ tn+1 = b. Ifγ is not compact, its length is the supremum of the lengths of the compact subcurvesof γ. A curve γ is rectifiable if its length is finite, otherwise it is nonrectifiable. Itis locally rectifiable if all its compact subcurves are rectifiable. We usually write γalso when we mean γ(I), the image of I.

The line integral over a rectifiable curve is defined using the fact that for eachsuch γ there is a length function sγ : I → [0, `(γ)] and a unique 1-Lipschitz mapγs : [0, `(γ)] → X such that γ = γs sγ. We usually assume that γ = γs, that is, γis parametrized by the arc length.

39

For a rectifiable curve γ and a Borel function ρ : X → [0,∞] the line integral ofρ over γ is ∫

γ

ρ ds =

∫ `(γ)

0

ρ(γs(t)

)dt.

For the details concerning curves in metric spaces we refer to [40] and [41, Chapter4.1]. The monograph [79] of Vaisala is a classical reference for the Euclidean case.

Definition 6.3 (Upper gradient). A Borel function g : X → [0,∞] is an uppergradient of a function u : X → [−∞,∞], if

(6.1) |u(x)− u(y)| ≤∫γ

g ds,

for all x, y ∈ X and for every rectifiable curve γ joining points x and y, wheneverboth u(x) and u(y) are finite, and

∫γg ds =∞ otherwise. If (6.1) holds for p-almost

every rectifiable curve, then g is a p-weak upper gradient of u. Upper gradients andp-weak upper gradients in open set Ω ⊂ X are defined naturally using rectifiablecurves contained in Ω.

Example 6.4 (From [41] Chapter 5).

(1) the function g = ∞ is an upper gradient of each function u. Of course, weare, in general, interested in upper gradients for which the right side of (6.1)is finite.

(2) If gu is an upper gradient of u and g ≥ 0 is a Borel function, then the functiongu+g is an upper gradient of u, as well as a pointwise maximum of two uppergradients.

(3) If X has no nonconstant rectifiable curves, then g = 0 is an upper gradientof each function u.

(4) If u : X → R is L-Lipschitz, then g = L is an upper gradient of u.(5) If g is an upper gradient of a function u : X → R and v : R → R is L-

Lipschitz, then Lg is an upper gradient of the function v u : X → R.(6)(7) If gu and gv are upper gradients of functions u, v : X → R and c ∈ R, then

gu + gv is an upper gradient of u+ v, |c|gu is an upper gradient of cu and guis an upper gradient of |u|.

(8) If Ω ⊂ Rn is an open set and u ∈ C∞(Ω), then the fundamental theorem ofcalculus implies that |∇u| is an upper gradient u in Ω. By the characteri-zation of W 1,p(Ω) discussed above, (every Borel representative of) |∇u| is ap-weak upper gradient of (a quasicontinuous representative of) u.

24.2 =============================

Example 6.5 (Pointwise Lipschitz constants).Let u : X → R. The pointwise lower Lipschitz constant function lipu : X → R

and the pointwise upper Lipschitz constant function Lipu : X → R of u are definedas

lipu(x) = lim infr→0

supy∈B(x,r)

|u(x)− u(y)|r

, for all x ∈ X

40

and

Lipu(x) = lim supr→0

supy∈B(x,r)

|u(x)− u(y)|r

, for all x ∈ X.

If u is continuous, then lipu and Lipu are Borel functions and if u is (locally)Lipschitz, then they are upper gradients of u, see [41, Lemmas 5.2.7, 5.2.8].

Note that if u is only continuous, then lipu is not necessarily an upper gradient:”the Cantor-staircase function” is continuous and u′ = 0 almost everywhere in [0, 1]and hence lipu is not an upper gradient.

Exercise 3. Let F ⊂ X be a closed set. Let g ≥ 0 be a bounded Borel functionand let u : X → R,

u(x) = inf

∫γx

g ds,

where the infimum is taken over all rectifiable curves γx joining x to F . Show thatg is an upper gradient of u.

In the definition of p-weak upper gradients, we ignore a set of curves with zerop-modulus, and require that upper gradient inequality (6.1) holds for the remainingcurves. The p-weak upper gradients turn out to be more flexible than upper gradi-ents; they can, for example, be modified in a set of measure zero so that the weakupper gradient property still holds, see Lemma 6.12 below.

6.1. Modulus of a curve family. In the definition of p-weak upper gradients, weneed the p-modulus, a tool to measure curve families. The abstract definition ofmodulus appeared in the paper [21] of Fuglede. Vaisala’s lecture notes [79] containa careful treatment of line integrals and modulus in Rn. Modulus in a more generalmetric space was defined in [40], for the properties and proofs of the results that arenot proved below, see also, for example, [41, Chapter 4.2] and [71].

Definition 6.6. Let Γ be a family of curves in X and let 1 ≤ p <∞. The p-modulusof Γ is

Modp(Γ) = inf

∫X

ρp dµ,

where the infimum is taken over all Borel-functions ρ : X → [0,∞] satisfying

(6.2)

∫γ

ρ ds ≥ 1

for all locally rectifiable curves γ ∈ Γ. If a function satisfies (6.2), it is said to beadmissible for Γ.

Basic properties of the p-modulus:

(1) The family of curves that are not locally rectifiable has zero modulus. If Γcontains a constant curve, then Modp(Γ) =∞ for all p.

(2) Modp is an outer measure on the collection of all curve families in X.(3) If Γ and Γ0 are curve families such that each γ ∈ Γ has a subcurve γ0 ∈ Γ0,

then Modp(Γ) ≤ Modp(Γ0). ”Existence of long curves in the family impliessmall modulus.”

41

A property of curves holds for p-almost every curve, if the p-modulus of theset of curves for which the property does not hold is zero. A family Γ of curveswith Modp(Γ) = 0 is sometimes called p-exceptional. Moreover, a set E ⊂ X isp-exceptional, if the family of all nonconstant curves that meet E is p-exceptional.

The following lemma characterizes curve families of zero p-modulus. If we canfind an admissible function that is in Lp(X) and for which the line integral over allcurves of Γ is infinite, we know that the p-modulus of Γ is zero. The lemma is ageneralization of one of Fuglede’s results, [21, Thm 2].

Lemma 6.7. Let 1 ≤ p < ∞ and let Γ be a curve family in X. Then Modp(Γ) =0 if and only if there is a Borel function ρ : X → [0,∞], ρ ∈ Lp(X), such that∫γρ ds =∞ for all γ ∈ Γ.

Proof. Assume first that there is a Borel function ρ ≥ 0, ρ ∈ Lp(X), such that∫γρ ds = ∞ for all curves γ ∈ Γ. Then the function ρi = 2−iρ is admissible for Γ

for all i ∈ N and ∫X

ρip dµ = 2−ip

∫X

ρp dµ→ 0 as i→∞.

Hence Modp(Γ) = 0.Assume then that Modp(Γ) = 0. For all i ∈ N, there is ρi, an admissible function

for Γ with∫Xρip dµ ≤ 2−ip. Then the function

ρ(x) =∞∑i=1

ρi(x)

is a Borel function and ρ ∈ Lp(X). As each ρi is admissible,∫γρi ds ≥ 1 for all

curves γ in Γ. The integral∫γρ ds is thus infinite by the definition of ρ.

Lemma 6.8. Let E ⊂ X be a set with µ(E) = 0 and let

Γ+E = γ : L1(γ−1

s (E)) > 0be a family of curves for which the length of γ in E is positive. Then Modp(Γ

+E) = 0

for all 1 ≤ p <∞.

Proof. Let F be a Borel set such that E ⊂ F and µ(F ) = 0. Since for the functionρ =∞χF , ∫

γ

ρ ds =

∫ `(γ)

0

ρ(γs(t)) dt =∞L1(γ−1s (F )) =∞

for all γ ∈ Γ+E and

∫Xρp dµ = 0, the claim follows from Lemma 6.7.

Note that, by using the fact that H1(γ ∩ E) ≤ L1(γ−1s (E)), the lemma above

shows that H1(γ ∩ E) = 0 for p-almost every curve if µ(E) = 0, see [41, Lemma4.2.18].

Example 6.9. It is usually difficult to obtain the precise value of the p-modulus ofthe given curve family. By the definition of the p-modulus, it is easier to find anupper bound. The following examples are from [41, Chapter 4.3].

42

(1) Let A ⊂ X be a Borel set and let Γ be family of curves for which γ ⊂ A and`(γ) ≥ L > 0 for all γ ∈ Γ. Then the function ρ = L−1χA is admissible forΓ and hence Modp(Γ) ≤ L−pµ(A).

(2) Spherical shells: Let 0 < r < R and let x ∈ Rn, n ≥ 2. Let Γr,R be thecollection of all locally rectifiable curves that connect B(x, r) to Rn\B(x,R).Then

Modp(Γr,R) =

∣∣n−pp−1

∣∣p−1ωn−1

∣∣r p−np−1 −Rp−np−1

∣∣1−p, if p /∈ 1, n,ωn−1

(log R

r

)1−nif p = n

ωn−1rn−1 if p = 1,

where ωn−1 is the surface measure on Sn−1. This implies that the p-modulusof all nonconstant curves in Rn that go through a fixed point is zero if andonly if 1 ≤ p ≤ n.

(3) A metric space version of spherical shells case ”p = n”: Let x0 ∈ X. Assumethat there exist constants C0, R0 > 0 such that µ(B(x0, r)) ≤ C0r

p for all 0 <r < R0. Let Γr,R be the collection of all locally rectifiable curves that connectB(x0, r) to X \ B(x,R). Then there exists a constant C1 = C1(C0, p) > 0such that

Modp(Γr,R) ≤ C1

(log

R

r

)1−p,

whenever 0 < 2r < R < R0.If there exist constants R,CR > 0 andQ > 1 such that µ(B(x0, r)) ≤ CRr

Q

for all 0 < r < R, then the p-modulus of all nonconstant curves in X thatgo through x0 is zero for all 1 ≤ p ≤ Q.

3.3 =============================

Fuglede’s lemma shows that if a sequence converges in Lp(X), then it has asubsequence that converges on p-almost every locally rectifiable curve.

Lemma 6.10 (Fuglede’s Lemma). Let 1 ≤ p < ∞ and let g, gi, i ∈ N, be Borelfunctions such that gi → g in Lp(X).

Then there is a subsequence (gik) of (gi) such that

limk→∞

∫γ

|gik − g| ds = 0

for p-almost all curves γ in X. (Indeed, the claim holds for all Borel representativesof the Lp(X) limit of (gi)).

Proof. There is a subsequence (gik) of (gi) such that∫X

|gik − g|p ≤ 2−(p+1)k for each k ∈ N.

Define functions

ρk = |gik − g|,43

and families of locally rectifiable curves by setting

Γ =γ : lim

k→∞

∫γ

ρk ds does not exist or limk→∞

∫γ

ρk ds 6= 0

and

Γk =γ :

∫γ

ρk ds > 2−k, k ∈ N.

Then Γ ⊂⋃∞k=j Γk for each j ∈ N. Since for each k, the function 2kρk is admissible

for Γk, we have, by the selection of the subsequence, that

Mod(Γk) ≤ 2kp∫X

ρpk dµ = 2kp∫X

|gik − g|p dµ ≤ 2−k.

The subadditivity of p-modulus implies that

Modp(Γ) ≤∞∑k=j

Modp(Γk) ≤ 2−j+1

for each j ∈ N. The lemma follows by letting j →∞.

6.2. Properties of p-weak upper gradients. In this subsection, we prove someimportant properties of weak upper gradients. We are usually interested in func-tions having p-integrable p-weak upper gradients but the first two results hold evenwithout integrability.

Lemma 6.11. If a Borel function g is a p-weak upper gradient of a function u, thenthere is a decreasing sequence (gi) of upper gradients of u such that

‖gi − g‖Lp(X) → 0 as i→∞.

Proof. Let Γ be the curve family with zero p-modulus for which upper gradientinequality (6.1) does not hold for the pair (u, g). By Lemma 6.7, there is a Borelfunction ρ ≥ 0 such that ρ ∈ Lp(X) and

∫γρ ds =∞ for all γ ∈ Γ. Define for each

i ∈ N, gi : X → [0,∞],

gi = g + 2−iρ.

Then each gi is an upper gradient of u, the sequence (gi) is decerasing and

‖gi − g‖Lp(X) = 2−i‖ρ‖Lp(X) → 0 as i→∞.

Weak upper gradients can be modified in a set of measure zero.

Lemma 6.12. Let g be a p-weak upper gradient of u. If h : X → [0,∞] is a Borelfunction such that h = g µ-almost everywhere, then h is also a p-weak upper gradientof u.

Proof. Let

gi = |g − h|44

for all i ∈ N. Then the sequence (gi) trivially converges to 0 in Lp(X) and, byFuglede’s lemma 6.10,

(6.3)

∫γ

|g − h| ds = 0

for p-almost all curves γ in X. Let γ be a rectifiable curve with endpoints x andy such that (6.3) holds and that inequality (6.1) holds for u and g (it holds forp-almost curves). Then

|u(x)− u(y)| ≤∫γ

g ds =

∫γ

g ds−∫γ

|g − h| ds ≤∫γ

h ds,

which shows that h is a p-weak upper gradient of u.

Remark 12. Lemma 6.12 implies that if E is a Borel set with µ(E) = 0 and g is ap-weak upper gradient of u, then gχX\E is a p-weak upper gradient of u.

Note that a corresponding modification on a set of measure zero cannot be donefor upper gradients. Namely, if u : R2 → R is an 1-Lipschitz function, then g = 1is an upper gradient of u but gχR2\L, where L is a line, is not necessarily an uppergradient.

Next we will show that the existence of a p-integrable weak upper gradient impliesthat the function is absolutely continuous on p-almost every curve. This is of coursenatural because p-weak upper gradients are defined starting from Theorem 6.2 forclassical Sobolev spaces.

Example 6.13. If u : Rn → R is (locally) Lipschitz, then it is differentiable almosteverywhere by the Rademacher theorem. If x ∈ Rn is a point of differentiability ofsuch a function u, r > 0 and y ∈ B(x, r), then

|u(x)− u(y)| = |∇u(x) · (y − x) + o(|y − x|)| ≤ r|∇u(x)|+ o(r).

This together with the upper gradient property of lipu and Lemma 6.12 shows that|∇u| is an upper gradient of u.

Definition 6.14. A function u : X → R is absolutely continuous on p-almost everycurve in X, u ∈ ACCp(X), if u γs is absolutely continuous on [0, `(γ)] for p-almostevery rectifiable curve γ. Then we sometimes say that u is absolutely continuous onγ.

Lemma 6.15. If u : X → R has a p-weak upper gradient g ∈ Lp(X), then u ∈ACCp(X).

Proof. Define families of rectifiable curves,

Γ0 =γ : (6.1) does not hold for u and g

, Γ∞ =

γ :

∫γ

g ds =∞,

and

Γ =γ : γ has a subcurve in Γ0 ∪ Γ∞

.

45

By the definition of the p-weak upper gradient, Modp(Γ0) = 0. Since g ∈ Lp(X),Lemma 6.7 implies that Modp(Γ∞) = 0. The properties of p-modulus show thatalso Modp(Γ) = 0.

The absolute continuity of u on all rectifiable curves γ /∈ Γ follows from uppergradient inequality (6.1) and from the absolute continuity of the integral

∫γg ds <

∞.

Exercise 4. Let u ∈ ACCp(X) be a function such that u = c0 µ-almost everywherein X. Show that the function g = 0 is a p-weak upper gradient of u.

6.3. How to find p-weak upper gradients. In this subsection, we give somemethods to find p-weak upper gradients. For the proof of Lemmas 6.16 and 6.17,see [41] or [8, Chapter 2].

Lemma 6.16 (Lemma 5.3.10 in [41]). Let g, h ∈ Lp(X) be p-weak upper gradientsof a function u : X → R. If F ⊂ X is a Borel set, then the function

ρ = gχF + hχX\F

is a p-weak upper gradient of u.

4.3 =============================

Remark 13. By selecting E = x ∈ X : g(x) < h(x) in Lemma 6.16, we seethat the function ming, h is a p-weak upper gradient of the function u : X → Rwhenever g, h ∈ Lp(X) are p-weak upper gradients of u. This together with thebasic properties of p-weak upper gradients and Fuglede’s lemma 6.10 shows that foreach function u : X → R, the collection

U =g : g ∈ Lp(X), g is a p–weak upper gradients of u

is a closed, convex lattice in Lp(X) (Check the details as an exercise). Recall thata lattice in Lp(X) is a set of functions that is closed under pointwise minimum andmaximum.

