QUESTIONS RELATED WITH - THE INVERSION OF THE FOURIER ...

QUESTIONS RELATED WITH -

THE INVERSION OF THE FOURIER TRANSFORM

IN DIMENSIONS GREATER THAN 1

FOR FUNCTIONS IN ZP

Emmanuel Montini

Presented for the degree of Doctor of Philosophy

University of Edinburgh 1998

-It iln U.1 an

Declaration

This thesis was composed by myself and has not been submitted for any other degree

or professional qualification. All work not otherwise attributed is original.

Acknowledgements

First of all I would like to express my gratitude to my supervisor, Prof. A. Carbery,

for all his help and encouragement during the last three years. I also want to thank him

and his co-author Prof. F. Soria of the Universidad Autónoma de Madrid to give me

the right to use some of their unpublished results [CS31 in this thesis.

My gratitude also goes to the Commonwealth Scholarship Commission in The United

Kingdom. By funding my studies and my expenses, they gave me the extraordinary

chance not only to study mathematics but also to discover a country and its culture.

To the people of the department, secretaries, librarians, computing officers, fellow

students, I say thanks for making these three years easier for me. I am specially

grateful to J. Bennett for putting a bit of life in our office and helping me correcting my

English.

On a more personal level, I want to thank my parents, my brother and my sister as

well as my friends in Canada and UK. Their constant support and affection (e-mails,

letters, visits, etc.) made me feel as if I never left Canada to pass these three years far

away from most of them.

11

Abstract

In this thesis, we will study different questions related with the inversion of the

Fourier transform for some special subspaces of LP (R') when n ~! 2.

It is known [CS 1] that SRf(x) --> f(x) almost everywhere if f is in the space of

Bessel potentials Z(1R") for 2 :! ~ p < and a > 0. It is also known that in dimen-

sion one a better result holds ([B], [Hu] and [SZ]). More precisely, SRf(x) - f(x)

C p quasieverywhereif f is in ZP (1R) with l<p<oo and 0<a ~ . Most of this

thesis will be dedicated to the study of these results and to the explanation of how

some generalisations can be obtained in higher dimensions.

When p =2, we will in particular extend Beurling's result to all dimensions (i.e. we

will show that the pointwise convergence of SRf takes place Ca/k 2 -quasieverywhere

for all k >1 if f Z(") with 0< a ~ -). After proving a ç 2 -localisation

principle which reduces the study in the sharp case (k = 1) to functions in Z,(1R")

with bounded support, we will also extend Salem and Zygmund's result to all dimen-

sion under the stronger condition < a !!~ a (i.e. we will show that the previous

result is also true for k = 1 if < a :!~ fl.

When 2 <p < , the Col/k, 2 and C,1 ,2 results just mentioned are also true for functions

in Z(1R"). Moreover, it is possible to extend the Ca2 result to a C theorem like in

IR, even though Hunt's extension of Carleson's theorem can not exist when n> 1. In

order to prove this Cap theorem, we will in particular construct an everywhere

localisation principle for functions in Z 1 (IR') when 2 ~ p < r and ~ a ~

With a technique similar to this last localisation result, we will also see that different

improvements are possible for the C a2 -localisation principle mentioned above.

Moreover, with a different technique and under an additional support condition, we

will realise that some improvement can be made not only on our evaluation of the size

of the set of divergence in the localisation problem, but also on the generality of the

class of functions and operators considered.

When p =2 and a> , it is common knowledge that SRf(x) converges uniformly to

f(x) on IR". Here f is the continuous representative of f. A natural question is then:

Ill

"how uniformly" is the convergence taking place when 0 < a :!~ -? We will in this

thesis try to provide some intuition on that problem and on what should be the correct

conjecture in this case. With a similar frame of mind, we will also study rapidly some

other related problems (e.g. the case 1 1 <p < 2 of all the questions raised up to now).

