Atomic norms and finite sums of atoms

Fulvio Ricci

Scuola Normale Superiore di Pisa

Bardonecchia, June 16, 2009

Convegnetto in honour of Poor Little Guido

This is joint work with J. Verdera, to appear in TAMS

Everything is done in good old Euclidean Rn.

(p, q)-atoms

Let 0 < p ≤ 1 ≤ q ≤ ∞ and p < q.

A (p,q)-atom is an Lq-function supported on a ball B and suchthat• ‖a‖q ≤ m(B)

•∫

xαa(x) dx = 0 for |α| ≤ n(1

p − 1).

(p, q)-atoms

Let 0 < p ≤ 1 ≤ q ≤ ∞ and p < q.

A (p,q)-atom is an Lq-function supported on a ball B and suchthat• ‖a‖q ≤ m(B)

•∫

xαa(x) dx = 0 for |α| ≤ n(1

p − 1).

Atomic norms

With p,q as above, Hp,q is the space of distributions f(functions if p = 1) that can be represented by atomic series ofthe form

f =∞∑

λjaj , (in the sense of distributions)

where the aj are (p,q)-atoms, and the sequence λ = (λj) isin `p.

The (p,q)-atomic norm of f is

‖f‖Hp,q = inf{‖λ‖`p : f =

∞∑j=0

λjaj , aj (p,q)-atoms}.

Atomic norms

With p,q as above, Hp,q is the space of distributions f(functions if p = 1) that can be represented by atomic series ofthe form

f =∞∑

λjaj , (in the sense of distributions)

where the aj are (p,q)-atoms, and the sequence λ = (λj) isin `p.The (p,q)-atomic norm of f is

‖f‖Hp,q = inf{‖λ‖`p : f =

∞∑j=0

Equivalence of atomic norms

For fixed p, all Hp,q spaces coincide with the Hardy space Hp

defined in terms of maximal operators, Riesz transforms, etc.(Coifman&Weiss, Latter).

Finite atomic norms

In 1985, Meyer, Taibleson and Weiss observed a specialfeature of (p,∞)-atomic norms:a single (p,∞)- atom a can have arbitrary small Hp,∞-norm.

In fact they showed that the ratio between the atomic norm of a(p,∞)-atom a and its finite (p,∞)-atomic norm,

‖a‖F p,∞ = inf{‖λ‖`p : a =

∑finite

λjaj , aj (p,∞)-atoms},

can be arbitrary small.

This shows that the simple fact that a linear operator isuniformly bounded on atoms may not be sufficient to imply thatthe operator is bounded on Hp.

Finite atomic norms

‖a‖F p,∞ = inf{‖λ‖`p : a =

∑finite

Finite atomic norms

‖a‖F p,∞ = inf{‖λ‖`p : a =

∑finite

Recent developments

In 2005 Bownik exhibited a linear functional, defined on(1,∞)-atoms and uniformly bounded on them, which is notcontrolled by the H1,∞-norms of the same atoms, and thereforedoes not admit a continuous extension to H1.

It was recently shown by Yang and Zhou for q = 2, and byMeda, Sjögren and Vallarino (MSV) in general, that thisphenomenon is typical of (p,∞)-atoms, and it does not occurfor any p if q <∞.

Recent developments

In 2005 Bownik exhibited a linear functional, defined on(1,∞)-atoms and uniformly bounded on them, which is notcontrolled by the H1,∞-norms of the same atoms, and thereforedoes not admit a continuous extension to H1.

It was recently shown by Yang and Zhou for q = 2, and byMeda, Sjögren and Vallarino (MSV) in general, that thisphenomenon is typical of (p,∞)-atoms, and it does not occurfor any p if q <∞.

F p,q versus Hp,q norms

It is natural to introduce at this point the spaces F p,q of finitelinear combinations of (p,q)-atoms, with the norm

‖f‖F p,q = inf{‖λ‖`p : f =

∑finite

A linear functional that is uniformly bounded on single(p,q)-atoms automatically extends boundedly to F p,q.