The next lemma shows that we can cut and paste p-weak upper gradients.

Lemma 6.17 (Lemma 5.3.16 in [41]). Let u ∈ ACCp(X) and let v, w : X → R befunctions with p-weak upper gradients g, h ∈ Lp(X). If E ⊂ X is a Borel set suchthat u|E = v and u|X\E = w, then the function

ρ = gχE + hχX\E

is a p-weak upper gradient of u.

Proof. The main task is to show that p-integrable functions ρ1 = g + hχX\E andρ2 = gχE + h are p-weak upper gradients of u. The claim then follows from Lemma6.16. For details, see [41, Lemma 5.3.16].

Note that Lemma 6.17 for set E = x ∈ X : v(x) < w(x) shows that the functiongχE + hχX\E is a p-weak upper gradient of the function minv, w.

Since the zero function is a p-weak upper gradient of each constant function, weobtain also the following corollary to Lemma 6.17.

46

Corollary 6.18. Let g ∈ Lp(X) be a p-weak upper gradient of the function u : X →R and let c ∈ R. Then the function gχX\E is a p-weak upper gradient of u for allBorel sets E ⊂ x ∈ X : u(x) = c.

Sometimes, for example while studying removability questions for Sobolev spacesand Sobolev spaces with zero boundary values, we want to glue p-weak upper gradi-ents that are defined on open sets. For the proof of the next lemma, see [77, Lemma4.11].

Lemma 6.19. Let U, V ⊂ X be open sets. If gU ∈ Lp(U) and gV ∈ Lp(V ) arep-weak upper gradients of u in U and V respectively, then the function

g(x) =

gU(x), if x ∈ U \ V,gV (x), if x ∈ V \ U,gU(x) + gV (x), if x ∈ U ∩ V

is a p-weak upper gradient of u in U ∪ V .

Lemma 6.20. Let (ui) be a sequence of measurable functions with a correspondingsequence of p-weak upper gradients (gi). If u = supi ui is finite almost everywhere,then g = supi gi is a p-weak upper gradient of u.

Note that assumption about finiteness of u almost everywhere is needed. If ui = ifor all i ∈ N, then zero-function is a p-weak upper gradient of each ui but it is nota p-weak upper gradient of supui =∞.

Proof. For each i ∈ N, let Γi be the family of rectifiable curves with Modp(Γi) = 0such that (6.1) holds for the pair ui, gi for all curves γ /∈ Γi and let Γ = ∪iΓi. Bythe subadditivity of p-modulus, Modp(Γ) = 0.

Let ε > 0 and let γ /∈ Γ be a curve with endpoints x and y. Assume first that|u(x)| and |u(y)| are finite. We may assume that u(y) ≤ u(x) <∞. Let i ∈ N suchthat

ui(x) + ε > u(x).

Since u(y) ≥ ui(y), gi is a weak upper gradient of ui, and g ≥ gi, we have that

|u(x)− u(y)| = u(x)− u(y) < ui(x) + ε− ui(y)

≤∫γ

gi ds+ ε ≤∫γ

g ds+ ε.

The claim follows by letting ε→ 0 .For the general case, let Γ∞ be a family of curves γ, for which |u| = ∞ on some

subcurve of γ, and let

A = z ∈ X : |u(z)| =∞ and h =∞χA.Since u is finite almost everywhere, µ(A) = 0 and h ∈ Lp(X). As

∫γh ds = ∞ for

all curves γ ∈ Γ∞, Lemma 6.7 implies that Modp(Γ∞) = 0. By the subadditivity ofmodulus, Modp(Γ ∪ Γ∞) = 0.

Let now γ /∈ (Γ ∪ Γ∞) with endpoints x and y. We may assume that u(x) = ∞and |u(y)| <∞, since otherwise, by the selection of Γ∞, there is a point z ∈ γ such

47

that |u(z)| < ∞, and we would estimate subcurves connecting x to z and z to yseparately.

Let ui(x) be a subsequence such that ui(x)→ u(x) as i→∞. By the selection ofthe subsequence, there is i0 ∈ N such that ui(x) > u(y) as i ≥ i0. Then, for i ≥ i0,we have that

|ui(x)− u(y)| = ui(x)− u(y) ≤ ui(x)− ui(y) ≤∫γ

gi ds ≤∫γ

g ds.

The claim follows by letting i→∞.

Lemma 6.20 implies that if ui : X → [0,∞], i ∈ N, are measurable functions withp-weak upper gradients gi and

∑∞i=1 ui(x) converges for almost every x ∈ X, then∑∞

i=1 gi is a p-weak upper gradient of∑∞

i=1 ui.The following lemma is a useful tool while proving a Leibniz rule for p-weak upper

gradients.

Lemma 6.21. Let u ∈ ACCp(X).

(1) If g ∈ Lp(X) is a p-weak upper gradient of u, then for p-almost every recti-fiable curve γ,

(6.4) |(u γs)′(t)| ≤ g(γs(t)) for almost all t ∈ [0, `(γ)].

(2) If g : X → [0,∞] is a measurable function and (6.21) holds for p-almostrectifiable curve, then g is a p-weak upper gradient of u.

Proof. We begin with the first claim. By the assumptions and Lemma 6.7, p-almostevery curve γ satisfies the following properties: u is absolutely continuous on γ,the upper gradient inequality (6.1) holds for u and g (also on subcurves of γ), and∫γg ds <∞.

Let γ be one of those curves with endpoints x, y ∈ X. As an absolutely continuousfunction, uγs is differentiable on almost every t ∈ [0, `(γ)]. Moreover, almost everyt ∈ [0, `(γ)] is a Lebesgue point of g γs. For each such t, we have

|(u γs)′(t)| = limh→0

∣∣∣∣u(γs(t+ h))− u(γs(t))

h

∣∣∣∣≤ lim

h→0

1

h

∫ t+h

t

g(γs(τ)) dτ = g(γs(t)).

For the second claim, let γ be a rectifiable curve with endpoints x and y such thatu is absolutely continuous on γ, inequality (6.4) holds and

∫γg ds is well-defined.

These properties hold for p-almost rectifiable curve, (see Lemma [8, 1.42] for thelast property). Then

|u(x)− u(y)| ≤∫ `(γ)

0

(u γs)′(t) dt ≤∫ `(γ)

0

g(γs(t)) dt =

∫γ

g ds.

Hence g is a p-weak upper gradient of u. 48

Lemma 6.22 (Leibniz rule for p-weak upper gradients). Let u, v : X → R be mea-surable functions with p-weak upper gradients gu, gv ∈ Lp(X). Then the function|u|gv + |v|gu is a p-weak upper gradient of uv.

Proof. By the assumptions on u, v, gu and gv, the function

g = |u|gv + |v|guis nonnegative and measurable. By Lemma 6.15, u, v ∈ ACCp(X), and hence alsouv ∈ ACCp(X). Now p-almost every rectifiable curve γ in X satisfies the followingproperties: u and v are absolutely continuous on γ and the upper gradient inequality(6.1) holds for pairs u, gu and v, gv.

Let γ be one of those curves. By the absolute continuity of u, v and uv on γand by Lemma 6.21, derivatives (u γs)′(t), (v γs)′(t) and ((uv) γs)′(t) exist and|(u γs)′(t)| ≤ gu(γs(t)) and |(v γs)′(t)| ≤ gv(γs(t)) for almost all t ∈ [0, `(γ)].Hence

|((uv) γs)′(t)| = |(u(γs(t))(v γs)′(t) + (v(γs(t))(u γs)′(t)|≤ |(u(γs(t))|gv(γs(t)) + |(v(γs(t))|gu(γs(t))= g(γs(t)),

from which the claim follows by Lemma 6.21.

6.4. Minimal p-weak upper gradient. Each function u : X → R has infinitelymany upper gradients. If it has an p-integrable one, then it has a minimal one.

Definition 6.23. A p-weak upper gradient gu ∈ Lp(X) is a minimal p-weak uppergradient of of a function u : X → R, if gu ≤ g almost everywhere for all p-weakupper gradients g ∈ Lp(X) of u.

Note that the minimal p-weak upper gradient is unique (up to a set of measurezero) and ‖gu‖Lp(X) ≤ ‖g‖Lp(X) for all p-weak upper gradients g ∈ Lp(X) of u.

Theorem 6.24. Let u : X → R and let 1 ≤ p <∞. If u has a p-weak upper gradientg ∈ Lp(X), then it has a minimal p-weak upper gradient gu ∈ Lp(X). Moreover,

(6.5) gu(x) = infρ

lim supr→0

∫B(x,r)

ρ dµ,

where the infimum is taken over (p-weak) upper gradients ρ ∈ Lp(X) of u.

10.3 =============================

One way to prove the existence of the minimal p-weak upper gradient is to useconvergence results for p-weak upper gradients. We will give a proof from [41, Thm5.3.23] which is based on the lattice property discussed in Remark 13. Note alsothat using the uniform convexity of Lp(X) for p > 1 and the standard result offunctional analysis: a closed, convex subset of a uniformly convex Banach spacecontains a unique, norm minimizing element one can show that for each functionu with a p-weak upper gradient in Lp(X), p > 1, there is a unique p-weak uppergradient that minimizes the norm. The proof below holds also for p = 1.

49

Proof. We begin by showing that u has a norm minimizing p-weak upper gradient.Let

U =g : g ∈ Lp(X), g is a p-weak upper gradient of u

,

and let

m = infg∈U‖g‖Lp(X).

Let (gi) ⊂ U be a sequence such that limi→∞ ‖gi‖Lp(X) = m. We may assume that thesequence (gi) is decreasing. (If it is not, define a new decreasing sequence by settinggi = min1≤j≤i gj. Then (gi) ⊂ U by the lattice property of U and limi→∞ ‖gi‖Lp(X) =m.)

Since the sequence (gi) is nonnegative and decreasing, it converges to a (Borel)function gu. By the Lebesgue monotone convergence theorem for the sequence (fi),fi = (gi − gu)p, ∫

X

|gi − gu|p dµ =

∫X

(gi − gu)p dµ→ 0 as i→∞

and hence gi → gu in Lp(X). Moreover, ‖gu‖Lp(X) = m and Fuglede’s lemma 6.10implies that gu is a p-weak upper gradient of u.

To show the pointwise minimizing property, assume that there is h ∈ U such thatµ(Eh) > 0 for the set

Eh = x ∈ X : gu(x) > h(x).By Lemma 6.16, the function

ρ = hχEh + guχX\Eh

belongs to U . Now ‖ρ‖Lp(X) < ‖gu‖Lp(X) = m, which is impossible by the definitionof m. Hence gu ≤ h almost everywhere for each h ∈ U .

Next we show that (6.5) holds. Let g ∈ U . By the Lebesgue differentiationtheorem 3.11 and the pointwise minimizing property of gu, we have that

gu(x) = limr→0

∫B(x,r)

gu dµ ≤ lim supr→0

∫B(x,r)

g dµ

for almost all x ∈ X. This proves the ”≤”-direction of (6.5).By Lemma 6.11, there is a nonnegative function ρ ∈ Lp(X) such that the function

gi = gu + 2−iρ

is an upper gradient of u and belongs to Lp(X) for each i ∈ N. Hence, for almostall x ∈ X, we have

infρ

lim supr→0

∫B(x,r)

ρ dµ ≤ lim supr→0

∫B(x,r)

gi dµ

≤ gu(x) + 2−i lim supr→0

∫B(x,r)

ρ dµ,

from which the ”≥”-direction of (6.5) follows by letting i→∞. 50

Example 6.25. Minimal p-weak upper gradient is a substitute for |∇u|. Namely,if Ω ⊂ Rn is open and u : Ω → R is locally Lipschitz, then gu = |∇u| almosteverywhere in Ω if it is p-integrable. It suffices to show the minimality because byExample 6.13, |∇u| is an upper gradient of u. By (6.5),

gu(x) = infρ

lim supr→0

∫B(x,r)

ρ dx,

where the infimum is taken over all upper gradients ρ ∈ Lp(Ω) of u. Since almostall points are Lebesgue points of |∇u|,

|∇u(x)| = limr→0

∫B(x,r)

|∇u(y)| dy

for almost every x ∈ Ω. Hence, it suffices to show that |∇u| ≤ g almost everywherein Ω whenever g ∈ Lp(Ω) is an upper gradient of u. For the rest of the proofthat uses the Rademacher theorem, the Fubini theorem and Lebesgue points, see [8,Prop. A.3]

Remark 14. If u, v : X → R are functions with minimal p-weak upper gradients guand gv, λ ∈ R, and f : R → R is L-Lipschitz, then, we have formulas for minimalp-weak upper gradients:

g−u = gu, gu+v ≤ gu + gv, gλu ≤ |λ|gu and gfu ≤ Lgu.

Moreover, using the lattice properties of p-weak upper gradients in Lp(X), we havethat g|u| = gu for each measurable function u : X → R, see [41, Chapter 5.3]

What happens to minimal p-weak upper gradients in subsets of X. If gu is theminimal p-weak upper gradient of u in X, can it happen that there is a functiong0, that is a p-weak upper gradient of u in Ω such that g0 < gu in a set of positivemeasure in Ω. For open sets that does not happen. The proof is from [8, Le 2.22].

Lemma 6.26. Let Ω ⊂ X be an open set. Let gu ∈ Lp(X) be the minimal p-weakupper gradient of u in X. Then gu|Ω is the minimal p-weak upper gradient of u inΩ.

Proof. Let g0 be the minimal p-weak upper gradient of u in Ω. Then g0 ≤ gu almosteverywhere in Ω.

Suppose that g0 < gu on a set of positive measure. Then there is a ball B such that2B ⊂ Ω and g0 < gu on a set of positive measure in B. The function g : X → [0,∞],

g(x) =

gu if x ∈ X \B,g0 if x ∈ B,

is a p-weak upper gradient of u in Ω by Lemma 6.17.Let Γ be the curve family for which the upper gradient inequality (6.1) does not

hold for u and g (containing subcurves) and let

Γ1 = γ ∈ Γ : γ ⊂ Ω and Γ2 = γ ∈ Γ : γ ⊂ X \B.Then Modp(Γ1) = 0, and since g is a p-weak upper gradient of u in X \ B, alsoModp(Γ2) = 0.

51

Let γ ∈ Γ. Since 2B ⊂ Ω and γ is rectifiable, we can split it into finite subcurvesγ1, γ2, . . . , γn such that each γi is contained in X \ B or in Ω (note that between Band 2B, the length of subcurve of γ is always at least the radius of B).

By the choice of Γ, there is γi such that the upper gradient inequality does nothold for u and g on γi. Hence γi ⊂ Γ1∪Γ2. By the basic properties of the p-modulus,Modp(Γ) ≤ Modp(Γ1 ∪ Γ2) = 0, which implies that g is a p-weak upper gradient ofu in X. This is not possible because g < gu on a set of positive measure.

6.5. Convergence properties of p-weak upper gradients. If we do not knowthat the space is reflexive, we cannot use weak compactness results. The Sobolevspace N1,p(X), defined in Section 6.6, p > 1, is reflexive if X is complete by a recentresult of Ambrosio, Colombo and Di Marino [4] or if X is a geodesic space andsupports a p-Poincare inequality, see [41, Thm 12.5.7.].

There are several convergence results for sequences of functions and correspondingp-weak upper gradients. We present two of them. Theorem 6.28 is an importanttool to show that weak convergence implies strong convergence. It was first provedby Kallunki and Shanmugalingam, [43, Lemma 3.1] for p > 1. The proof whichworks also for p = 1 is from [26, Lemma 7.8].

Lemma 6.27. Let ui : X → R, i ∈ N, be a measurable functions with correspondingp-weak upper gradients gi ∈ Lp(X). Let E ⊂ X be a p-exceptional set, that is,Modp(ΓE) = 0 for the curve family

ΓE =γ : γ rectifiable, γ ∩ E 6= ∅

.

If there exist a measurable function u : X → R and a Borel function g : X → [0,∞]such that ui(x)→ u(x) for all x ∈ X \E and gi → g in Lp(X) as i→∞, then g isa p-weak upper gradient of u.

Proof. By the assumptions, the basic properties of the p-modulus, and by Fuglede’slemma 6.10, p-almost every curve γ satisfies the following properties: upper gradientinequality (6.1) holds for ui and gi for all i ∈ N, γ does not intersect E, and∫

γ

gi ds→∫γ

g ds <∞ as i→∞.

For such a curve γ with endpoints x, y ∈ X we have that

|u(x)− u(y)| = limi→∞|ui(x)− ui(y)| ≤ lim

i→∞

∫γ

gi ds =

∫γ

g ds.

Hence g is a p-weak upper gradient of u.