Lk'A

"La vraie faute est celle qu'on ne corrige pas." (Confucius)

V

Contents

Declaration I

Acknowledgements II

Abstract III

Contents VI

List of symbols viii

1 Introduction and Background 1

1.1 Notation and Conventions .................................................................... 2

1.2 General Tools .......................................................................................... 2

1.3 On Two Function Spaces ........................................................................ 5

1.3. 1 Riesz Kernel and Spaces of Riesz Potentials 5

1.3.2 Bessel Kernel and Spaces of Bessel Potentials ........................... 6

1.4 Introduction to Capacities (Potential Theory) ........................................ 8

1.4.1 Some Definitions ..................................................................... 9

1.4.2 Basic Properties and Useful Results ............................................ 11

1.4.3 Quasicontinuity ........................................................................... 13

1.5 Related Classical Results in Fourier Analysis ......................................... 15

1.5. 1 The Historical Results ................................................................. 15

1.5.2 The One Dimensional Capacitarian Study ................................... 18

2 The Z 2 Case 21

2.1 Carbery and Soria's Result ................................................................21

2.2 Motivations in Higher Dimensions ...............................................25

VI

2.3 Some Direct Computations 31

2.3. 1 An Extension of Beurling's Result ........................................31

2.3.2 A Ca2 Localisation Principle ...................................................... 35

2.4 The Sharp Capacitarian Result when p =2 and a > '........................ 40

3 Localisation Principles 50

3.1 On Some Known Localisation Principles ............................................. 50

3.2 An Approach to Localisation Based on Fefferman's Folk Computation 58

3.2. 1 Fefferman's Folk Computation ................................ 58

3.2.2 On a Closed Sets Condition ....................................................... 64

3.2.3 An Everywhere Localisation Principle ...................................... 70

3.3 Some Improved Localisation Results ...................................................... 72

3.4 A Capacitarian Study for a General Class of Functions .......................... 74

4 A Collection of Related Results 78

4.1 On a Question Related with Uniformity ..................................................78

4.2 Interpolation (The Case when 2 < p < and 0< a ~ !!iI) . 84

4.3 Some Connections with the Bochner-Riesz Operators ..........................87

4.3. 1 Some "Forecasts" Based on Bochner-Riesz ...............................87

4.3.2 The ZP Case when p <2 ..........................................88

4.4 Retrospective .......................................................................................... 95

Appendix 97

Bibliography 102

List of Symbols

The following symbols will be used throughout this thesis. The notions represented by

these symbol will be defined when they will first appear in the thesis if more than one

definition exists or if they are not part of a standard mathematical background.

Spaces of numbers and related operations

IN Natural numbers (0 excluded).

IR" Euclidean space of dimension n.

IR The positive real numbers (0 excluded).

T" The n dimensional cube with side 2,r in IR" (centred at the origin).

Integers.

S' 1 Unit sphere in IR".

Re(z) Real part of z E C (where C is the complex plane).

Im(z) Imaginary part of z E C.

Max(x,y) Maximum of x and y.

Min(x,y) Minimum of x and y.

x y Standard Euclidean scalar product (x 1 y 1 + x2 y2 +. . . +xy).

Measure snaces and capacities

dx Lebesgue measure.

Ii' L -space with respect to the Lebesgue measure.

If II L -norm of f with respect to the Lebesgue measure.

L Subspace of LP composed of all the functions in L" positive a.e..

L (dii) L -space with respect to the measure dji.

L(E,dp) Spaces of functions f such that ZEf is in L"(d1u).

ii! IIL(d/1) L -norm of f with respect to the measure dp.

uuI

L''(L2 ) The mixed norm space LA' radial and L2 angular.

IIfIILp(m LP(L2 )-norm of f.

LP. Sobolev space of order a in L.

ZP Space of Bessel potentials of order a in L".

ZP Space of Riesz potentials of order a in L".

Lp, q Lorentz space of order p. q.

L Space of functions whose derivatives up to order a are in L"

ITlI, q Norm of the operator T: L' - L'.

C (E) Capacity of order a, p of E with respect to the Bessel kernel.

ea, ( E) Capacity of order a,p of E with respect to the Riesz kernel.

Other spaces

C(E) Space of continuous functions on E.

Ck Space of functions with all their derivatives up to order k continuous.