MSV proved that the F p,q-norm and the Hp,q-norm areequivalent on F p,q when q is finite, in contrast with the q =∞case.Therefore, for q <∞, the completion of F p,q is Hp and thisexplains the absence of Bownik-type examples for q <∞.

F p,q versus Hp,q norms

It is natural to introduce at this point the spaces F p,q of finitelinear combinations of (p,q)-atoms, with the norm

‖f‖F p,q = inf{‖λ‖`p : f =

∑finite

A linear functional that is uniformly bounded on single(p,q)-atoms automatically extends boundedly to F p,q.

MSV proved that the F p,q-norm and the Hp,q-norm areequivalent on F p,q when q is finite, in contrast with the q =∞case.Therefore, for q <∞, the completion of F p,q is Hp and thisexplains the absence of Bownik-type examples for q <∞.

Sums of continuous atoms

Let Hp,c (resp. F p,c) be the space of finite linear combinations(resp. finite l.c.) of continuous (p,∞)-atoms, with the norm

‖f‖Hp,c = inf{‖λ‖`p : f =

∞∑j=0

λjaj , aj continuous (p,q)-atoms},

‖f‖F p,c = inf{‖λ‖`p : f =

∑finite

λjaj , aj continuous (p,q)-atoms}.

The completion F p,∞ of F p,∞

Lemma (MSV). The Hp-, F p,∞-, F p,c-norms are all equivalenton F p,c , and F p,c is dense in Hp. Hence the completion of F p,c

is Hp.

Consider the inclusion maps

F p,c T−→ F p,∞ U−→ Hp ,

and their continuous extension to the completions,

Hp eT−→ F p,∞ eU−→ Hp .

Since U ◦ T = I, we have the following

Lemma. T (Hp) is closed in F p,∞, Np = ker U is nontrivial, and

F p,∞ = T (Hp)⊕ Np .

is Hp.

F p,c T−→ F p,∞ U−→ Hp ,

F p,∞ = T (Hp)⊕ Np .

is Hp.

F p,c T−→ F p,∞ U−→ Hp ,

F p,∞ = T (Hp)⊕ Np .

is Hp.

F p,c T−→ F p,∞ U−→ Hp ,

F p,∞ = T (Hp)⊕ Np .

The atomic structure of F p,∞

Theorem. Given any sequence of (p,∞) atoms aj and any`p-sequence of scalars λj , the series

∑∞j=0 λj aj converges in

F p,∞ to an element ξ such that ‖ξ‖peF p,∞ ≤∑∞

j=0 |λj |p.

Conversely, each ξ ∈ F p,∞ can be written as

ξ =∞∑

λj aj , (∗)

with each aj a (p,∞) atom and the sum converging in F p,∞.Moreover,

‖ξ‖fF p = inf( ∞∑

|λj |p) 1

where the infimum is taken over all decompositions (*) of ξ.

j=0 |λj |p.

ξ =∞∑

λj aj , (∗)

with each aj a (p,∞) atom and the sum converging in F p,∞.

Moreover,

|λj |p) 1

j=0 |λj |p.

ξ =∞∑

λj aj , (∗)

with each aj a (p,∞) atom and the sum converging in F p,∞.Moreover,

|λj |p) 1

Atomic series in F p,∞ and in Hp

Corollary. If a series∑∞

j=0 λj aj of (p,∞)-atoms converges inthe F p,∞-norm to ξ, then it converges to Uξ in the Hp-norm.

The non-trivial elements of Np are those ξ =∑∞

j=0 λj aj suchthat the series converges to 0 in Hp.

Formally, consider the “free atomic space” `p(Ap), where Apthe set of (p,∞)-atoms.Both Hp and F p are quotient spaces of `p(Ap), and theprojection onto Hp factors through F p.

Atomic series in F p,∞ and in Hp

Corollary. If a series∑∞

j=0 λj aj of (p,∞)-atoms converges inthe F p,∞-norm to ξ, then it converges to Uξ in the Hp-norm.