Theorem 6.28. If (uj) ⊂ Lp(X) and (gj) ⊂ Lp(X), 1 ≤ p <∞, are a sequence offunctions and a corresponding sequence of p-weak upper gradients, such that uj → uand gj → g weakly in Lp(X), then g is a p-weak upper gradient of a representativeof u.

Remark 15 (Lower semicontinuity). The above theorem together with the lowersemicontinuity of norms imply that if (ui) is a bounded sequence in N1,p(X), 1 <

52

p < ∞ and ui → u weakly in Lp(X), then u has a representative in N1,p(X) suchthat

‖u‖N1,p(X) ≤ lim infi→∞

‖ui‖N1,p(X),

see also the discussion in [41, Chapter 6.3].

Proof. By Lemma 6.11, we may assume that the functions gj are upper gradients ofthe functions uj.

By the Mazur’s lemma [83, p.120] and the fact that each convex combination∑nj=1 αjgj is an upper gradient of the corresponding convex combination

∑nj=1 αjuj,

we may assume that uj → u and gj → g in Lp(X), see the details from [77, Thm4.17]. Moreover, by taking a subsequence of (uj) and using Fuglede’s lemma 6.10for (gj), we may assume that ui(x)→ u(x) as i→∞ for µ-almost every x ∈ X, and

(6.6)

∫γ

|gi − g| ds→ 0 as i→∞

for p-almost every curve γ. Denote the curve family where (6.6) fails by Γg. Let

Γ∞ =γ :

∫γ

g ds =∞ or

∫γ

gj ds =∞ for some j ∈ N.

Then Modp(Γ∞) = 0 by Lemma 6.7 and by the subadditivity of the p-modulus.Since the measure of the set

E =x ∈ X : lim

i→∞ui(x) 6= u(x) or |u(x)| =∞

is zero, Lemma 6.8 implies that Modp(ΓE) = 0 for the curve family

ΓE =γ : γ rectifiable, L1

(γ−1(E)

)> 0.

The subadditivity of the p-modulus shows that the Modp(Γ) = 0 also for the curvefamily

Γ = Γg ∪ Γ∞ ∪ ΓE.

Let then γ /∈ Γ be a rectifiable curve (parametriced by arc length). Since γ /∈ ΓE,γ(t) /∈ E for L1-almost every t ∈ [0, `(γ)] and hence

(6.7) uj(γ(t))→ u(γ(t)) <∞ as j →∞

for almost every t ∈ [0, `(γ)]. Moreover, since γ /∈ Γg, (6.6) shows that

(6.8)

∫γ

|gi − g| ds→ 0 as j →∞.

Claim: uj γj∈N is an equicontinuous family of functions on [0, `(γ)].To prove the claim, let ε > 0. Since gj is an upper gradient of uj, we have that

(6.9)∣∣uj(γ(t))− uj(γ(h))

∣∣ ≤ ∫ t

h

gj(γ(τ)) dτ

53

for all 0 ≤ h < t ≤ `(γ). It is enough to show that there is a δ > 0 such that ifA ⊂ [0, `(γ)] and L1(A) < δ, then

supj

∫A

gj(γ) ds < ε.

For that, let A ⊂ [0, `(γ)]. By the triangle inequality, we have that∫A

gj(γ) ds ≤∫A

g(γ) ds+

∫γ

|gj − g| ds,

and by (6.8), there is an integer j0 ∈ N such that

(6.10)

∫γ

|gj − g| ds <ε

2when j ≥ j0.

Since γ does not belong to Γ∞, (6.10) and the absolute continuity of the integralimply that there is a δ > 0 such that if L1(A) < δ, then∫

A

g(γ) ds <ε

2and

∫A

gj(γ) ds < ε for all j = 1, 2, . . . , j0 − 1.

Hence uj γj∈N is an equicontinuous family of functions and the claim follows.

We already noticed in (6.7) that the sequence (ui(γ)) converges pointwise on[0, `(γ)] almost everywhere. By equicontinuity, it converges uniformly on [0, `(γ)].Next we define a representative of u by setting

u(x) =

limj→∞ uj(x), if the limit exists,

0, otherwise.

Then, for each γ /∈ Γ with endpoints x and y, (uj) converges uniformly to u on γ.Letting h→ 0, t→ `(γ), and i→∞ in (6.9), we obtain

|u(x)− u(y)| ≤∫γ

g ds.

Therefore g is a p-weak upper gradient of u.

6.6. Definition and basic properties of N1,p(X). We define the Sobolev spacesN1,p(X) using a standard procedure where one first defines a seminormed space andthe normed space is then obtained by taking a quotient space.

Let N1,p(X), 1 ≤ p < ∞, be the collection of all p-integrable functions u havinga p-weak upper gradient g ∈ Lp(X). Then N1,p(X), equipped with a seminorm

(6.11) ‖u‖N1,p(X) = ‖u‖Lp(X) + ‖gu‖Lp(X),

where gu is the minimal p-weak upper gradient of u guaranteed by Theorem 6.24, isa vector space. Note that the equivalent seminorm

(6.12) ‖u‖1,p =(∫

X

|u|p dµ+

∫X

gpu dµ)1/p

is sometimes used. The latter seminorm is better when one wants to define Sobolevcapacity using the norm of the space because it guarantees that the capacity issubadditive. However, a triangle inequality for (6.12) requires little work, see [8,

54

Chapter 1]. Of course one can define Sobolev spaces using (6.11) and capacity using(6.12) as it is done in [41].

Definition 6.29. We define an equivalence relation in N1,p(X) by setting u ∼ v,if ‖u − v‖N1,p(X) = 0. The space N1,p(X) is formed from the equivalence classes

of N1,p(X), N1,p(X) = N1,p(X)/ ∼ and equipped with the norm ‖u‖N1,p(X) =‖u‖N1,p(X).

For simplicity, we will refer to the elements of N1,p(X) as functions instead ofequivalence classes. Sometimes spaces N1,p(X) are called Newtonian spaces. Thespaces N1,p(Ω) for open sets Ω ⊂ X are defined naturally by looking p-weak uppergradients in Ω of u ∈ Lp(Ω).

Note that in [8], N1,p(X) and N1,p(X) are reversed; there N1,p(X) refers to func-tions defined everywhere, and N1,p(X) is the space of equivalence classes. Moreover,they study the class

N1,p(X) = u : u = v almost everywhere for some v ∈ N1,p(X),

(with their definition of N1,p(X)).

Remark 16. Note that the properties proved earlier in this section for functionshaving p-weak upper gradients in Lp(X), hold for functions in N1,p(X).

If a function of a classical Sobolev space W 1,p(Rn), is modified in a set of measurezero, the new function also belongs to W 1,p(Rn). N1,p(X) functions do not have thisproperty. If µ(E) = 0 but the p-modulus of the collection of all rectifiable curvesthat intersect E is positive, then χE is not in N1,p(X). However, if two functionsthat differ in a set of measure zero both belong to N1,p(X), then they generate thesame equivalence class of N1,p(X).

Corollary 6.30. If u1, u2 ∈ N1,p(X) and u1 = u2 µ-almost everywhere in X, thenu1 ∼ u2, that is, u1 and u2 generate the same equivalence class of N1,p(X).

Proof. Since N1,p(X) is a vector space, the function u = u1−u2 belongs to N1,p(X).By Lemma 6.15, u ∈ ACCp(X). Moreover, as we also have that

µ(x ∈ X : u(x) 6= 0

)= 0,

the function g = 0 is a p-weak upper gradient of u by Exercise 4. Consequently,‖u1 − u2‖N1,p(X) = ‖u‖N1,p(X) = 0, and so u1 ∼ u2.

By the definition of N1,p(X) spaces and the properties of the p-modulus, we seethat the functions of N1,p(X) are well defined outside of p-exceptional set (the familyof all nonconstant curves that meet E is of p-modulus zero) of measure zero, see thediscussion in [41, Chapter 6.1]. Another way (and important) way to characterizethe equivalence classes and small sets for N1,p(X) is to use capacity, see [41, Chapter6.2] and [8]. This approach is important for example while studying the pointwiseproperties of Sobolev functions and in nonlinear potential theory. We will probablyhave no time for that in this course.

55

Corollary 6.31 (Prop 6.1.35 in [41]). Let u ∈ N1,p(X), let v : X → R be a mea-surable function, and let E = x ∈ X : u(x) = v(x). If the set E is p-exceptionaland µ(E) = 0, then v ∈ N1,p(X) and u and v belong to the same equivalence classin N1,p(X).

Conversely, if u, v ∈ N1,p(X) and if µ(E) = 0, then E is p-exceptional.

6.7. Nontriviality of spaces N1,p(X). By the definition, N1,p(X) ⊂ Lp(X) andthe corresponding embedding given by the identity map is continuous. If there areno nonconstant rectifiable curves in X, then the function gu = 0 is the minimalp-weak upper gradient of each function u and hence N1,p(X) = Lp(X), and we saythat N1,p(X) is trivial. This is the case for example when X is totally disconnectedor ”snowflaked” space. If the embedding is not surjective, we say that N1,p(X) isnontrivial.

Theorem 6.32. The space N1,p(X) is nontrivial if and only if the p-modulus of thecollection of all nonconstant curves in X is positive.

Proof. The necessity direction follows from the definitions. In the sufficiency direc-tion, one uses separability of X, properties of the p-modulus and absolute continuityto show that there is a ball whose characteristic function does not belong to N1,p(X).For details, see [41, Prop 6.1.38].

There are at least two ways to show that the space N1,p(X) is a Banach space. Oneof them uses capacity and the second one convergence properties proved in Theorem6.28. The main difficulty in showing that Cauchy sequences converge in N1,p(X)comes from the definition of N1,p(X) using p-weak upper gradients. Namely, if guand gv are p-weak upper gradients of functions u and v, the function |gu− gv| is notnecessarily a p-weak upper gradient of u − v. The proof is from Haj lasz, [26, Thm7.12].

Theorem 6.33. N1,p(X) is a Banach space.

Proof. Let (uj) be a Cauchy sequence in N1,p(X). We have to show that there is afunction u such that uj → u in N1,p(X). By the definition of the norm of N1,p(X),(uj) is a Cauchy sequence also in the Banach space Lp(X), and hence there is afunction u ∈ Lp(X) such that uj → u in Lp(X).

It suffices to show that (uj) contains a subsequence that converges to u in N1,p(X)because if a Cauchy sequence contains a convergent subsequence, then the originalCauchy sequence also converges.

Let (ujk) be a subsequence of (uj) such that for each k ∈ N,

(6.13) ‖ujk − ujk+1‖N1,p(X) < 2−k.

Next we use Lemma 6.11 to obtain ρk, an upper gradient of ujk − ujk+1, with

(6.14) ‖ρk‖Lp(X) < 2−k.

Then the function

(6.15) gk =∞∑i=k

ρi

56

is an upper gradient of ujk − ujk+l for each l ≥ 1. (For example, if k = 2 and l = 2,and γ is a rectifiable curve with endpoints x and y, then∣∣(uj2(x)− uj4(x)

)−(uj2(y)− uj4(y)

)∣∣ ≤ ∣∣(uj2(x)− uj3(x))−(uj2(y)− uj3(y)

)∣∣+∣∣(uj3(x)− uj4(x)

)−(uj3(y)− uj4(y)

)∣∣≤∫γ

ρ2 ds+

∫γ

ρ3 ds ≤∫γ

g2 ds.)

By the definition of gk and the selection of upper gradients ρk, we have that

‖gk‖Lp(X) < 21−k → 0 as k →∞.Note that

ujk − ujk+l → ujk − uin Lp(X) as l → ∞. By setting gkl = gk, an upper gradient of ujk − ujk+l from(6.15), we also have that gkl → gk in Lp(X) when l→∞. Theorem 6.28 shows thatgk is a p-weak upper gradient of a representative of ujk − u. Since functions ujk , u,and gk belong to Lp(X), the representative of ujk − u is in N1,p(X). As N1,p(X) isa norm space, u ∈ N1,p(X) also. The estimates above give

‖ujk − u‖N1,p(X) ≤ ‖ujk − u‖Lp(X) + ‖gk‖Lp(X) → 0 as k →∞,and hence ujk → u in N1,p(X).

11.3 =============================

The following lemmas correspond the density of smooth functions with boundedsupport and the convergence of truncations in W 1,p(Rn).

Lemma 6.34. Bounded functions with bounded support are dense in N1,p(X).

Proof. Let u ∈ N1,p(X) and let gu ∈ Lp(X) be the minimal p-weak upper gradientof u.

For the density of functions with bounded support, we may assume that X isunbounded. Let x0 ∈ X, Bj = B(x0, j), and let ϕj : X → [0, 1], j ∈ N,

ϕj(x) =

1, if x ∈ Bj,

j + 1− d(x0, x), if x ∈ Bj+1 \Bj,

0, if x ∈ X \Bj+1,

be 1-Lipschitz cut-off functions. We start by showing that the functions

uj = ϕju

with bounded supports belong to N1,p(X) and that uj → u in N1,p(X) as j →∞.The convergence uj → u in Lp(X) follows from the dominated convergence theo-

rem because uj → u as j →∞, |u− uj|p ≤ 2p|u|p and u ∈ Lp(X).Concerning p-weak upper gradients, by Lemmas 6.22 and 6.18, and the definition

of ϕj, the function ρj = (|u| + gu)χBj+1is a p-weak upper gradient of uj. This

together with the fact that u− uj = 0 in Bj, implies that

gj = (|u|+ 2gu)χX\Bj57

is a p-weak upper gradient of u− uj. Now∫X

gpj dµ =

∫X\Bj

(|u|+ 2gu)p dµ ≤ 2p+1

∫X\Bj

(|u|p + gpu

)dµ→ 0,

as j → ∞ because u, gu ∈ Lp(X). The density of functions with bounded supportfollows from this.

To show the density of bounded functions, we will show that the two sided trun-cations of u to level j ∈ N,

(6.16) uj = max

minu, j,−j

convergence to u in N1,p(X) as j →∞.As in the first claim, the convergence uj → u in Lp(X) follows from the dominated

convergence theorem because uj → u as j →∞, |u− uj|p ≤ 2p|u|p and u ∈ Lp(X).By the definition of uj and the basic properties of p-weak upper gradients, the

functionρj = gu + 3gu = 4gu,

a sum of p-weak upper gradients of u and uj, is a p-weak upper gradient of u− uj.Since uj = u in the set |u| ≤ j, using the Borel regularity of µ, Lemmas 6.18,6.12, and Exercise 4, we have that

gj = 4guχ|u|>j

is a p-weak upper gradient of u − uj. Since µ(|u| > j) ≤ j−p∫X|u|p dµ and

u ∈ Lp(X), we have that ∫X

gpj dµ =

∫|u|>j

(4gu)p dµ→ 0,

as j → ∞. Hence uj → u in N1,p(X) and we conclude that bounded functions aredense in N1,p(X).

As the last results of this section, we study the connections of N1,p(X), M1,p(X)and the classical Sobolev spaces W 1,p(Rn) in the Euclidean setting. We constructedthe space N1,p(X) by mimicking the characterization of the classical Sobolev spacevia functions that are absolutely continuous on p-almost every curves. Thus theequality between W 1,p(Ω) and N1,p(Ω) follows using Theorem 6.2 (for open sets)and Lemma 6.15. The equality below means that each function has a representativein the other space and that there is a linear isometry between the Banach spaces.See also [8, Appedix A] and [41, Chapter 6.4].

Theorem 6.35. Let Ω ⊂ Rn be an open set and let 1 ≤ p < ∞. Then W 1,p(Ω) =N1,p(Ω).

Combining Theorem 6.35 with the discussion in Remark 8, we see that if a domainΩ ⊂ Rn is ”nice enough” or Ω = Rn and 1 < p < ∞, then W 1,p(Ω) = N1,p(Ω) =M1,p(Ω).

Theorem 6.36. M1,p(X) ⊂ N1,p(X) for all 1 ≤ p < ∞ and the embedding iscontinuous.

58

One possible way to prove Theorem 6.36 is to use density of Lipschitz functionsin M1,p(X) (Theorem 5.8) and then show, using similar methods as in the proof ofTheorem 5.3, that 3g is a weak upper gradient whenever u ∈M1,p(X) is a continuousfunction with a generalized gradient g ∈ Lp(X). We will give the proof from [26,Thm 8.6].

Proof. Let u ∈ M1,p(X) and g ∈ D(u) ∩ Lp(X). We will show that u has a repre-sentative that belongs to N1,p(X) and has 2g as a p-weak upper gradient.