C'° Space of functions with all their derivatives of any order continuous.

C Space of continuous functions with compact support.

C' Subspace of C where all the functions have compact support.

C° Subspace of C° where all the functions tend to 0 at infinity.

fl1. 4 (E) Space of positive Borel measures supported on the set E.

S Schwartz class.

Basic sets and operations onsets

B(x, r) Ball of centre x and radius r in IR".

B(r) Ball centred at the origin of radius r in R.

B(x, r) Closed ball of centre x and radius r in IR".

AC Complement of the set A.

dia,n(A) Diameter of the set A.

dist(A, B) Distance between the sets A and B.

A Interior of the set A.

Ix

IAI n-dimensional Lebesgue measure of the set A c IR" -

IAL k-dimensional Lebesgue measure of the set A c IR" where k <n.

Operations on functions

0(f (x)) Big order of f (g = 0(f) at a if urn sup = x—)a

f* g Convolution of f and g.

f Fourier transform of f.

f Inverse Fourier transform of f.

f A quasicontinuous representative of f.

AE Restriction of f to E.

o(f(x)) Small order of f (g = o(f) at a if lim.%- = 0). x-4a

Special functions

k (x) Bessel function of the first kind of order k.

G. Bessel kernel of order a.

F(x) Gamma function.

'a Riesz kernel of order a.

Special operators

K R Dirichiet kernel of radius R.

K 45 Kernel of the Bochner-Riesz mean of order 5 and radius R.

SNf , Skf N th (kth) partial sum of the Fourier series of f.

SRf Rth spherical partial integral of the Fourier integral of f.

Sf R th cubic partial integral of the Fourier integral of f.

Sf Rth rectangular partial integral of the Fourier integral of f.

Sf Rth partial integral of the Bochner-Riesz mean of order 8 of f.

Mf Hardy-Littlewood maximal operator applied to f.

Sf The maximal operator associated with SRf or SNf (Skf).

S., f The maximal operator associated with SRf (R k is a lacunary sequence).

Chapter 1

Introduction and Background

In 1915, Lusin conjectured that

(1.0.*) lim SNf(x) = urn I j(k)e = f(x) N- N_k_N

almost everywhere for every f € L(T). Here f(k) = f(x)edx. This difficult 2n fir

problem stayed open for half a century before being finally proven by Carleson [Cal]

in 1966. In the process to solve it, different natural questions were raised. One of

these questions asks whether the size of the set where 1.0.* holds is "bigger" in some

sense when f "has some smoothness".

In one dimension, this question was positively answered partially by Beurling [B] in

1939 (Theorem 1.5.7) and completely by Salem and Zygmund [SZ] seven years later

(Theorem 1.5.8). This thesis will generalise to all dimensions the result of Beurling

and provide different extensions of Salem and Zygmund's theorem in higher

dimensions. Some new questions which arise naturally in the higher dimensional

setting will also be considered.

To avoid confusing the reader, we choose in this thesis to put as many details as

possible. This should help in providing a good picture of what is happening. The only

prerequisites necessary to follow this text are a standard background in analysis and a

basic knowledge of measure theory (definitions, properties of L" spaces, etc.).

The thesis has four main divisions. This first chapter will provide the general

background needed. Chapter 2 will present different motivations and partial results for

the question raised at the beginning in the context of the space Z (see Section 1.3.2),

while Chapter 3 will study the localisation principles related with the inversion

problem (1.0.*). Finally, Chapter 4 will extend the study made in Chapter 2 to

(p # 2) and also raise some closely related questions.

1

1.1 Notation and Conventions

The notation used for the definitions, lemmas, theorems, etc. is a sequence of three

numbers separated by dots. The two first numbers represent respectively the chapter

and the section where this element appear while the third digit is there to differentiate

chronologically two elements in the same section. Occasionally, the third digit will be

replaced or followed by a symbol to denote an important element inside a proposition.

When a computation required in the text to explain an idea is also needed later, we will

highlight this calculation by giving it the name "Computation".

In this text, the almost everywhere property will be assume to hold with respect to the

Lebesgue measure if there is no contrary specification. This property will be

sometimes represented by "a.e.".