The non-trivial elements of Np are those ξ =∑∞

j=0 λj aj suchthat the series converges to 0 in Hp.

Formally, consider the “free atomic space” `p(Ap), where Apthe set of (p,∞)-atoms.Both Hp and F p are quotient spaces of `p(Ap), and theprojection onto Hp factors through F p.

The Stone space Rn

Denote by L∞0 (Rn) the space of elements of L∞(Rn) vanishingat infinity.Under pointwise multiplication, L∞0 (Rn) is a commutativeC∗-algebra without unit.

The Gelfand-Mazur theorem provides the existence of a locallycompact Hausdorff space Rn and of a canonical isometricisomorphism of C∗-algebras

L∞0 (Rn) 3 f 7−→ f ∈ C0(Rn) .

Rn is defined as the set of non-zero multiplicative linearfunctionals ϕ on L∞0 (Rn) with the weak-* topology.

The Stone space Rn

Denote by L∞0 (Rn) the space of elements of L∞(Rn) vanishingat infinity.Under pointwise multiplication, L∞0 (Rn) is a commutativeC∗-algebra without unit.The Gelfand-Mazur theorem provides the existence of a locallycompact Hausdorff space Rn and of a canonical isometricisomorphism of C∗-algebras

L∞0 (Rn) 3 f 7−→ f ∈ C0(Rn) .

The Stone space Rn

Denote by L∞0 (Rn) the space of elements of L∞(Rn) vanishingat infinity.Under pointwise multiplication, L∞0 (Rn) is a commutativeC∗-algebra without unit.The Gelfand-Mazur theorem provides the existence of a locallycompact Hausdorff space Rn and of a canonical isometricisomorphism of C∗-algebras

L∞0 (Rn) 3 f 7−→ f ∈ C0(Rn) .

The projection of Rn onto Rn

By the Riesz representation theorem, ϕ|C0(Rn)is given by

evaluation at a point x ∈ Rn.

The map π : ϕ 7−→ x is a continuous projection of Rn onto Rn.

For f ∈ C0(Rn), f = f ◦ π.

The projection of Rn onto Rn

By the Riesz representation theorem, ϕ|C0(Rn)is given by

evaluation at a point x ∈ Rn.

The map π : ϕ 7−→ x is a continuous projection of Rn onto Rn.

For f ∈ C0(Rn), f = f ◦ π.

A description of Rn - I

A continuous linear functional ϕ on L∞0 (Rn) is uniquelydetermined by the values it takes on characteristic functions ofbounded sets.

If ϕ is multiplicative, ϕ(χE) is either 0 or 1.

Then ϕ is uniquely determined by the family of boundedmeasurable sets (modulo sets of measure 0)Eϕ = {E : ϕ(χE) = 1}.

A description of Rn - I

A continuous linear functional ϕ on L∞0 (Rn) is uniquelydetermined by the values it takes on characteristic functions ofbounded sets.

If ϕ is multiplicative, ϕ(χE) is either 0 or 1.

Then ϕ is uniquely determined by the family of boundedmeasurable sets (modulo sets of measure 0)Eϕ = {E : ϕ(χE) = 1}.

A description of Rn - II

The set E = Eϕ = {E : ϕ(χE) = 1} satisfies the followingproperties:

• E is nonempty;• E ∈ E , E ′ ⊃ E =⇒ E ′ ∈ E ;• E ,E ′ ∈ E =⇒ E ∩ E ′ ∈ E ;• if E ∈ E is the disjoint union of E1 and E2, then one and

only one between E1 and E2 is in E .

Conversely, for any family E with the above properties, there isa multiplicative functional ϕ such that E = Eϕ.

(It is easy to prove that, given E as above, there is a uniquepoint x ∈ Rn such that E contains every neighborhood of x .)