By Theorem 5.8, there is a sequence (ui) of Lipschitz functions such that ui → u inM1,p(X). Let hi ∈ D(u−ui) such that ‖hi‖Lp(X) → 0 as i→∞. As ui = u−(u−ui),the function gi = g+hi is a generalized gradient of ui and gi → g in Lp(X) as i→∞.

If necessary, we modify each gi in a set of measure zero such that gi is a Borelfunction and that

|ui(x)− ui(y)| ≤ d(x, y)(gi(x) + gi(y)

)for all x, y ∈ X.

Let γ : [0, `(γ)] → X be a rectifiable curve such that∫γgi ds < ∞ for all i ∈ N.

By Lemma 6.7, this is true for p-almost every curve. An application of the Lusintheorem shows that there is a set E ⊂ [0, `(γ)] such that L1([0, `(γ)] \ E) = 0 andthat for each t ∈ E, there is a sequence (hn), hn → 0, for which

gi(γ(t+ hn))→ gi(γ(t)) as n→∞.As a Lipschitz continuous function, ui γ is differentiable at almost every pointt ∈ [0, `(γ)]. At such points t ∈ E, we have, using the fact that gi ∈ D(ui) and the1-Lipschitz continuity of γ, that

|(ui γ)′(t)| =∣∣∣∣ limn→∞

ui(γ(t+ hn))− ui(γ(t))

hn

∣∣∣∣≤ lim sup

n→∞

∣∣∣∣d(γ(t+ hn), γ(t))

hn

∣∣∣∣(gi(γ(t+ hn)) + gi(γ(t)))

≤ 2gi(γ(t)).

Hence

|ui(γ(`(γ)))− ui(γ(0))| =∣∣∣∣ ∫ `(γ)

0

(ui γ)′(t) dt

∣∣∣∣ ≤ ∫ `(γ)

0

2gi(γ(t)) dt,

which implies that 2gi is a p-weak upper gradient of ui. The claim follows now fromthe convergence theorem 6.28.

17.3 =============================

7. About Poincare inequalities

In this section, we discuss Poincare inequalities in metric measure spaces. APoincare inequality connects the metric, the measure and the gradient and it canbe used to transfer information from small scales to larger scales. As we will see,the validity of a Poincare inequality in the metric measure space guarantees, forexample, that X has plenty of rectifiable curves.

59

The classical Poincare inequality in the Euclidean setting is an important tool forexample in harmonic analysis, in calculus of variations and in the theory of partialdifferential equations. A simplest version of the Poincare inequality says that foreach ball B = B(x, r) ⊂ Rn and for each u ∈ W 1,1(B),

(7.1)

∫B

|u− uB| dx ≤ Cr

∫B

|∇u| dx,

where the constant C > 0 depends only on n. The stronger version,

(7.2)(∫

B

|u− uB|p∗dx)1/p∗

≤ Cr(∫

B

|∇u|p dx)1/p

,

which holds for each u ∈ W 1,p(B), 1 ≤ p < n, p∗ = np/(n − p), is a kind oflocal version of the Sobolev embedding theorem. It is often called Sobolev-Poincareinequality. Note that the embedding ‖u‖Lp∗ (B) ≤ C‖|∇u|‖Lp(B) cannot hold for allSobolev functions in the ball; if u is constant in B, then ∇u = 0, but (7.2) holds.

Definition 7.1. We say that X supports a p-Poincare inequality, 1 ≤ p < ∞, ifthere exists constants cP > 0 and τ ≥ 1 such that

(7.3)

∫B

|u− uB| dµ ≤ cP diam(B)(∫

τB

gp dµ)1/p

for every ball B ⊂ X, for each u ∈ L1loc(X) and for every upper gradient g of u.

We begin with some remarks about the p-Poincare inequality.

(1) Poincare inequality can be defined also for 0 < p < 1, see [30] and thereferences therein.

(2) Sometimes inequality (7.3) with τ > 1 is a called a weak p-Poincare in-equality to emphasize the difference to the Euclidean setting. If the spaceX supporting p-Poincare inequality is nice enough (geodesic suffices), theninequality (7.3) holds with τ = 1 possibly with a constant bigger than cP ,see [41, Thm 8.1.15], [29], [30].

(3) By the Holder inequality, the 1-Poincare inequality is the strongest one. IfX supports p-Poincare inequality, then it supports q-Poincare inequality foreach q ≥ p.

(4) There are several versions of the Poincare inequality in the literature. Boththe class of functions for which the inequality is required to hold and the sub-stitute for the gradient vary. If p ≥ 1 and X is complete, then for examplethe validity of inequality (7.3) for Lipschitz functions with pointwise Lips-chitz constants is equivalent to our definition, see [44, Thm 2], [8, Chapter4].

(5) It follows from the definition of the p-Poincare inequality and Lemma 6.11that if u ∈ L1

loc(X) has a p-integrable p-weak upper gradient and X supportsa p-Poincare inequality, then (7.3) holds for u and the minimal p-weak up-per gradient gu. Moreover, using the two sided truncations (6.16) and theLebesgue monotone convergence theorem, it is easy to see that if X supportsa p-Poincare inequality, then each measurable function u : X → R having alocally p-integrable upper gradient is locally integrable, see [41, Chapter 7.1].

60

The validity of the p-Poincare inequality tells us something about the geometryof the space X.

Theorem 7.2. If X supports a p-Poincare inequality, then it is connected.

Proof. Suppose that there are nonempty, disjoint open sets U, V such that X =U ∪ V . Then there are no curves connecting U and V and hence g = 0 is an uppergradient of the function χU .

If B is a ball centered in U such that B∩V 6= ∅, then by the p-Poincare inequality,

0 <

∫B

|u− uB| dµ ≤ cP diam(B)(∫

τB

gp dµ)1/p

= 0,

which is a contradiction. Hence X is connected.

Connectedness together with the reverse doubling lemma 3.5 gives the followingcorollary.

Corollary 7.3. Let x ∈ X and let 0 < r < 12

diamX. If X supports a p-Poincareinequality, then µ(x) = 0 and ∂B(x, r) 6= ∅.

If X is complete, (which is equivalent to being proper by Lemma 3.3), then ap-Poincare inequality ensures the existence of short curves in X. Recall that X isquasiconvex if there is a constant C ≥ 1 such that each pair of points x, y ∈ X canbe joined by a rectifiable curve γxy for which

(7.4) `(γxy) ≤ Cd(x, y).

Theorem 7.4. If X is complete and supports a p-Poincare inequality, then it isquasiconvex. The constant C in (7.4) depends on cd and on the constant of thep-Poincare inequality.

Proof. Maybe later...

Actually, even a stronger connectivity property called annular quasiconvexityholds if the measure is Q-regular, see [52, Thm 3.3], [41, Thm 8.3.8]. For Poincareinequalities and Loewner type connectivity properties, see [40], [41, Chapter 13.2.18].

Exercise 5. When using Poincare inequalities, we integrate the power of the dif-ference of a function and its integral average over a ball. In many cases, it is notessential, over which subset the integral average is taken:

Show that if A ⊂ Ω ⊂ X with µ(A) > 0 and µ(Ω) < ∞, 1 ≤ p < ∞, c ≥ 0 andu ∈ L1(Ω), then ∫

Ω

|u− uA|p dµ ≤ 2pµ(Ω)

µ(A)

∫Ω

|u− uΩ|p dµ,

and that the mean oscillation is ”the smallest possible”,∫Ω

|u− uΩ|p dµ ≤ 2p∫

Ω

|u− c|p dµ.

61

7.1. Bi-Lipschitz invariance of the Poincare inequality. A p-Poincare inequal-ity is quantitatively invariant under bi-Lipschitz mappings. This is a useful propertyin many proofs. Usually we endow a complete space X with the length metric

ρ(x, y) = inf `(γ),

where the infimum is taken over all rectifiable curves connecting x and y. Then thespace (X, ρ, µ) is bi-Lipschitz homeomorphic to X, geodesic (see Sebastiano’s talk)space that supports a p-Poincare inequality. See also [41, Chapter 7.3], [37, Chapter9] and note that also spaces N1,p(X) are invariant under bi-Lipschitz mappings(check it as an exercise). The following proof is from [8, Prop 4.11].

Theorem 7.5. Let (X, d, µ) and (Y, d′, ν) be two metric measure spaces and let µbe doubling. Let f : X → Y be a bi-Lipschitz mapping with constant L such that

M−1µ(E) ≤ ν(f(E)) ≤Mµ(E)

for some constant M > 0 for all measurable sets E ⊂ X. If X supports a p-Poincareinequality with constants cP and τ , then Y supports supports a p-Poincare inequalitywith constants c and L2τ , where c depends only on cP , L, M , and cd.

Proof. Let B′ = B(y, r) ⊂ Y be a ball, v ∈ L1loc(Y ) and let h be an upper gradient

of v. In the proof we will use the p-Poincare inequality in X. For that, let

x = f−1(y), u = v f and B = B(x, Lr).

The bi-Lipschitz property implies that the function g = L(hf) is an upper gradientof u. Now, using the p-Poincare inequality in X, the assumptions on f and thecomparability of the measures, we obtain∫

B′|v − vB′ | dν ≤ 2

∫B′|v − uB| dν ≤

2

ν(B′)

∫f−1(B′)

|u− uB| d(ν f)

≤ M2

µ(f−1(B′))

∫f−1(B′)

|u− uB| dµ

≤ c

∫B

|u− uB| dµ ≤ c diam(B)(∫

τB

gp dµ)1/p

≤ c diam(B′)(∫

f(τB)

hp dν)1/p

≤ c diam(B′)(∫

B(y,L2τr)

hp dν)1/p

,

which proves the claim.

The validity of p-Poincare inequality (7.3) for a locally integrable function u andfor a measurable function g ≥ 0 is connected to a pointwise estimate for u and therestricted maximal function of g. The first versions of the pointwise characterizationare from [30, Thm 3.2, 3.3.] and [40, Lemma 5.15]. Recall that we already discussedsuch inequalities while studying spacesM1,p(X) and their connections to the classicalSobolev spaces.

62

Theorem 7.6. If the pair u ∈ L1loc(X) and a measurable function g ≥ 0 satisfies

inequality (7.3) with 0 < p <∞, then there exists a constant C = C(cd, cP , p) suchthat

|u(x)− u(y)| ≤ Cd(x, y)((M2τd(x,y)g

p(x))1/p

+(M2τd(x,y)g

p(y))1/p)

for almost all x, y ∈ X.Conversely, if 1 ≤ p < ∞, τ ≥ 1, and the pair u ∈ L1

loc(X), g ∈ Lploc(X), g ≥ 0,satisfies inequality

(7.5) |u(x)− u(y)| ≤ Cd(x, y)((Mτd(x,y)g

p(x))1/p

+(Mτd(x,y)g

p(y))1/p)

for almost all x, y ∈ X, then there is a constant CP = CP (C, p, cd) such that Poincareinequality

(7.6)

∫B

|u− uB| dµ ≤ CP diam(B)(∫

3τB

gp dµ)1/p

holds in all balls B.

18.3 =============================

Proof. The first claim is proved using a telescoping argument and it is an easymodification of the proof of Theorem 5.2.

For the second claim, we prove only the case p > 1. Let B = B(x0, r) be a ball.For almost all x, y ∈ B, we have

|u(x)− u(y)| ≤ Cd(x, y)((M(gpχ3τB)(x)

)1/p+(M(gpχ3τB)(y)

)1/p)

because B(x, s) ⊂ 3τB whenever s ≤ τd(x, y) (and similarly for B(y, s)).Now, using the Cavalieri principle (3.12) and the weak type inequality (3.9) for

the Hardy-Littlewood maximal function, we have that∫B

|u(y)− uB| dµ(y) ≤∫B

∫B

|u(y)− u(x)| dµ(x) dµ(y)

≤ C

∫B

∫B

d(x, y)((M(gpχ3τB)(x)

)1/p

+(M(gpχ3τB)(y)

)1/p)dµ(x) dµ(y)

≤ Cr

∫B

(M(gpχ3τB)

)1/pdµ

= Crµ(B)−1

∫ ∞0

µ(M(gpχ3τB) > tp

)dt

≤ Crµ(B)−1

(∫ t0

0

µ(B) dt+ C

∫ ∞t0

t−p∫

3τB

gp dµ dt

)= Crµ(B)−1

(t0µ(B) + Ct1−p0

∫3τB

gp dµ)

for each t0 ∈ (0,∞). The claim follows by selecting t0 =(µ(B)−1

∫3τB

gp dµ)1/p

andusing the doubling property of µ.

63

The technique used in the previous proof, where an integral over an interval, givenfor example by the Cavalieri principle, is dived to two integrals and the division pointis selected at the end of the proof so that the two integrals are comparable, is usefulin many proofs.

Remark 17. Using Sobolev-Poincare embedding, Theorem 5.10, for spaces M1,p,one can show that the latter claim of Theorem 7.6 holds actually for all p > s/(s+1),see [26, Thm 9.5].

Theorem 7.6 and the validity of a Poincare inequality for functions in M1,p(X)(Theorem 5.7) provides us a characterization of M1,p(X) via Poincare inequalities.

Corollary 7.7. Let s/(s + 1) < p < ∞ and let u ∈ Lp(X). Then u ∈ M1,p(X) ifand only if there exist constants C > 0, τ ≥ 1, a nonnegative function g ∈ Lp(X)and 0 < q < p such that

(7.7)

∫B

|u− uB| dµ ≤ C diam(B)(∫

τB

gq dµ)1/q

.

for all balls B.

Proof. Assume first that u ∈ M1,p(X). If p > 1, then inequality (7.7) followsfrom Theorem 5.7 and from the Holder inequality. If s/(s + 1) < p ≤ 1, thenu ∈ M1,s/(s+1)(2B), and Theorem 5.10 together with the fact that (s/(s + 1))∗ = 1shows that ∫

B

|u− uB| dµ ≤ C diam(B)(∫

τB

gs/(s+1) dµ)(s+1)/s

,

which shows that (7.7) holds with q = s/(s+ 1).If inequality (7.7) holds for the pair u, g, then Theorem 7.6 implies that

|u(x)− u(y)| ≤ Cd(x, y)((Mgq(x)

)1/q+(Mgq(y)

)1/q).

Since g ∈ Lp(X), we have that gq ∈ Lp/q(X). As p/q > 1, the maximal functiontheorem 3.9 implies that Mgq ∈ Lp/q(X). Hence u ∈ M1,p(X) with (Mgq)1/q ∈D(u) ∩ Lp(X).

7.2. p-Poncare inequality - improvement and dependence of p. In this sub-section, we will discuss improvement properties of the p-Poincare inequality. Wecan ask for example the following questions for a space that supports a p-Poincareinequality:

(1) Does a version of Sobolev-Poincare inequality (7.2) hold in the metric set-ting?

(2) What is the dependence on p? We already noticed that a p-Poincare inequal-ity implies a q-Poincare inequality for all q ≥ p by the Holder inequality. Howabout the other direction: does there exists any q < p such that X supportsa q-Poincare inequality?

We begin with the first question, which was originally studied by Haj lasz and Koskelain [29], [30]. We will show that p-Poincare inequality for a pair of functions u, g

64

implies a seemingly stronger inequality with p. integral on the left side of (7.3).After that, we will formulate a stronger result, a counterpart of (7.2).

Theorem 7.8. If 1 ≤ p < ∞ and a pair u ∈ Lploc(X), g ≥ 0, g ∈ Lploc(X) satisfiesp-Poincare inequality (7.3), then there exists a constant C = C(cP , cd, p, τ) such that

(7.8)(∫

B

|u− uB|p dµ)1/p

≤ C diam(B)(∫

4τB

gp dµ)1/p

for all balls B.

Proof. Let B = B(x0, r) be a ball in X. We may assume that uB = 0. (If not, definea function v = u− uB, for which vB = uB − uB = 0 and v − vB = u− uB.)

The idea of the proof is first to apply a chaining argument to find an upper boundfor the measure of the level sets

At = x ∈ B : |u(x)| > t,and then to use the Cavalieri principle and this weak type estimate to find a suitableupper bound for

∫B|u|p dµ. To estimate µ(At), we first use a telescoping argument

at Lebesgue points, then make a useful trick for sums and finally use a coveringargument for good radii given by the previous step.

By the Lebesgue differentiation theorem 3.11, almost every point is a Lebesguepoint of u. Let x ∈ At be one of those.