When the range of values for n, the dimension of IR", is not specified, this will mean

that the result under study holds for all possible values (i.e. n ~! 1).

C and c will denote constants which can change value from one line to the next one.

If an index is also given (e.g. C) it will signify that the constant is dependent on that

quantity. In the proof of the classical results, we will often forget the normalisation

constants (e.g. the constant for the Fourier transform on T) as they are not important

for our study.

1.2 General Tools

In order to obtain the pointwise results desired, we will have to study some "maximal

functions" naturally associated with our problems and try to show that they satisfy

some boundedness property. A maximal function is generally obtained by replacing a

limit in the problem under study by a supremum. An example of such a function is the

Hardy-Littlewood maximal function

Mf(x) = SD IB(x,ñI SB(x ,r) If (Y )IY r>O

which is naturally associated with the problem in Corollary 1.2.2 below. The kind of

boundedness property wanted in this case is given by (2) and (3) of the following

theorem:

2

THEOREM 1.2. 1. (Hardy and Littiewood's maximal theorem)

Let 1 :5 p S: 00• If f c LP(1R) then Mf is finite almost everywhere.

If fE L'(") then {xE]R:Mf(x)>a}!~ -IIfII 1 for all a>O. C,, is a

constant depending only on n.

Let l<p!!~ oo. If f LP W) then liMfIlp 5; 11P p with a constant C

I

depending only on p and n.

COROLLARY 1.2.2. (Lebesgue's differentiation theorem) If f is a locally integrable

function then

lim IB(x,r)l 5B(ryf(Y)y = f(x) r-+O

almost everywhere on

PROOF.

The proof of Theorem 1.2.1 is done by obtaining (2) first using a covering lemma

argument and then (1) and (3) are deduced from (2) by splitting f into a large and a

small parts (for the details see [St3, pp. 6 -711).

The corollary's proof is more interesting in the context of this thesis as it shows the

classical way to deduce a pointwise result from the boundedness property of a

maximal operator.

We assume without loss of generality that f E L' because, if it is not the case, we can

cover IR" with a countable union of balls, {B 1}, and replace f by xf on each

B.

To simplify the notation let f (x) = IB(x,r)J f(y)dy for r> 0. Now, JIfr - f01 - 0

when r - 0 because IB(x,r)r1xB(x,r) is an approximation of the identity [St3, pp.62-

65]. Consequently, it is possible to find a subsequence f,. (x) - f(x) almost

everywhere when , - 0. Hence, the result will be proven if it can be shown that

lim f, (x) exists almost everywhere. r-O

In order to show this, we split f in two parts f E C. and f E L' where the L' -norm

of f can be made as small as we want by redefining an appropriate

Let where gEL' and gr is defined like f• r-O r-O

'I

f e C imply that Af, = 0 as (f,), (x) - f, (x) uniformly. While, by Theorem 1. 2.2

(2) and Af1 (x) !!~ 2Mf1 , we have

I{x E IR":Af1 (x)> 2L}I !~ IfX € IR":Mf1 (x) > !~ LS-11f,11 1 . Consequently, I{x € IR":Af(x) > ~ %-f1 as Af !!~- Af, + Af1 . By making f smaller

and smaller, this implies that urn f, (x) exists almost everywhere. r-O

For what follows, we also need to define three important tools:

DEFINITION 1.2.3. The Fourier transform of a suitable function f is given on IR"

by

f()=$f(x)e_2dx, EIR",

and on 7 by

f(k) = J f(x)edx, k E Z. ir

DEFINITION 1.2.4. The inverse Fourier Transform of a suitable function f is

defined on IR" by

f() = f(—) = .L f(x)e 2'd., for € ]R".

When f E L (IR") these two operations are related by Plancherel's theorem which

says that for such functions: hf 112 = 1142 - DEFINITION 1.2.5. The convolution of tvo suitable functions f and g is given by

(f*g)(x) = SE" f(x - y)g(y)dy.