The set E = Eϕ = {E : ϕ(χE) = 1} satisfies the followingproperties:• E is nonempty;

• E ∈ E , E ′ ⊃ E =⇒ E ′ ∈ E ;• E ,E ′ ∈ E =⇒ E ∩ E ′ ∈ E ;• if E ∈ E is the disjoint union of E1 and E2, then one and

The set E = Eϕ = {E : ϕ(χE) = 1} satisfies the followingproperties:• E is nonempty;• E ∈ E , E ′ ⊃ E =⇒ E ′ ∈ E ;

• E ,E ′ ∈ E =⇒ E ∩ E ′ ∈ E ;• if E ∈ E is the disjoint union of E1 and E2, then one and

The set E = Eϕ = {E : ϕ(χE) = 1} satisfies the followingproperties:• E is nonempty;• E ∈ E , E ′ ⊃ E =⇒ E ′ ∈ E ;• E ,E ′ ∈ E =⇒ E ∩ E ′ ∈ E ;

• if E ∈ E is the disjoint union of E1 and E2, then one andonly one between E1 and E2 is in E .

The set E = Eϕ = {E : ϕ(χE) = 1} satisfies the followingproperties:• E is nonempty;• E ∈ E , E ′ ⊃ E =⇒ E ′ ∈ E ;• E ,E ′ ∈ E =⇒ E ∩ E ′ ∈ E ;• if E ∈ E is the disjoint union of E1 and E2, then one and

The Stone space of a ball

Consider now a ball B ⊂ Rn. By the Gelfand-Mazur theoremagain,

L∞(B) ∼= C(B) ,

where now B is a compact space. In fact, there is a naturalinclusion of B into Rn that makes the following diagramcommutative:

B −→ Rn

−→πB π

−→

B −→ Rn

About B

B consists of those multiplicative functionals ϕ such thatϕ(χB) = 1.

B is closed and open in Rn, π(B) = B, and

π−1(oB) ⊂ B ⊂ π−1(B) ,

with proper inclusions.

The measure on Rn

For f ∈ L∞0 (Rn), supp f ⊂ B if and only if supp f ⊂ B. Inparticular, L∞c (Rn) = Cc(Rn).

There is a unique positive Borel measure m on Rn such that∫Rn

f dm =

∫cRn

for every f ∈ L∞c (Rn).

Lemma. The map f 7−→ f extends to an isomorphism fromL1

loc(Rn) onto L1

loc(Rn). In particular, every locally boundedm-measurable function on Rn coincides m-a.e. with acontinuous function.

The measure on Rn

f dm =

∫cRn

loc(Rn) onto L1

The measure on Rn

f dm =

∫cRn

loc(Rn) onto L1

The dual space of F 1,∞

Theorem.• Let ` be a bounded linear functional on F 1,∞.

There exist a function b ∈ BMO(Rn) and a Radon measureµ on Rn, singular with respect to m, satisfying

|µ|(B) ≤ C m(B), for each ball B, (1)

such that, if f =∑

j λjaj ,

`(f ) =∑

∫aj(b − bBj ) dm +

∫f dµ, f ∈ F 1,∞. (2)

• Conversely, if b and µ are as above, then the identity (??)defines a bounded linear functional on F 1,∞ and

‖`‖(eF 1,∞)∗

∼= ‖b‖BMO + supB

|µ|(B)

The dual space of F 1,∞

Theorem.• Let ` be a bounded linear functional on F 1,∞.

There exist a function b ∈ BMO(Rn) and a Radon measureµ on Rn, singular with respect to m, satisfying

|µ|(B) ≤ C m(B), for each ball B, (1)

such that, if f =∑

j λjaj ,

`(f ) =∑

∫aj(b − bBj ) dm +

∫f dµ, f ∈ F 1,∞. (2)

• Conversely, if b and µ are as above, then the identity (??)defines a bounded linear functional on F 1,∞ and

‖`‖(eF 1,∞)∗

∼= ‖b‖BMO + supB

|µ|(B)

Let B be a ball in Rn. Any function f ∈ L∞(B) with mean valuezero on B, is a scalar multiple of a (1,∞)-atom. More precisely,

‖f‖F 1,∞ ≤ ‖f‖∞m(B) .