Let B0 = B and for each i ∈ N, let Bi = B(x, 2−ir). Then Bi ⊂ 2B and Bi = 2Bi

for all i ∈ N and B0 ⊂ 2B1.Since uBi → u(x) as i → ∞ and uB0 = 0, the triangle inequality, the doubling

property of µ, and a p-Poincare inequality for u and g give

t ≤ |u(x)− uB0| ≤∞∑i=0

|uBi − uBi+1| ≤∫B0

|u− uB1| dµ+∞∑i=1

∫Bi+1

|u− uBi | dµ

≤ C

∫2B1

|u− u2B1 | dµ+ C

∞∑i=1

∫Bi

|u− uBi | dµ

≤ C

∞∑i=1

∫2Bi

|u− u2Bi | dµ ≤ C

∞∑i=1

ri

(∫2τBi

gp dµ)1/p

,

where ri = 2−ir.24.3 =============================

Let 1/2 < ε < 1. Then∞∑i=1

ri

(∫2τBi

gp dµ)1/p

≥ Ct = Ct(2ε − 1)∞∑i=1

2−iε ≥ Ctr−ε∞∑i=1

rεi .

Now for some index i0, the ith0 term on the left sum is as least as large as thecorresponding term on the right. Hence for each x ∈ At, there exists 0 < rx ≤ rsuch that

(7.9) rpε∫B(x,2τrx)

gp dµ ≥ Ctprp(ε−1)x µ(B(x, 2τrx)).

65

By the measure estimate (3.4), implied by the doubling property of µ, we have thatµ(B(x, rx)) ≥ C(rx/r)

δµ(B) for such an x and rx, for all δ ≥ log2 cd. Applying theestimate

rx ≤ Cµ(B(x, rx))1/δµ(B)−1/δr,

the fact that p(ε− 1) < 0, and the doubling property of µ to (7.9), we obtain

rpε∫B(x,2τrx)

gp dµ ≥ Ctpµ(B(x, rx))1+p(ε−1)/δµ(B)−p(ε−1)/δrp(ε−1),

and hence

(7.10) µ(B(x, rx))1+p(ε−1)/δ ≤ Ct−pµ(B)p(ε−1)/δrp

∫B(x,2τrx)

gp dµ.

Using the 5R-covering Lemma 3.7, we obtain a collection of balls Bxi = B(xi, rxi),i ∈ N, such that (7.10) holds for each Bxi , balls 2τBxi are disjoint, and At ⊂∪i∈N10τBxi . As we only required δ ≥ log2 cd, we can assume that δ > p. Then0 < 1 + p(ε− 1)/δ < 1, and we can move it as an exponent inside the sum. Since µis doubling, (7.10) holds and 2τBxi ⊂ 4τB for all i, we have

µ(At)1+p(ε−1)/δ ≤ µ

( ∞⋃i=1

10τBxi

)1+p(ε−1)/δ

≤ C∞∑i=1

µ(Bxi)1+p(ε−1)/δ

≤ Ct−pL∞∑i=1

∫2τBxi

gp dµ ≤ Ct−pL

∫4τB

gp dµ,

where L = rpµ(B)p(ε−1)/δ.Let q = δ/(δ + p(ε− 1)). Then q > 1 and

(7.11) µ(At) ≤ Ct−pqLq(∫

4τB

gp dµ)q.

Next we do the same trick as in the proof of Theorem 7.6. Now we use theCavalieri principle (3.12) to estimate

∫B|u|p dµ. For that, let 0 < a < ∞. In the

second row, we use two different upper bounds for µ(At). In the first term, we takethe trivial upper bound µ(B) and in the second, term we use (7.11),∫

B

|u|p dµ = p

∫ ∞0

tp−1µ(At) dt

= p

∫ a

0

tp−1µ(At) dt+ p

∫ ∞a

tp−1µ(At) dt

≤ pµ(B)

∫ a

0

tp−1 dt+ pCLq(∫

4τB

gp dµ)q ∫ ∞

a

tp−pq−1 dt

= α + β.

By calculating the corresponding integrals, we obtain

α = µ(B)ap and β = CLq(∫

4τB

gp dµ)q 1

q − 1ap(1−q),

66

and then we choose a such that α = β. This holds for

a = Cµ(B)−1/(pq)+(ε−1)/δr(∫

4τB

gp dµ)1/p

.

Finally, as ε−1δ− 1

pq= −1

p, we have∫

B

|u|p dµ ≤ Cµ(B)1+p·(−1/p)rp∫

4τB

gp dµ = Crp∫

4τB

gp dµ,

from which the theorem follows using the doubling property of µ.

The above results can be improved if the space satisfies a suitable chaining condi-tion, for example when X is geodesic. By considering small and large balls the chainseparately and using maximal function and weak type arguments, one obtains, aftera long calculation, the following Sobolev-Poincare embedding theorem in the metricsetting, see [41, Chapter 8.1]. The first version of this result in [30] was provedusing a discrete version of the Riesz potential. Below, s = log2 cd is the doublingdimension of X (or any exponent that is at least 1 and satisfies inequality (3.4)).

Theorem 7.9. Let X be a geodesic space that supports a p-Poincare inequality,1 ≤ p < ∞. There are constants c, C > 0 such that for each ball B, each functionu ∈ L1(B) and for each upper gradient g ∈ Lp(B),

(1) if p < s, then, with p∗ = ps/(s− p),(∫B

|u− uB|p∗dµ)1/p∗

≤ C diam(B)(∫

B

gp dµ)1/p

.

(2) if p = s, then∫B

exp

((|u− uB|

c diam(B)(∫

Bgs dµ

)1/s

)s/(s−1))dµ ≤ C.

(3) if p > s, then

‖u− uB‖L∞(B) ≤ C diam(B)(∫

B

gp dµ)1/p

.

Note that if X is complete, we can use these inequalities via bi-Lipschitz homeo-morphic, geodesic space as discussed earlier in this section. A version of Theorem7.9 holds without the assumption that X is geodesic, in that case the balls on theright sides of the inequalities are larger, see [30], [41, Corollary 8.1.36].

Concerning the second question of this subsection, the dependence of p-Poincareinequalities on p, Keith and Zhong proved in [45] that in complete spaces, thevalidity of a Poincare inequality is a self-improving property. Note that such a self-improvement property holds also for some other scale invariant conditions; the mostfamous (at least in Finland and for people working with PDEs), is the Gehringlemma [22], which says that a reverse Holder inequality for some exponent impliesa similar inequality for some larger exponent. As a consequence of the Gehringlemma, we that the Ap-condition for weights is an open-ended condition.

67

Theorem 7.10. Let 1 < p < ∞ and let X be a complete space (with a doublingmeasure) that satisfies a p-Poincare inequality. Then there is ε > 0 such that Xsupports a q-Poincare inequality for all q > p − ε. The constants depend on thedoubling constant of µ and on the constants of the p-Poincare inequality.

Remark 18. In [45], the p-Poincare inequality was defined for Lipschitz functions u

and their pointwise Lipschitz constants lim supy→x|u(x)−u(y)|d(x,y)

. As we discussed earlier,

the different definitions are equivalent in complete spaces with a doubling measure.For our definition of the p-Poincare inequality, the claim is not true without thecompleteness of the space, a counterexample is given in [54]. For the definition us-ing Lipschitz functions the claim holds without the assumption that X is completebecause this definition is preserved under taking the completion of the metric mea-sure space. Moreover, one can remove any null set with dense complement withoutaffecting the Poincare inequality. The main tools used in the long and technicalproof are estimates (and self-improvement) for the level sets of maximal functions,extensions of Lipschitz functions, Whitney type covering theorem and the Cavalieriprinciple. In addition to the original paper, see also [41, Chapter 11].

Now, using Theorems 6.36 and 7.6 and Corollory 7.7 and the self-improving prop-erty of the p-Poincare inequality, we obtain a result to the question: When thespaces M1,p(X) and N1,p(X) are the same?

Corollary 7.11. Assume that X supports a q-Poincare inequality for 1 ≤ q < p.Then M1,p(X) = N1,p(X). If X is complete, then it suffices to assume a p-Poincareinequality.

The validity of a p-Poincare inequality implies that Lipschitz functions are densein N1,p(X). In a complete space this follows from Theorem 5.8 and Corollary 7.11,the general case in proved for example in [41, Thm 7.2.1].

Theorem 7.12. Let 1 ≤ p < ∞. If X supports a p-Poincare inequality, thenLipschitz functions are dense in N1,p(X).

Some words about the proof: By Lemma 6.34, we may assume that u ∈ N1,p(X)vanishes outside a ball B0. By Theorem 7.6, u and its minimal p-weak upper gradientgu satisfy inequality

|u(x)− u(y)| ≤ Cd(x, y)((Mgpu(x)

)1/p+(Mgpu(y)

)1/p)

for all x, y ∈ X \ E, where µ(E) = 0, and hence u is Ct-Lipschitz in X \ (E ∪ Et),where

Et =x ∈ X : Mgpu(x) > tp

.

By the McShane extension Theorem 2.5, there is ut, a Ct-Lipschitz extension of u.The remaining steps are to show that Et ⊂ 2B0 when t is big enough, that ut → u

in Lp(X) as t→∞ and to find a p-weak upper gradient gut−u for ut − u such thatgut−u → 0 in Lp(X) as t→∞.

Remark 19. A recent research concerning ∞-Poincare inequality, where the rightside of (7.3) is replaced with ‖g‖L∞(|tauB) and geometric properties connected toPoincare inequalities, see [16]-[19] and the references therein.

68

Remark 20. Long lists of examples of spaces with doubling measure supporting ap-Poincare can be found for example in [41, Chapter 13], [8], [40] and [5].

Remark 21. For other definitions of Sobolev spaces in the metric setting, for ex-ample Cheeger’s definition using relaxation, Korevaar–Schoen spaces and the spacesconsisting of pairs of functions supporting a Poincare inequality and the correspond-ing references, see [41, Chapter 9]. Note that Cheeger showed in [13] that a spacesupporting a p-Poincare inequality admits a differentiable structure for which theRademacher theorem holds.

8. Removability for Sobolev spaces and Poincare inequality

In this section, we first shortly discuss the classical results about removable setsfor Sobolev spaces and then show that the validity of a p-Poincare inequality isconnected to the removability question for Sobolev functions. Koskela’s result in [54,Thm C] says that given a compact set E ⊂ Rn of measure zero, equip X = Rn \ Ewith the restrictions of the Euclidean distance and the Lebesgue measure, then Xsupports a p-Poincare inequality if and only if E is removable for the Sobolev spaceW 1,p. A generalization of this result holds also in the metric setting for spaces N1,p,see [57], [77].

Definition 8.1. A compact set E ⊂ Rn with Ln(E) = 0 is removable for W 1,p, ifW 1,p(Rn) = W 1,p(Rn \ E) as sets.

As the measure of E is zero, and Sobolev functions are defined only almost every-where, each function u ∈ W 1,p(Rn \ E) as well as the weak partial derivatives ∂ju,j = 1, . . . , n, of u in Rn \ E, belong to Lp(Rn). Hence we have to check that theintegration by parts formula

(8.1)

∫Rnu∂jϕdx = −

∫Rnϕ∂ju dx

holds for all test functions ϕ ∈ C∞0 (Rn) and for all j = 1, . . . , n. Thus it is enoughto show that each function u ∈ W 1,p(Rn \ E) belongs to W 1,1(Rn).

Using for example the ACL-characterization of Sobolev spaces on an open sub-set of Rn, [84, Thm 2.1.4], we see that removability is a local question: a set Eis removable for W 1,p if and only if for each x ∈ E there is r > 0 such thatW 1,p(B(x, r)\E) = W 1,p(B(x, r)) as sets. If E ⊂ Ω for some open set Ω ⊂ Rn, thenE is removable for W 1,p, if and only if E is removable for W 1,p(Ω).

Moreover, as smooth and bounded functions are dense in W 1,p(Ω\E) and W 1,p(Ω)is a Banach space, it suffices consider bounded or smooth Sobolev functions, thatis, to show that (8.1) holds for functions u ∈ C∞(Ω \E)∩W 1,p(Ω \E) and for eachϕ ∈ C∞0 (Ω). (Check the details as an exercise.)

8.1. Classical results. The removability question in Rn is relatively well under-stood. For example, using the small results above, the Fubini theorem and in-tegration by parts for smooth functions, it is easy see that E is removable forW 1,p for all p ≥ 1, if the projections of E along the coordinate axes have (n − 1)-Hausdorff measure zero. As the orthogonal projections along the coordinate axes

69

are 1-Lipschitz mappings, they can not increase the Hausdorff measure. Hence setswith Hn−1(E) = 0 are removable for all p ≥ 1.

The removability result connected to projections can be used to show that thereexist removable sets of Hausdorff dimension n. Let F ⊂ R be a set with dimH(F ) = 1and H1(F ) = 0. If we define E = F × · · · × F , where the Cartesian product of Fis taken n times, then E is removable for W 1,p, and by properties of product sets,[65, Thm 8.10], n = n dimH(F ) ≤ dimH(E) ≤ n. This fact dates at least back tothe work of Ahlfors and Beurling [3]. On the other hand, there are non-removableCantor sets of Hausdorff dimension n− 1. For more information about the classicalresults, see [54] and the references in that paper.25.3 =============================

8.2. How does the removability property of a fixed set depend on p? Re-movability for p = 1 is the strongest one; the fact that the removability is a localproperty, implies that if a set E is removable for W 1,p, then it is removable for W 1,q

for all q > p.Using the fact that it suffices to consider smooth functions, the Holder continuity

of Sobolev functions when p > n, the Fubini theorem, and integration by parts, onecan rather easily show that any compact set E ⊂ Rn−1 without interior is removablefor W 1,p for all p > n. Hence the case p ≤ n is more interesting.

By [54, Thm A], the condition q > p above is sharp, for each 1 < p ≤ n thereexists a compact set E ⊂ Rn−1 such that E is removable for W 1,p but not for anyq < p. That set is p-porous with a certain definition and there are no “big holescompared with q”. The same theorem says that each p-porous set E ⊂ Rn−1 isremovable for W 1,p in Rn. This theorem together with Theorem 8.2 gives [54, CorD]: for each 1 < p ≤ n, there exists a locally compact n-regular space that supportsa p-Poincare inequality but does not support any q-Poincare inequality, 1 < q < p.

Theorem 8.2 ([54], Thm C). Let E ⊂ Rn be a compact set with |E| = 0. Then E isremovable for W 1,p, p > 1, if and only if X = Rn\E, equipped with the restriction ofthe Euclidean distance and the Lebesgue measure, supports a p-Poincare inequality.

The proof below uses discrete convolution and partition of unity, familiar for usfrom the beginig of the course.

Proof. Assume first that E is removable for W 1,p. Let u ∈ L1loc(X) and let g be an

upper gradient of u. Let B = B(x, r) be a ball. We may assume that g ∈ Lp(B),since otherwise there is nothing to show. Lemma 6.15 shows that u ∈ ACCp(X) andusing the results of Section 6, we see that u belongs to W 1,p(B \ E), which equalsW 1,p(B) by the removability of E. Moreover, |∇u| ≤ g almost everywhere in B.Now a p-Poincare inequality follows from the usual Poincare inequality (7.1) in Rn

using the Holder inequality.To see the converse, assume that X supports a p-Poincare inequality. Since re-

movability is a local property, it suffices to show that W 1,p(B) = W 1,p(B \ E)as sets for each ball B. Moreover, it suffices to consider bounded functions u ∈W 1,p(Rn \ E) ∩ C∞(Rn \ E). Let u be such a function.

70

Let B = B(x0, r), x0 ∈ E. First we define approximation functions uj to u usingdiscrete convolution.

Let j ∈ N. By the 5R-covering lemma 3.7, there are disjoint balls Bi = B(xi, 2−j),

i ∈ N, such that xi ∈ X and Rn = ∪∞i=15Bi. As in Sections 2.3 and 4.3, correspondingto this covering, we take a partition of unity ϕii∈N consisting of smooth functionsϕi : Rn → [0, 1] for which

∑i ϕi(x) = 1 for all x ∈ Rn, ϕi = 0 in Rn \ 10Bi, and

|∇ϕi| ≤ C2j.A discrete convolution uj : Rn → R of u,

uj(x) =∑i

u10Biϕi(x),

is a finite sum at every point x ∈ Rn, actually we sum over i for which x ∈ 10Bi.

Claim: uj is a bounded sequence in W 1,p(B).We begin with estimates for the gradient of uj. Let k ∈ N. Using the properties

of functions ϕi, we have

(8.2) |∇uj(x)| =∣∣∇∑

i

(u10Bi − u10Bk)ϕi(x)∣∣ =

∣∣∑i

(u10Bi − u10Bk)∇ϕi(x)∣∣.