These three operations (1.2.3, 1.2.4 and 1.2.5) are defined for functions for which

they make sense (e.g. f € 8) and are related by (f*g)"() = f(),). A simple and

useful estimate for the product of convolution is Young's inequality:

THEOREM 1.2.6. Let 1 15 p,q, r oo and 1 = .i + -- —1 then r p q

11f* 9 1Ir !' IIfIIpII&II q

This result can be obtained by interpolating between three easy (p, q, r) -cases: (1,1,1),

(1,00,00) and (co,1,00). As we will see in Section 1.5, the Fourier transform can be

used to represent some functions on T. This representation is a series:

DEFINITION 1.2.7. Let f be a real function defined on (—r,ir). The Fourier series of

f at the point x is f(k)e th or, equivalently, - + (ak cos(kx) + k sin(kx)) k= 2

ir

where a ff(y)cos(ky)dy and bk = 7Jf(y)sin('Y)dY

1.3 On Two Function Spaces

Throughout this thesis, we will be working extensively with the spaces of Bessel

Potentials and the closely related spaces of Riesz Potentials. We will define in this

section these spaces and some of their properties which will be needed later.

1.3.1 Riesz Kernel and Spaces of Riesz Potentials

DEFINITION 1.3.1. For 0< a <n, the Riesz kernel of order a in IR", 'a' is defined

to be the inverse Fourier transform of i•r" or, equivalently,

Ia(x) = r(n_a)(rI22ar()IxI )

A simple computation using this last definition shows the following properties of 'a :

PROPOSITION 1.3.2. Let 11

(I)(x)dx<oo only if 0<a<.

The convolution of I. with a function in an appropriate function space (e.g. an LP

space) defines an operator 6 ,, called the Riesz potential of order a. This operator is

bounded2 from L"(IR") to 1Y(]R")ifO<a<n, 1!!~ p<q<zoo and

DEFINITION 1.3.3. The space of Riesz potential of order a in IR" is defined by

(lR") = {f: f = (g) = I*g a. e. for some g € Jf(IR")}, 0< a < n.

In what follows, we will see that even if the Riesz potentials are suitable to 'smooth" a

function, their kernels have an important lack of decay for our purposes. This will be

particularly apparent in Propositions 2.3.1 and 2.3.3 of Chapter 2. To overcome this

problem we will most of the time use the Bessel potentials. These are closely related

1 In this thesis, the coefficient of normalisation, -i-, will often be forgotten.

2 This is known as Hardy, Littlewood and Sobolev's theorem of fractional integration [St3, pp.119.1211.

5

operators which preserve the nice 'local behaviour of the Riesz potentials with an

improved global behaviour.

1.3.2 Bessel Kernel and Spaces of Bessel Potentials

DEFINITION 1.3.4. For 0< a !!~ n, the Bessel kernel of order a in , Ga , is 2 -a/2

defined to be the inverse Fourier transform of (i +1.1 ) or, equivalently,

Ga (x) = (4)_a/2 (F(.-))' J e'2 /ze_z/41r7(a_n)12_ldZ 0

For our purposes, we will need the following properties of the Bessel kernel:

PROPOSITION 1.3.5.

Ga EL'(R)forall a>0, moreover IlGaIIi =lforali a>0,

G*Gjj = Ga,jj if a, fl > 0 (the same is true for the Riesz kernel),

Ga (X) = Ia (X)+ o(IxI") near the origin for 0< a <n,

G(x) = o(log) near the origin,

G0 (x) = O(e 4 ) with c >0 when lxi ----> 00,

Let 11

(G)(x)dxazooforaii

The first definition above gives (1) as IVa Iii = Ga (0) = 1 and (2) as (f*g)A = f. (3)

and (4) are consequence of the second definition because 'a can be expressed in the

form

(4)_a/2 (F(..))1 5 e_2zz_n2)2_1dz.

(5) is also obtained by studying the second definition, while (6) uses (3) and (5) (For

detailed proofs see [St3, pp.131-1133] or [AH, pp. 10-12]). Some equivalent

definitions as well as some other properties of Ga can be found in chapter 1 of [AH]

and in chapter 5 of [St3]. We should point out that it is (5) which provides us with the

nicer behaviour alluded above.