Hence ` induces a linear functional on the space ofL∞-functions on B with mean value zero, with norm not greaterthan m(B)‖`‖.

By one-dimensional extension, this is the same as saying aone-parameter family of linear functionals on L∞(B), alldiffering one from the other by a scalar multiple of f 7−→

∫B f .

‖f‖F 1,∞ ≤ ‖f‖∞m(B) .

∫B f .

‖f‖F 1,∞ ≤ ‖f‖∞m(B) .

∫B f .

Proof, cont.dIdentify now L∞(B) with C(B).

By the Riesz representation theorem, ` can be represented on(1,∞)-atoms supported on B as an integral

`(a) =

∫bB a dνB ,

with νB a Borel measure on B, and as well by any othermeasure νB + cχbBm with c ∈ C.

By a standard construction, we obtain a measure ν on Rn,unique up to contant multiples of m, such that

`(a) =

∫cRn

a dν ,

for every (1,∞)-atom a, and

‖`‖ ∼= supB

∣∣∣ν − ν(B)

m(B)m∣∣∣(B) .

Proof, cont.dIdentify now L∞(B) with C(B).By the Riesz representation theorem, ` can be represented on(1,∞)-atoms supported on B as an integral

`(a) =

∫bB a dνB ,

`(a) =

∫cRn

a dν ,

‖`‖ ∼= supB

∣∣∣ν − ν(B)

m(B)m∣∣∣(B) .

Proof, cont.dIdentify now L∞(B) with C(B).By the Riesz representation theorem, ` can be represented on(1,∞)-atoms supported on B as an integral

`(a) =

∫bB a dνB ,

`(a) =

∫cRn

a dν ,

‖`‖ ∼= supB

∣∣∣ν − ν(B)

m(B)m∣∣∣(B) .

Proof, cont.d

Let ν = G m + µ be the Lebesgue decomposition of ν w.r. to m(with G unique up to constants). Then

‖`‖ ∼= supB

∫bB∣∣∣G − ν(B)

∣∣∣dm + supB

|µ|(B)

By the Lemma, G = g for some g ∈ L1loc(R

Proof, cont.d

Let ν = G m + µ be the Lebesgue decomposition of ν w.r. to m(with G unique up to constants). Then

‖`‖ ∼= supB

∫bB∣∣∣G − ν(B)

∣∣∣dm + supB

|µ|(B)

By the Lemma, G = g for some g ∈ L1loc(R

The dual space of F p,∞, p < 1

Theorem. The bounded linear functionals on F p,∞, 0 < p < 1,extend uniquely to Hp(Rn). Thus (F p,∞)∗ = Hp(Rn)∗,0 < p < 1.

Proof. A repetition of the above argument leads to theexistence of a measure ν = G m + µ on Rn, where now

‖`‖ ∼= supB

m(B)1p

∫bB |G − P

ν,bB|dm + supB

|µ|(B)

m(B)1p

The condition |µ|(B) ≤ Cm(B)1p implies that µ = 0.

The dual space of F p,∞, p < 1

Theorem. The bounded linear functionals on F p,∞, 0 < p < 1,extend uniquely to Hp(Rn). Thus (F p,∞)∗ = Hp(Rn)∗,0 < p < 1.

Proof. A repetition of the above argument leads to theexistence of a measure ν = G m + µ on Rn, where now

‖`‖ ∼= supB

m(B)1p

∫bB |G − P

ν,bB|dm + supB

|µ|(B)

m(B)1p

The condition |µ|(B) ≤ Cm(B)1p implies that µ = 0.

Concluding remarks

In particular, the dual space of F p,∞ does not separate points.

Corollary. For p < 1, let T be a linear operator with values in aBanach space, defined on (p,∞)-atoms, and uniformlybounded on them. Then T admits a continuous extension toHp.

Concluding remarks

In particular, the dual space of F p,∞ does not separate points.

Corollary. For p < 1, let T be a linear operator with values in aBanach space, defined on (p,∞)-atoms, and uniformlybounded on them. Then T admits a continuous extension toHp.

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