If x ∈ 10Bk and Ix = i : 10Bi ∩ 10Bk 6= ∅, then 10Bi ⊂ 30Bk for each i ∈ Ix andthe number of indices in Ix is bounded (with a bound depending only on n). Now,using (8.2), Exercise 5, the p-Poincare inequality, and the fact that |∇u| is an uppergradient of u, we have

|∇uj(x)| ≤ C2j∑i∈Ix

|u10Bi − u10Bk | ≤ C2j∑i∈Ix

∫10Bi

|u− u10Bk | dx

≤ C2j∑i∈Ix

∫30Bk\E

|u− u30Bk\E| dx ≤ C(∫

30τBk\E|∇u|p dx

)1/p

,

and hence

(8.3)

∫10Bk

|∇uj(x)|p dx ≤ C

∫30τBk

|∇u|p dx.

To find an upper bound for the integrals of |∇uj|p over B, we use a collection ofballs

B = Bi : 10Bi ∩B 6= ∅.The closure of B is compact and thus there is m ∈ N such that B ⊂ ∪mi=15Bi. Using(8.3), we have

(8.4)

∫B

|∇uj(x)|p dx ≤m∑i=1

∫10Bi

|∇uj|p dx ≤ C

m∑i=1

∫30τBi

|∇u|p dx.

Since radius of Bi is 2−j and 10Bi ∩B 6= ∅ for all i = 1, . . . ,m, we have that

|y − x0| ≤ (30τ + 10) + r ≤ 40τ + r71

for all y ∈ 30τBi. Hence, r ≥ 1, 30τBi ⊂ 41τB. If 0 < r < 1, let k ∈ N be suchthat 41τ2−k ≤ r < 41τ2−k+1. Then 30τBi ⊂ 2k+1τB. In both cases, denote theenlarging constant of B by λ, which is independent of j. Now (8.4) implies that∫

B

|∇uj(x)|p dx ≤ C

m∑i=1

∫30τBi

|∇u|p dx ≤ C

∫λB

|∇u|p dx ≤M <∞,

where M is independent of j.By Lemma 4.7, the discrete convolutions uj converge to u in Lp(X). Hence, as

a convergent sequence, (uj)j is bounded in Lp(B) (recall that |E| = 0). Thus theclaim about the boundedness of uj in W 1,p(B) follows.

By weak compactness of W 1,p(B) for 1 < p <∞, there exists a subsequence (ujk)of (uj), and a function v ∈ W 1,p(B) such that ujk converges weakly to v in W 1,p(B).Since uj → u in Lp(B), uj(x) → u(x) for almost all x ∈ B. As the weak limit isunique, we have u = v and the theorem follows.

About the metric setting. Note that in spaces M1,p(X), the removability is notan interesting question because inequality (5.1) defining the space is required to holdonly almost everywhere.

As in the classical case, we say that a compact set E ⊂ X with µ(E) = 0 isremovable for N1,p, if N1,p(X \ E) = N1,p(X) as sets. If u belongs to N1,p(X \ E)and µ(E) = 0, then u and each p-weak upper gradient g ∈ Lp(X\E) of u also belongto Lp(X). Hence the main task is to show that u has a p-weak upper gradient inLp(X) that satisfies the upper gradient inequality (6.1) in X. Using the propertiesof (minimal) p-weak upper gradients, one can show that for a function u ∈ N1,p(X),the minimal p-weak upper gradient in X is a minimal p-weak upper gradient inX \ E, see Lemma 6.26, and [77, Chapter 8].

Also in the metric setting, removability is a local question and it suffices to studydense sets of N1,p(X \E), for example bounded functions with compact support byLemma 6.34 and functions with upper gradients in X \ E.

The following theorem is a metric space version of Thorem 8.2. The proof ofthe sufficiency part can be done by modifying the proof of Thorem 8.2. Instead ofestimating the gradient, we estimate |uj(x) − uj(y)| for points that belong to thesame ball of the covering and find, using the properties of the partition of unity andthe p-Poincare inequality in X \E, an upper bound for the pointwise lower Lipschitzconstant of uj. By Example 6.5, it is a p-weak upper gradient of uj.

Below, we will give a proof from [77, Thm 8.13], originally for Orlicz–Sobolevspaces.

Theorem 8.3. Let X be a metric measure space that supports a p-Poincare inequal-ity, p > 1, and let E ⊂ X be a compact set with µ(E) = 0. Then E is removablefor N1,p, if and only if Y = X \ E supports a p-Poincare inequality.

Proof. Assume first that E is removable for N1,p. Let u ∈ L1loc(Y ) and let g be an

upper gradient of u in Y . Let B = B(x0, r) be a ball in Y . We may assume thatg ∈ Lp(τB), since otherwise there is nothing to show.

72

For each j ∈ N, letAj =

x ∈ τB : |u(x)| > j

,

and let uj be and the two-sided truncation function

uj(x) =

u(x), x ∈ τB \ Aj,j, u(x) > j,

−j, u(x) < −j.Then Aj+1 ⊂ Aj for each j ∈ N and

µ(Aj) ≤∫Aj

|u(x)|j

dµ ≤ 1

j

∫τB

|u(x)| dµ → 0 as j →∞.

Then g is an upper gradient of each uj and, as bounded functions, uj belong toLp(τB). Hence uj ∈ N1,p(τB \ E) for each j ∈ N. The fact that removability is alocal property, we have that uj ∈ N1,p(τB) with a minimal p-weak upper gradientgj. Now, by the p-Poincare inequality in X for uj and gj,∫

B

|uj − (uj)B| dµ ≤ cP diamB(∫

B

gpj dµ)1/p

≤ cP diamB(∫

B

gp dµ)1/p

,

from which the p-Poincare inequality for u and g follows using the dominated con-vergence theorem.

Assume then that Y = X \E supports a p-Poincare inequality. By Theorem 7.12,Lipschitz functions are dense in N1,p(Y ) and hence it suffices to show that eachLipschitz function of N1,p(Y ) belongs also to N1,p(X).

Let U be an open set of finite measure such that E ⊂ U . Let u ∈ N1,p(Y ) be anL-Lipschitz function with a p-weak upper gradient g ∈ Lp(X) in Y .

By the McShane extension theorem 2.5, there is an L-Lipschitz extension uX ofu on X. Now the function

gX = LχU + gχX\E

belongs to Lp(X) because g is in Lp(X) and the measure of U is finite. It is a p-weakupper gradient of uX in X by Lemma 6.19. From that, we conclude that uX belongsto N1,p(X) and hence the theorem follows.

Since a q-Poincare inequality follows from a p-Poincare inequality for all 1 < p < q,the previous theorem implies that sets removable for N1,p are removable also for N1,q

when 1 < p < q. Note that the validity of a p-Poincare inequality is essential. Inthe following example, a p-Poincare inequality cannot hold for u and g and E isremovable for N1,p only if 1 ≤ p < 2.

Example 8.4. Let X =x = (x1, x2) ∈ R2 : |x2| ≤ |x1| ≤ 2

be equipped with the

Euclidean metric of R2 and with the Lebesgue measure and let E = (0, 0). ThenX consists of two closed sectors with a common vertex at E. Denote the left sectorby X− and the right sector by X+. Let u be a function on X \ E,

u(x) =

0, x ∈ X− \ E,1, x ∈ X+ \ E.

73

Then u ∈ N1,p(X \ E) for all 1 ≤ p < ∞, and the zero function g = 0 is an uppergradient of u in X \ E. If 1 ≤ p ≤ 2, then, by Example 6.9, the p-modulus ofrectifiable curves that go through E is zero. Hence u ∈ N1,p(X) with a p-weakupper gradient g. On the other hand, the q-modulus of rectifiable curves that gothrough E is positive for each 2 < q <∞. The function g cannot be a q-weak uppergradient of u in X. Hence E is not removable for N1,q if 2 < q <∞.

31.3 =============================

9. Extension results for Sobolev spaces in the metric setting

In this last section, we discuss about extension results for Sobolev spaces in themetric setting. In particular, we are interested in the connection of a measuredensity condition of the set where the original functions are defined and the extensionproperty. As earlier in these notes, we assume that the measure µ is doubling. Theresults for are from [32], for the classical Sobolev spaces, also with higher derivatives,see [31].

Extension domains in the metric case are defined similarly as in the Euclideansetting. A domain Ω ⊂ X is an M1,p-extension domain if there exists a bounded(linear) operator E : M1,p(Ω) → M1,p(X) such that Eu|Ω = u for all u ∈ M1,p(Ω).The extension property for measurable sets F ⊂ X and for spaces N1,p are definedsimilarly. When studying extension property for values 0 < p < 1, one usually hasto use techniques that do not give a linear extension, see for example [85].

Definition 9.1. A measurable set F ⊂ X satisfies a measure density condition, ifthere is a constant cF > 0 such that

(9.1) µ(F ∩B(x, r)) ≥ cFµ(B(x, r))

for all x ∈ F , and for all 0 < r ≤ 1.

Exercise 6. Show, using the doubling property of µ, that the upper bound in (9.1)is not essential, any finite upper bound works as well.

We will start with spaces M1,p(X) and show that (9.1) characterizes extensiondomains for M1,p(X) in Q-regular, geodesic spaces.

The Q-regularity (sometimes called Ahlfors-David regularity) means that there isa constant cQ ≥ 1 such that

(9.2) c−1Q rQ ≤ µ(B(x, r)) ≤ cQr

Q

for each x ∈ X, 0 < r < diam(X). Condition (9.2) for a Borel measure µ impliesthat (9.2) holds for the Hausdorff measure HQ (usually with different comparabilityconstant) and that dimHX = Q. See for example [40], [68], [15] about regular spacesand regular subsets in Rn. The article [59] shows that for each Q > 1, there is aQ-regular metric space that supports a 1-Poincare inequality.

74

9.1. Measure density from extension.

Theorem 9.2. Let X be a Q-regular, geodesic metric measure space. A domainΩ ⊂ X is an M1,p-extension domain, 1 ≤ p <∞, if and only if (9.1) holds.

Note that in a geodesic space, boundaries of balls are of zero measure. To seethat, let B = B(x, r) be a ball. Let y ∈ ∂B and 0 < R < r. Since X is geodesic, xand y can be connected by a curve γ with `(γ) = d(x, y), which implies that thereis a ball of radius R/2 in B ∩B(y,R). Hence, by the doubling property of µ,

µ(B(y,R) ∩B) ≥ cdµ(B(y,R)).

This shows that B satisfies the measure density condition, which by the Exercisebelow, proves the claim.

Exercise 7. Let F ⊂ X be a set that satisfies (9.1). Show that any point ofF \ F is not a point of density for X \ F . Then (a general version of) the Lebesguedifferentiation theorem shows that µ(F \ F ) = 0. In particular, if a domain Ωsatisfies the measure density condition, then µ(∂Ω) = 0.

Proof of Theorem 9.2. The extension property follows from the measure density bythe more general result, Theorem 9.5 below.

Assume that Ω is an M1,p-extension domain. Let x ∈ Ω, let 0 < r ≤ 1 and letB = B(x, r). If Ω ⊂ B, then condition (9.1) holds. Thus we may assume thatΩ \B 6= ∅.

We will use embedding theorems and hence the cases 1 ≤ p < Q, p = Q andp > Q require different proofs.

Case: 1 ≤ p < Q. Since the boundaries of balls have zero measure, there are radii0 < ˜r < r < r such that

µ(B(x, ˜r) ∩ Ω) = 12µ(B(x, r) ∩ Ω) = 1

4µ(B(x, r) ∩ Ω).

Let

A(r, ˜r) = B(x, r) \B(x, ˜r), A(r, r) = B(x, r) \B(x, r),

and define a function u on Ω as

u(y) =

1, if y ∈ B(x, ˜r) ∩ Ω,r−d(x,y)

r−˜r, if y ∈ A(r, ˜r) ∩ Ω,

0, if y ∈ Ω \B(x, r).

Then u is (r− ˜r)−1-Lipschitz. This together with the fact that u = 0 in Ω \B(x, r)shows that the function

g = (r − ˜r)−1χB(x,r)∩Ω

is a generalized gradient of u. Since 0 < r − ˜r < 1, we have that

(9.3)‖u‖M1,p(Ω) ≤

(1 + (r − ˜r)−1

)(µ(B(x, r) ∩ Ω)

)1/p

≤ 2(r − ˜r)−1(µ(B(x, r) ∩ Ω)

)1/p.

75

We want to find a lower bound for ‖u‖M1,p(Ω). Let v = Eu be the extension of u(may assume that 0 ≤ v ≤ 1) and let gv ∈ D(v) ∩ Lp(X) be such that

(9.4) ‖gv‖Lp(X) ≤ 2‖v‖M1,p(X) ≤ C‖u‖M1,p(Ω).

By the weak-type version of Theorem 5.10 and the Q-regularity, we have

(9.5)(µ(x ∈ B : |v(x)− vB| > t

))1/p∗ ≤ C

t

(∫2B

gpv dµ)1/p

.

Using the facts that v = u = 1 on B(x, ˜r) ∩ Ω and v = u = 0 on A(r, r) ∩ Ω, andthe selection of the radii r and ˜r, we see that

|v − vB| ≥1

2

on at least one of the sets B(x, ˜r) ∩Ω and A(r, r) ∩Ω. The two sets have measurescomparable to the measure of B(x, r) ∩ Ω and hence inequality (9.5) with t = 1/2together with (9.3) and (9.4) implies that

µ(B(x, r) ∩ Ω)1/p∗ ≤ C‖v‖M1,p(X) ≤ C(r − ˜r)−1µ(B(x, r) ∩ Ω)1/p,

and hence

(9.6) r − ˜r ≤ C(µ(B(x, r) ∩ Ω)1/Q ≤ C(µ(B(x, r) ∩ Ω)1/Q.

Now we define a sequence of radii,

r0 = r, rj+1 = rj,

for whichµ(B(x, rj) ∩ Ω) = 2−jµ(B(x, r) ∩ Ω),

and hence rj → 0 as j →∞.By applying (9.6) for B(x, rj), we obtain

rj+1 − rj+2 ≤ C2−j/Qµ(B(x, r) ∩ Ω)1/Q.

Now

r = r1 =∞∑j=0

(rj+1 − rj+2) ≤ C( ∞∑j=0

2−j/Q)µ(B(x, r) ∩ Ω)1/Q

= Cµ(B(x, r) ∩ Ω)1/Q.

If r ≥ Cr for some constant C, then the measure density condition follows using theQ-regularity. Lemma 9.3 shows, that we may assume that r ≥ 1/10r.

Case: p > Q. The function

u(y) =

1− d(x,y)

r, if y ∈ B ∩ Ω,

0, if y ∈ Ω \B,

is r−1-Lipschitz and the function

g = r−1χB∩Ω

is a generalized gradient of u.76

Let v = Eu be the extension of u and let gv ∈ D(v) ∩ Lp(X) be such that‖gv‖Lp(X) ≤ 2‖v‖M1,p(X). Using the fact that 0 < r ≤ 1, we have

‖v‖M1,p(X) ≤ C‖u‖M1,p(Ω) ≤ C(µ(B(x, r) ∩ Ω)1/p + r−1µ(B(x, r) ∩ Ω)1/p

)≤ Cr−1µ(B(x, r) ∩ Ω)1/p.

Since v(x) = u(x) = 1 and v(y) = u(y) = 0 for some y ∈ (Ω \ B) ∩ 2B, usingTheorem 5.10 and the Q-regularity, we have

1 = |v(x)− v(y)| ≤ CrQ/pr1−Q/p(∫

2B

gpv dµ)1/p

≤ Cr1−Q/p(∫

2B

gpv dµ)1/p

≤ Cr1−Q/p‖v‖M1,p(X) ≤ Cr1−Q/pr−1µ(B(x, r) ∩ Ω)1/p,

from which inequality (9.1) easily follows using the Q-regularity.

Case: p = Q.Let A = 2

3B \ 1

3B. The function

u(y) =

1, if y ∈ 1

3B ∩ Ω,

2− 3d(x,y)r

, if y ∈ A ∩ Ω,

0, if y ∈ Ω \ 23B

is 3/r-Lipschitz, and the function g = 3r−1χ23B∩Ω is a generalized gradient of u in

Ω.Let v = Eu be the extension of u, and let gv ∈ D(v) ∩ LQ(X) be such that‖gv‖LQ(X) ≤ 2‖v‖M1,Q(X). Similarly as in other cases, using the fact that 0 < r ≤ 1,we obtain

‖v‖M1,Q(X) ≤ C‖u‖M1,Q(Ω) ≤ C(µ(

23B ∩ Ω

)1/Q+ 3r−1µ

(23B ∩ Ω

)1/Q)

≤ Cr−1µ(B ∩ Ω)1/Q.