As in the previous section, the Bessel potential of order a in IR" is defined to be the

convolution of Ga with some appropriate functions.

Al

DEFINITION 1. 3.6. The space of Bessel potentials of order a in IR" is defined by

(1R") = {f: f = G*g a.e. for some g E LP(lRn)}, a> 0.

It is sometimes also useful to define Z(lR") = L"(IR") (G0 *f = f in this case). The

Js given by IIgII.

Proposition 1.3.5 (2) directly implies that ZP c ZP when /3> a, while Young's

inequality implies the L" -boundedness of the Bessel potential (i.e. IIGa * fII :!~ 11f II) as

IVa IL = 1. A really important feature of these spaces is their interaction with the

classical Sobolev spaces.

aa DEFINITION 1.3.7. The weak derivative = a in the sense of distribu

dx a dx1 '••• dx

tions of a locally integrable function f on IR" is the locally integrable function g on

IR" (defined up to a set of measure 0) such that

.L f (x)9(x)dr = (_1)aSR, g(x)ço(x)thc, Vq E C°,

where 4j. is the classical - derivative of q. The weak derivative is equal to thedXa

classical derivative if this derivative of f exists.

DEFINITION 1.3.8. The Sobolev space of order k in L"(IR") is defined as follows:

LPk = {f E L': weak 2Lf exists and is in L, Va such that IaI :!~ k}. dxa

The norm, 11fli p., on LP, is given by IIfII + I 1~ do' fl~ _ 1:5IaI5

The most important result about these spaces is Sobolev's imbedding theorem:

TI{EOREM1.3.9. Let k>O and l<p<oo

If pk <n then L(IR") IY(]R") Vq E[p,kI with a continuous imbedding.

If pk = n then Lpk LY(IR"), Vq E [p,00), with a continuous imbedding.

If pk > n then L (IR") C(IR") with a continuous imbedding.

The proof of Theorem 1.3.9 can be found in [SO, pp. 124-130] or in any classical

book on the subject. We can now return to our main concern the relation between these

spaces and the spaces of Bessel potentials.

VA

THEOREM 1.3.10. For k e IN and 1< p<-, f L(IR") if and only if f (IR")

and the corresponding norms are equivalent. For k even and n >1, f e (R") if

f € 4(]R") and, when k is even and n =1, this last part is an if and only if.

The proof is done by showing that if f € Z' then f and all its weak derivatives of

first order are in ZP and then the result follows by repeating this up to ZP = L" =

(for the details see [St3, pp. 135-138, 1 60]).

1.4 Introduction to Capacities (Potential Theory)

In this thesis, we will be mainly concerned with the behaviour of functions in some

"nice" spaces (spaces of Bessel potentials, spaces of Riesz potentials, ...) when

convolved with the Dirichlet kernel, KR(x) = .L<R Like for the Hardy-

Littlewood maximal function (Section 1.2), the partial sum operator (SRf = KR * f)

when applied to these functions tends to have a better pointwise convergence than

when it is applied to general If' functions. A natural question is thus to try to give the

best possible characterization of the set where such a function converges pointwise or,

equivalently, to characterise the set where we don't have such a convergence.

As we will see in Section 1.5 (Carleson's theorem), these sets are of null Lebesgue

measure for a general H' functions. We consequently need some tools to differentiate

between such sets. One way will be to look at the dimension of these sets using

techniques like the Hausdorff measure and dimension (see [Ro] for more information

about this subject). Another will be to use capacities. As we will see in Section 1.5 and

in the remaining of this thesis this is a better choice in our case.

Capacities have their origin in potential theory, a part of physics concerned with what

happens to a charged particle placed at a certain distance from the origin (see for this

any classical book of physics). The concepts of potential and energy of that theory

have been given general mathematical definitions which are the basis of the

mathematical part of potential theory.