The next step is to use ”a lower bound for Q-capacity, Lemma 9.4. For that, let

E = 13B ∩ Ω, F = (B \ 2

3B) ∩ Ω.

We have to show that

(9.7) minH1∞(E),H1

∞(F )≥ r

3.

By the connectivity of Ω, the 1-Lipschitz function p : Ω → [0,∞), p(y) = d(x, y)maps the sets E and F onto the intervals [0, r/3) and [2r/3, r), respectively. As a1-Lipschitz function does not increase the Hausdorff 1-content, we conclude that

H1∞(E) ≥ H1

∞(p(E)) =r

3, H1

∞(F ) ≥ H1∞(p(F )) =

r

3,

from which (9.7) follows.77

Since the pair v, gv satisfies a Q-Poincare inequality by Theorem 5.7 and theHolder inequality, we can use Lemma 9.4, and we obtain

C ≤∫

10B

gQv dµ ≤ ‖v‖QM1,Q(X)

≤ Cµ(B ∩ Ω)

rQ,

and inequality (9.1) follows from the Q-regularity.

Remark 22. The same proof with small modifications holds also for spaces N1,p.In that case, we assume that X is a Q-regular, complete and supports a p-Poincareinequality. Then we turn it a geodesic space as discussed in Section 7.1. For details,see [32, Thm 2].

Lemma 9.3 ([32], Lemma 14). If measure density condition (9.1) holds for all x ∈ Ωand all 0 < r ≤ 1 such that r ≤ 10r, then it holds for all x ∈ Ω and all 0 < r ≤ 1.

Idea of the proof. Let B = B(x, r), where x ∈ Ω, 0 < r ≤ 1 be a ball such thatΩ \B 6= ∅ and r > 10r. Then there is a big piece of Ω close to x.

Since Ω is connected, there is y ∈ B ∩ Ω such that d(x, y) = r + r/5 and letR = 2r + r/5. Show that for a ball B(y,R), R ≥ R/2, which implies that B(y,R)satisfies measure density condition. Use that and the Q-regularity to show thatµ(B ∩ Ω) ≥ Cµ(B).

We used the following lemma in the case p = Q of proof of Theorem 9.2. It canbe used there because the pair v, gv, satisfies a Q-Poincare inequality.

Lemma 9.4 ([40], Thm 5.9). Let Q ≥ 1. Let X be a Q-regular space that supportsa Q-Poincare inequality. Let E and F be disjoint subsets of a ball B = B(x, r) suchthat

(9.8) minH1∞(E),H1

∞(F ) ≥ λr

for some 0 < λ ≤ 1. Then there is a constant C ≥ 1, depending only on the dataassociated with X, such that

(9.9)

∫10τB

gQ dµ ≥ Cλ

whenever u ∈ L1loc(X), g is an upper gradient of u in 10τB, every x ∈ E ∪ F is a

Lebesgue point of u, u|E ≥ 1 and u|F ≤ 0.

1.4 =============================

Remark 23. Actually, the measure density property for a domain follows from aweaker property than the existence of a bounded extension operator, it suffices thatthe trace operator

T : N1,p(X)→ N1,p(Ω), T (v) = v|Ω,is surjective. By a standard functional analysis reasoning, we can show that thisproperty implies that there exists a constant C > 0 such that for each u ∈ N1,p(Ω),there is v ∈ N1,p(X) such that v|Ω = u and ‖v‖N1,p(X) ≤ C‖u‖N1,p(Ω); that is theproperty we needed in the proof of Theorem 9.2.

78

To see that the surjectivity of the trace implies the existence of such a v, letu ∈ N1,p(Ω) with ‖u‖N1,p(Ω) > 0 and let Q : N1,p(X) → N1,p(X)/ ker T be thequotient map,

Q(v) = v + ker T .N1,p(X) is a Banach space by Theorem 6.33, and hence also the quotient spaceN1,p(X)/ ker T is a Banach space with the norm

‖v + ker T ‖ = inf‖v + h‖N1,p(X) : h ∈ ker T .Since T is linear, bounded and surjective, there is a bounded, linear, one-to-onemapping T from N1,p(X)/ ker T onto N1,p(Ω) such that T = T Q. By the open

mapping theorem, T −1 : N1,p(Ω)→ N1,p(X)/ ker T is a bounded linear map.

Hence there is v ∈ N1,p(X) for which T −1(u) = v + ker T and

‖v + ker T ‖ ≤ C‖u‖N1.p(Ω),

where the constant C > 0 is independent of u and v. For each function h ∈ ker T ,we have that v + h ∈ N1,p(Ω), h|Ω = 0, and (v + h)|Ω = u. By the definition of thenorm in N1,p(X)/ ker T , there is h0 ∈ ker T such that

‖v + h0‖N1.p(X) < ‖v + ker T ‖+ ‖u‖N1.p(Ω) ≤ C‖u‖N1.p(Ω),

from which the claim follows. A similar proof holds for spaces M1,p(Ω).

For Sobolev spaces W k,p(Ω), a validity of a Sobolev type embedding theorem onΩ, implies the measure density by [31, Thm 1].

9.2. Extension from measure density. For the classical Sobolev spaces W k,p(Ω)and spaces N1,p(Ω), measure density condition (9.1) does not imply the extensionproperty because those functions have nice absolute continuity properties on curves.A slit disc

B(0, 1) \ x ∈ B(0, 1) : 0 ≤ x1 < 1, x2 = 0 ⊂ R2,

is an example of a domain satisfying measure density condition that is not an ex-tension domain for Sobolev spaces; take a function u ∈ W 1,p(Ω) that is 1 above theremoved radius and 0 below.

Spaces M1,p(F ), F ⊂ X, are defined only up to a sets of measure zero, and forthem, the measure density condition implies the extension property. The followingresult is from [32], the case p > 1 is proved also in [72].

Theorem 9.5. Let F ⊂ X be a measurable set which satisfies (9.1). Then there is abounded, linear extension operator of E : M1,p(F ) into M1,p(X) for all 1 ≤ p <∞.

Whitney balls and a partition of unity. As we discussed in Section 2.3, one of thebasic extension techniques is to cover the complement of the set where the originalfunctions are defined, by a Whitney type covering, take a corresponding partitionof unity, define the extension using a kind of discrete convolution and show thatthe extension satisfies the desired properties. In the metric setting, we cannot get adisjoint Whitney covering as Lemma 2.6 gives in Rn, but applying the 5R-coveringlemma 3.7, we obtain a covering consisting of balls whose radii are comparable to the

79

distance to the set where the functions are defined and enlarged balls have boundedoverlap, for the proofs see [14, Theorem III.1.3] and [63, Lemma 2.9].

Let U ( X be an open set. For each x ∈ U , let

r(x) = dist(x,X \ U)/10.

There exists a countable family B = Bii∈I of balls Bi = B(xi, ri), where ri = r(xi),such that B is a covering of U and the balls B(xi, ri/5) are disjoint. The nextlemma easily follows from the definition of the Whitney covering B and the doublingproperty of the measure µ, prove it as an exercise.

Lemma 9.6 (Whitney balls). There is M ∈ N such that for all i ∈ N,

(1) the balls 15Bi are disjoint,

(2) U = ∪i∈IBi,(3) 5Bi ⊂ U ,(4) if x ∈ 5Bi, then 5ri < dist(x,X \ U) < 15ri,(5) there is x∗i ∈ X \ U such that d(xi, x

∗i ) < 15ri,

(6)∑

i∈Iχ5Bi(x) ≤M for all x ∈ U .

Let ϕii∈I be a Lipschitz partition of unity subordinated to the covering B withthe following properties:

(i) suppϕi ⊂ 2Bi,(ii) ϕi(x) ≥M−1 for all x ∈ Bi,

(iii) there is a constant K such that each ϕi is Kr−1i -Lipschitz,

(iv)∑

i∈I ϕi(x) = χU(x).

Note that if 5Bi ∩ 5Bj 6= ∅, then 1/3ri ≤ rj ≤ 3rj and d(x∗i , x∗j) ≤ 80ri, where the

points x∗i , x∗j are as in Lemma 9.6 (5).

Proof of Theorem 9.5. We will prove only the case p > 1. The case p = 1 is muchmore technical because we cannot use the boundedness of the Hardy–Littlewoodmaximal function in Lp. One of the main tools in the case p = 1 is Theorem 5.10.

We may assume that F 6= X; otherwise the claim is trivial. Note that the measuredensity condition holds for balls of uniformly bounded radius because µ is doublingby Exercise 6. Since µ(F \ F ) = 0 by Exercise 7 (and [72, Le 2.1]), we may assumethat F is closed.

Let u ∈M1,p(F ) and let g ∈ D(u)∩Lp(F ) be a minimal generalized gradient of uprovided by Remark 11. We identify g with its zero extension by defining g(x) = 0for all x ∈ X \ F . Then, by the measure density condition, gB ≈ gB∩F for all ballsB = B(x, r), x ∈ F , 0 < r ≤ 1.

Let B = Bii∈I , Bi = B(xi, ri), be the Whitney covering of X \F and let ϕii∈Ibe the associated Lipschitz partition of unity. Let B1 = Bii∈J be the collection ofall balls from B with radius less than 1. For each i ∈ J , let x∗i be ”a closest point ofxi in F” as in Lemma 9.6, and let

B∗i = B(x∗i , ri).

For each x ∈ 2Bi, i ∈ J , let

Bx = B(x, 25r(x)) = B(x, 52

dist(x, F )).80

Then B∗i ⊂ Bx ⊂ 47B∗i and, by the measure density condition and the doublingproperty of µ,

µ(Bx) ≤ Cµ(B∗i ∩ F ).

In the proof below, we will show that the maximal function of g is a generalizedgradient of the extension of u. This is obtained by estimating the integral averagesof g over balls the balls Bx and By for x, y /∈ F .

We will first construct an extension of u with norm estimates to set

V = x ∈ X : dist(x, F ) < 8.For each x ∈ V \ F , let

Ix = i ∈ I : x ∈ 2Bi.By Lemma 9.6, there are at most M indices in Ix. Moreover, Ix ⊂ J (if i ∈ I \ J ,then ri ≥ 1 and hence dist(2Bi, F ) ≥ 8ri ≥ 8. Thus 2Bi ∩ V = ∅ and i 6∈ Ix) andtherefore ∑

i∈Ix

ϕi(x) =∑i∈I

ϕi(x) =∑i∈J

ϕi(x) = 1 for each x ∈ V \ F.

We define Eu, the local extension of u by

(9.10) Eu(x) =

u(x), if x ∈ F,∑

i∈J ϕi(x)uB∗i ∩F , if x ∈ X \ F.

We split the proof into several steps.

Claim 1: ‖Eu‖Lp(X) ≤ C‖u‖Lp(F ).Let x ∈ X \F . Using the properties of the functions ϕi, of the ball Bx and of the

set Ix, we have∣∣Eu(x)∣∣ =

∣∣∣∑i∈J

ϕi(x)uB∗i ∩F

∣∣∣ ≤∑i∈Ix

∫B∗i ∩F

|u| dµ ≤ C

∫Bx

|u| dµ ≤ CMu(x),

where u is extended to X \ F by zero, and hence∫Bx|u| dµ = µ(Bx)

−1∫Bx∩F |u| dµ.

The norm estimate follows now from the Hardy–Littlewood maximal theorem 3.9.

Claim 2: Eu ∈M1,p(V ). We will show that

(9.11) |Eu(x)− Eu(y)| ≤ Cd(x, y)(Mg(x) +Mg(y))

for almost every x, y ∈ V . This shows that Mg ∈ D(Eu) in V . The p-integrabilityofMg follows from Theorem 3.9. To show that (9.11) holds, we consider four cases.Case 1: x, y ∈ F . Since g ∈ D(u), we have

|Eu(x)− Eu(y)| = |u(x)− u(y)| ≤ d(x, y)(g(x) + g(y))

≤ d(x, y)(Mg(x) +Mg(y))

almost everywhere by Theorem 3.11 because g(x) ≤Mg(x) whenever x is a Lebesguepoint of g.

Case 2: x ∈ V \ F , y ∈ F .81

Now

|Eu(x)− Eu(y)| = |Eu(x)− u(y)| ≤ |Eu(x)− uBx∩F |+ |uBx∩F − u(y)|,and we begin with the first term. Using the properties of the functions ϕi and ofthe ball Bx and the assumption g ∈ D(u), we have

(9.12)

|Eu(x)− uBx∩F | =∣∣∣∑i∈Ix

ϕi(x)(uB∗i ∩F − uBx∩F )∣∣∣

≤∑i∈Ix

∫B∗i ∩F

∫Bx∩F

|u(w)− u(z)| dµ(w)dµ(z)

≤ C

∫Bx∩F

∫Bx∩F

|u(w)− u(z)| dµ(w)dµ(z)

≤ Cr(x)

∫Bx

∫Bx

(g(w) + g(z)) dµ(w)dµ(z)

≤ Cr(x)Mg(x) ≤ Cd(x, y)Mg(x)

because r(x) ≤ dist(x, F ) ≤ d(x, y). For the second term, we have, because g ∈ D(u)and d(z, y) ≤ Cd(x, y) for all z ∈ Bx ∩ F .

|uBx∩F − u(y)| ≤∫Bx∩F

|u(z)− u(y)| dµ(z)

≤∫Bx∩F

d(z, y)(g(z) + g(y)) dµ(z)

≤ Cd(x, y)(g(y) +Mg(x)).

Hence (9.11) holds in this case.

Case 3: x, y ∈ V \ F and d(x, y) ≥ mindist(x, F ), dist(y, F ).We start with estimate

|Eu(x)− Eu(y)| ≤ |Eu(x)− uBx∩F |+ |uBx∩F − uBy∩F |+ |Eu(y)− uBy∩F |= a+ b+ c.

We may assume that dist(x, F ) ≤ dist(y, F ). Then

r(y) = 110

dist(y, F ) ≤ 110

(d(x, y) + dist(x, F )) ≤ 15d(x, y)

and hence the estimates for a and c follow similarly as in (9.12).For b, note that d(z, w) ≤ Cd(x, y) for all z ∈ Bx, w ∈ By. Then, by similar

calculations as above,

|uBx∩F − uBy∩F | ≤∫Bx∩F

∫By∩F

|u(w)− u(z)| dµ(w)dµ(z)

≤∫Bx

∫By

d(w, z)(g(w) + g(z)) dµ(w)dµ(z)

≤ Cd(x, y)(Mg(x) +Mg(y)),

and hence (9.11) holds.82

Case 4: x, y ∈ V \ F and d(x, y) < mindist(x, F ), dist(y, F ).We may assume that dist(x, F ) ≤ dist(y, F ). Since each ϕi is Kr−1

i -Lipschitz, theselection of balls Bx and By and the assumption dist(x, F ) ≤ dist(y, F ) imply thatif i ∈ Ix ∪ Iy, then the Lipschitz constant of ϕi is bounded by Cr(x)−1. Moreover,B∗i ⊂ Bx for each i ∈ Ix ∪ Iy. Using similar arguments as in (9.12), we obtain

|Eu(x)− Eu(y)| =∣∣∣ ∑i∈Ix∪Iy

(ϕi(x)− ϕi(y))(uB∗i ∩F − uBx∩F )∣∣∣

≤ C∑

i∈Ix∪Iy

d(x, y)

ri

∫B∗i ∩F

∫Bx∩F

|u(w)− u(z)| dµ(w)dµ(z)

≤ C∑

i∈Ix∪Iy

d(x, y)

r(x)

∫B∗i ∩F

∫Bx∩F

|u(w)− u(z)| dµ(w)dµ(z)

≤ Cd(x, y)Mg(x).

Final step. The function Eu has good estimates in V , but we want estimates inX. To get the final extension, let Ψ: X → [0, 1] to be a cut-off function such thatΨ ≡ 1 in F , Ψ ≡ 0 in X \ V and Ψ is L-Lipschitz continuous. Now we define anextension operator E : M1,p(F )→M1,p(X) by

Eu = ΨEu.

By Lemma 5.6 and the first part of the proof, Eu ∈M1,p(X) and(CMg + L|Eu|

)χV ∈ D(Eu).

Hence

‖Eu‖M1,p(X) ≤ C(‖u‖Lp(F ) + ‖g‖Lp(F )

),

and the claim in the case p > 1 follows.

What happens to spaces N1,p(Ω)? We already discussed in Remarks 22 and 23,that if the underlying space is nice enough, then the extension property impliesmeasure density.

Theorem 9.7. Let X be a Q-regular, complete metric measure space that supportsa p-Poincare inequality, 1 < p <∞. A domain Ω ⊂ X is an N1,p-extension domainif and only if Ω satisfies measure density condition (9.1) and N1,p(Ω) = M1,p(Ω).

Proof. If Ω is an extension domain, then the measure density condition follows fromRemarks 22 and 23. We have to show that N1,p(Ω) = M1,p(Ω). By Theorem 6.36,it suffices to show that N1,p(Ω) ⊂M1,p(Ω).

Let u ∈ N1,p(Ω). By the assumptions on X and Corollary 7.11, N1,p(X) =M1,p(X). Hence Eu, the extension of u belongs to M1,p(X), and so u = Eu|Ω ∈M1,p(Ω). Therefore N1,p(Ω) ⊂M1,p(Ω).

Assume then that N1,p(Ω) = M1,p(Ω) and that Ω satisfies measure density condi-tion (9.1) . The claim follows from Theorem 9.5 and from the fact that N1,p(X) =M1,p(X).

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We end the course by noting that for Sobolev spaces W 1,p(Ω), we have correspond-ing characterization of the extension property. For more about the recent extensionresults in the Euclidean setting, see for example [31] and [73].

Theorem 9.8. Let Ω ⊂ Rn be a domain and let 1 < p < ∞. The followingproperties are equivalent.

(1) For each u ∈ W 1,p(Ω) there is v ∈ W 1,p(Rn) such that v|Ω = u.(2) The trace operator T : W 1,p(X)→ W 1,p(Ω) is surjective.(3) Ω is a W 1,p-extension domain with a linear extension operator.(4) Ω satisfies measure density condition (9.1) and W 1,p(Ω) = M1,p(Ω).

Open problem: What happens when p = 1? Does the surjectivity of the traceoperator for W 1,1(Rn) imply that there is a bounded, linear extension operatorE : W 1,1(Ω) → W 1,1(Rn)? See the discussion in [31] and tell me, if you know theanswer.

Thank you for attending the course/reading the notes. It was a great pleasure forme to give the lectures and write the notes.

References

[1] D. Aalto and J. Kinnunen: Maximal functions in Sobolev spaces, Sobolev spaces in mathe-matics. I, 25–67, Int. Math. Ser. (N. Y.), 8, Springer, New York, 2009.

[2] D. Aalto and J. Kinnunen: The discrete maximal operator in metric spaces, J. Anal. Math. 111(2010), 369–390.

[3] L. Ahlfors and A. Beurling: Conformal invariants and function-theoretic null-sets, Acta Math.83, (1950). 101–129.

[4] L. Ambrosio, M. Colombo, S. Di Marino: Sobolev spaces in metric measure spaces: reflexivityand lower semicontinuity of slope, http://arxiv.org/abs/1212.3779

[5] L. Ambrosio, M. Miranda, Jr., D. Pallara: Special functions of bounded variation in doublingmetric measure spaces, Calculus of variations: topics from the mathematical heritage of E. DeGiorgi, 1–45, Quad. Mat., 14, Dept. Math., Seconda Univ. Napoli, Caserta, 2004.

[6] L. Ambrosio and P. Tilli, Paolo: Topics on analysis in metric spaces, Oxford Lecture Series inMathematics and its Applications, 25. Oxford University Press, Oxford, 2004.

[7] Y. Benyamini and J. Lindenstrauss: Geometric nonlinear functional analysis. Vol. 1, AmericanMathematical Society Colloquium Publications, 48. American Mathematical Society, Provi-dence, RI, 2000.

[8] A. Bjorn and J. Bjorn: Nonlinear potential theory on metric spaces, EMS Tracts in Mathemat-ics, 17. European Mathematical Society (EMS), Zurich, 2011.

[9] A. Brudnyi and Y Brudnyi: Methods of geometric analysis in extension and trace problems.Volume 1, Monographs in Mathematics, 102. Birkhauser/Springer Basel AG, Basel, 2012.

[10] A. Brudnyi and Y Brudnyi: Methods of geometric analysis in extension and trace problems.Volume 2, Monographs in Mathematics, 102. Birkhauser/Springer Basel AG, Basel, 2012.

[11] S. M. Buckley: Is the maximal function of a Lipschitz function continuous?, Ann. Acad. Sci.Fenn. Math. 24 (1999), 519–528.

[12] A. -P. Calderon: Lebesgue spaces of differentiable functions and distributions 1961 Proc. Sym-pos. Pure Math., Vol. IV pp. 33–49

[13] J. Cheeger: Differentiability of Lipschitz functions on metric measure spaces Geom. Funct.Anal. 9 (1999), 428–517.

[14] R. R. Coifman and G. Weiss: Analyse harmonique non-commutative sur certains espaceshomogenes, Lecture Notes in Mathematics, Vol.242. Springer-Verlag, Berlin-New York, 1971.

84

[15] G. David and S. Semmes: Analysis of and on uniformly rectifiable sets, Mathematical Surveysand Monographs, 38. American Mathematical Society, Providence, RI, 1993.

[16] E. Durand-Cartagena, J. A. Jaramillo, and N. Shanmugalingam: The ∞-Poincare inequalityin metric measure spaces, Michigan Math. J. 61 (2012), no. 1, 63–85.

[17] E. Durand-Cartagena, J. A. Jaramillo, and N. Shanmugalingam: First order Poincare in-equalities in metric measure spaces, Ann. Acad. Sci. Fenn. Math. 38 (2013), no. 1, 287–308.

[18] E. Durand-Cartagena, J. A. Jaramillo, and N. Shanmugalingam: Geometric char-acterizations of p-Poincare inequalities in the metric setting, https://www.mittag-leffler.se/preprints/files/IML-1314f-15.pdf

[19] E. Durand-Cartagena, N. Shanmugalingam, and A. Williams: p-Poincare inequality versus∞-Poincare inequality: some counterexamples, Math. Z. 271 (2012), no. 1-2, 447–467.

[20] L.C. Evans and R.F. Gariepy: Measure Theory and Fine Properties of Functions - CRC Press,1992, Boca Raton-New York-London-Tokyo.

[21] B. Fuglede: Extremal length and functional completion, Acta Math., 98(1957), 171–219.[22] F. W. Gehring: The Lp-integrability of the partial derivatives of a quasiconformal mapping,

Acta Math. 130 (1973), 265–277.[23] D. Gilbarg and N. S. Trudinger: Elliptic partial differential equations of second order, Reprint

of the 1998 edition. Classics in Mathematics. Springer–Verlag, Berlin, 2001.[24] P. Haj lasz: Sobolev spaces on an arbitrary metric space, Potential Anal. 5 (1996), 403–415.[25] P. Haj lasz: Geometric approach to Sobolev spaces and badly degenerated elliptic equations,

Nonlinear analysis and applications (Warsaw, 1994), 141–168, GAKUTO Internat. Ser. Math.Sci. Appl., 7, Gakkotosho, Tokyo, 1996.

[26] P. Haj lasz: Sobolev spaces on metric-measure spaces, In Heat kernels and analysis on man-ifolds, graphs, and metric spaces (Paris, 2002), 173–218. Contemp. Math. 338. Amer. Math.Soc. Providence, RI, 2003.

[27] P. Haj lasz: A new characterization of the Sobolev space, Studia Math. 159 (2003), no. 2,263–275.

[28] P. Haj lasz and J. Kinnunen: Holder quasicontinuity of Sobolev functions on metric spaces,Rev. Mat. Iberoamericana 14 (1998), no. 3, 601–622.

[29] P. Haj lasz and P. Koskela: Sobolev meets Poincare. C. R. Acad. Sci. Paris Ser. I Math. 320(1995), no. 10, 1211–1215.

[30] P. Haj lasz and P. Koskela: Sobolev met Poincare. Mem. Amer. Math. Soc. 145 (2000), no.688.

[31] P. Haj lasz, P. Koskela, and H. Tuominen: Sobolev embeddings, extensions and measure densitycondition J. Funct. Anal. 254 (2008), no. 5, 1217–1234.

[32] P. Haj lasz, P. Koskela, and H. Tuominen: Measure density and extendability of Sobolev func-tions, Rev. Mat. Iberoam. 24 (2008), no. 2, 645–669.

[33] P. Haj lasz and J. Maly: On approximate differentiability of the maximal function Proc. Amer.Math. Soc. 138 (2010), no. 1, 165–174.

[34] P. Haj lasz and J. Onninen: On boundedness of maximal functions in Sobolev spaces, Ann.Acad. Sci. Fenn. Math. 29 (2004), no. 1, 167–176.

[35] G. H. Hardy and J. E. Littlewood: A maximal theorem with function-theoretic applications,Acta Math. 54 (1930), no. 1, 81–116.

[36] T. Heikkinen, J. Kinnunen, J. Nuutinen, and H. Tuominen: Mapping properties of the discretefractional maximal operator in metric measure spaces, Kyoto J. Math. 53 (2013), no. 3, 693–712.

[37] J. Heinonen: Lectures on analysis on metric spaces, Universitext, Springer-Verlag, New York,2001.

[38] J. Heinonen: Lectures on Lipschitz analysis Report. University of Jyvaskyla Department ofMathematics and Statistics, 100. University of Jyvaskyla, Jyvaskyla, 2005.

[39] J. Heinonen: Nonsmooth calculus, Bull. Amer. Math. Soc. (N.S.) 44 (2007), no. 2, 163–232.[40] J. Heinonen and P. Koskela: Quasiconformal maps in metric spaces with controlled geometry,

Acta Math. 181 (1998), 1–61.

85

[41] J. Heinonen, P. Koskela, N. Shanmugalingam, and J. Tyson: Sobolev spaces on metric measurespaces: an approach based on upper gradients

[42] P.W. Jones: Quasiconformal mappings and extendability of functions in Sobolev spaces, ActaMath. 147 (1981), 71–88.

[43] S. Kallunki (Rogovin) and N. Shanmugalingam: Modulus and continuous capacity, Ann. Acad.Sci. Fenn. Math. 26 (2001), no. 2, 455–464.

[44] S. Keith: Modulus and the Poincare inequality on metric measure spaces, Math. Z. 245 (2003),no. 2, 255–292.

[45] S. Keith and X. Zhong: The Poincare inequality is an open ended condition, Ann. of Math.(2) 167 (2008), no. 2, 575–599.

[46] J. Kinnunen: The Hardy-Littlewood maximal function of a Sobolev function, Israel J. Math.100 (1997), 117–124.

[47] J. Kinnunen and V. Latvala: Lebesgue points for Sobolev functions on metric spaces, Rev.Mat. Iberoamericana 18 (2002), no. 3, 685–700.

[48] J. Kinnunen and P. Lindqvist: The derivative of the maximal function, J. Reine Angew. Math.503 (1998), 161–167.

[49] J. Kinnunen and O. Martio: The Sobolev capacity on metric spaces, Ann. Acad. Sci. Fenn.Math. 21 (1996), no. 2, 367–382.

[50] J. Kinnunen and H. Tuominen: Pointwise behaviour of M1,1 Sobolev functions, Math. Z. 257(2007), no. 3, 613–630.

[51] M. D. Kirszbraun: Uber die zusammenziehende und Lipschitzsche Transformationen, Funda-menta Math. 22 (1934), 77–108

[52] R. Korte: Geometric implications of the Poincare inequality, Results Math. 50 (2007), no. 1-2,93–107.

[53] P. Koskela: Capacity extension domains Dissertation, University of Jyvaskyla, Jyvaskyla,1990. Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes No. 73 (1990)

[54] P. Koskela: Removable sets for Sobolev spaces, Ark. Mat. 37 (1999), no. 2, 291–304.[55] P. Koskela and P. MacManus: Quasiconformal mappings and Sobolev spaces, Studia Math.

131 (1998), no. 1, 1–17.[56] P. Koskela and E. Saksman: Pointwise characterizations of Hardy–Sobolev functions. Math.

Res. Lett. 15 (2008), no. 4, 727–744.[57] P. Koskela, N. Shanmugalingam, and H. Tuominen: Removable sets for the Poincare inequality

on metric spaces, Indiana Univ. Math. J. 49 (2000), no. 1, 333–352.[58] O. Kurka: On the variation of the Hardy–Littlewood maximal function,

http://arxiv.org/abs/1210.0496[59] T. J. Laakso: Ahlfors Q-regular spaces with arbitrary Q¿1 admitting weak Poincare inequality,

Geom. Funct. Anal. 10 (2000), no. 1, 111–123.[60] U. Lang and V. Schroeder: Kirszbraun’s theorem and metric spaces of bounded curvature

Geom. Funct. Anal. 7 (1997), no. 3, 535–560.[61] G. Leoni: A first course in Sobolev spaces, Graduate Studies in Mathematics, 105. American

Mathematical Society, Providence, RI, 2009.[62] J. Luukkainen and E. Saksman: Every complete doubling metric space carries a doubling

measure, Proc. Amer. Math. Soc. 126 (1998), no. 2, 531–534.[63] R. A. Macıas and C. Segovia: A decomposition into atoms of distributions on spaces of ho-

mogeneous type, Adv. in Math. 33 (1979), 271–309.[64] E. J. McShane: /it Extension of range of functions, Bull. Amer. Math. Soc. 40 (1934), no. 12,

837–842.[65] P. Mattila: Geometry of sets and measures in Euclidean spaces Fractals and rectifiability,

Cambridge Studies in Advanced Mathematics, 44. Cambridge University Press, Cambridge,1995.

[66] N. G. Meyers and J. Serrin: H = W , Proc. Nat. Acad. Sci. U.S.A. 51 1964 1055–1056.

86

[67] A. S. Romanov: On a generalization of Sobolev spaces, (Russian) Sibirsk. Mat. Zh. 39 (1998),no. 4, 949–953, translation in Siberian Math. J. 39 (1998), no. 4, 821–824

[68] S. Semmes: Finding curves on general spaces through quantitative topology, with applicationsto Sobolev and Poincare inequalities, Selecta Math. (N.S.) 2 (1996), no. 2, 155–295.

[69] N. Shanmugalingam: Newtonian spaces: an extension of Sobolev spaces to metric measurespaces, Rev. Mat. Iberoamericana 16 (2000), no. 2, 243–279.

[70] N. Shanmugalingam: Harmonic functions on metric spaces, Illinois J. Math. 45 (2001), no.3,1021–1050.

[71] N. Shanmugalingam: Introduction to p-modulus of path-families and Newtonian spaces, J.Anal. 18 (2010), 349–360.

[72] P. Shvartsman: On extensions of Sobolev functions defined on regular subsets of metric mea-sure spaces, J. Approx. Theory f144 (2007), no. 2, 139–161.

[73] P. Shvartsman: On Sobolev extension domains in Rn, J. Funct. Anal. 258 (2010), no. 7,2205–2245.

[74] E. M. Stein: Note on the class L logL, Studia Math. 32 1969 305–310.[75] E. M. Stein: Singular integrals and differentiability properties of functions, Princeton Mathe-

matical Series, No. 30 Princeton University Press, Princeton, N.J. 1970.[76] H. Tanaka: A remark on the derivative of the one-dimensional Hardy–Littlewood maximal

function, Bull. Austral. Math. Soc. 65 (2002), no. 2, 253–258.[77] H. Tuominen: Orlicz–Sobolev spaces on metric measure spaces, Dissertation, University of

Jyvaskyla, Jyvaskyla, 2004. Ann. Acad. Sci. Fenn. Math. Diss. No. 135 (2004),[78] F. A. Valentine: A Lipschitz condition preserving extension for a vector function, Amer. J.

Math. 67, (1945). 83–93.[79] J. Vaisala: Lectures on n-dimensional quasiconformal mappings, Lecture Notes in Mathemat-

ics, Vol. 229, Springer-Verlag, Berlin-New York, 1971.[80] H. Whitney: Analytic extensions of differentiable functions defined in closed sets, Trans. Amer.

Math. Soc. 36 (1934), no. 1, 63–89.[81] N. Wiener, Norbert The ergodic theorem, Duke Math. J. 5 (1939), no. 1, 1–18.[82] J-M. Wu: Removability of sets for quasiconformal mappings and Sobolev spaces, Complex

Variables Theory Appl. 37 (1998), no. 1-4, 491–506.[83] K. Yosida: Functional analysis, Reprint of the sixth (1980) edition. Classics in Mathematics.

Springer–Verlag, Berlin, 1995.[84] W. P. Ziemer: Weakly differentiable functions. Graduate Texts in Mathematics 120, Springer–

Verlag, 1989.[85] Y. Zhou, Fractional Sobolev extension and imbedding, to appear in Trans. Amer. Math. Soc.

Matematiikan ja tilastotieteen laitos, PL 35, 40014 Jyvaskylan yliopistoE-mail address: heli.m.tuominen@jyu.fi

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