This section does not intend to give a deep insight of these matters but just to explain

the basic tools which will be needed later, consequently, no proofs will be given but

references will be provided. For more details, the reader is invited to refer to classical

books like [Ca2, Sections I and 111], [KS, Chapter III] and [L] or to [AH] for a

ro]

modern approach. The notations and references will be essentially based on Adams

and Hedberg's book, [Al-I], and we will try to keep their presentation in mind to ease

such complementary readings. Consequently, and unlike most of the other parts of this

thesis, the references made are not to the original author.

1.4.1 Some Definitions

Before defining what a capacity is we need some definitions.

DEFINITION 1.4.1. Let X be a topological space. A function f : X ---> (_oo,00] is said

to be lower seinicontinuous on X if for every )i. E IR the set {x E X:f(x) !! ~ Al is

closed (for more information about this matter see [Br, p.8]).

DEFINITIONS 1.4.2. Let (X,3, v) be a measure space; by this we mean that X is a

space, v is a positive measure on X and 3 is an associated family of measurable

sets.

In what follows, a kernel on IR" x X will always mean a non negative function

K such that K(., y) is lower semicontinuous on IR" for each y E X and

K(x,.) is measurable on X for each x E IR".

Let p e 1fl., (]R") and f be a non negative v-measurable function.

The potential of f with respect to the kernel K is then defined by

9f(x) = JK(x,y)f(y)dv(y), XE

while the potential of p with respect to K is defined by

9p(y)=JK(X,y)dp(x). yEX.

Both potentials (b) and (c) have values in ]R = [O,00] and they are related by the

mutual energy of f and p. This important quantity is defined by

K(111f) = 5 (9f)dp =J(9p)fdv.

A particular case of it happens when f = ()q_1

In this case, (kj,(§P) q-1 )

is

called the energy of p and we will symbolise it by IK(P).

An important feature of (b), (c) and (d) is that they are all lower semicontinuous:

PROPOSITION 1.4.3. [AH, pp.24-25] In the setting of Definition 1.4.2, for f and y

fixed, we have

x i—* 9f (x) is lower semicontinuous on

u i-3 9/1(y) is lower semicontinuous on Tfl,(1R") in the weak* topology,

P i- K(u'f) is lower seinicontinuous on 1 4 (]R") in the weak* topology.

So what is a capacity? Informally, a good way to think of a capacity is to see it as a

kind of "fmer" measure. More precisely, the following definition can be given:

DEFINITION 1.4.4. For 1 :! ~ p <co and E c IR", the p-capacity of E with respect to

the kernel K, CKP(E), is defined by

CKP(E) = inf{$ fv : f E LP, (v) and 9f(x) ~! 1 Vx E E}.

In the special cases were K is the Bessel kernel, Ga, or the Riesz kernel, 'a' the

corresponding capacities will be denoted by Cap and by e.,p respectively.

DEFINITION 1.4.5. A Suslin set in IR" is a set obtained by taking the perpendicular

projection of a Borel set in n+1 While every Borel set is a Suslin set, the reverse is

true if and only if both the set and its complement are Suslin sets.

For these sets, some useful equivalent definitions of capacity can be established:

PROPOSITION 1.4.6. When 1 < p <00 and E c IR" is a Suslin set, Definition 1.4.4 is

equivalent to

CKP (E) =suP{/1(E): y TfL(A), A E and 11§11 11V(dv)!!~1 = i}). 'p q

PROPOSITION 1.4.7. When 1< p <00 and E ç IR" is a Suslin set, Definitions 1.4.4

and 1.4.6 are equivalent to

(CKP (E)) ' = (inf{IK (u): ji e 11t,(A), A E and p(A) = i}).

The equivalence between these definitions is covered in [AH, pp. 34-35] where 1.4.6

and 1.4.7 are obtained as a consequence of the minimax theorem. Some other

definitions are also given in [AH, Chapter 2] and for a more classical setting in [Ca2,

Chapter III]. In practice, Definition 1.4.4 gives generally shorter proofs for the kind of

problems covered by this thesis while Definition 1.4.6 is more easy to work with, but,

10

in any case, when one proof is found it is generally simple to transfer it to any of the

two other definitions.

Throughout this thesis, we will be concerned with a special class of kernels with more

properties than the one required in Definition 1.4.2. More precisely, we will ask our

kernels to be radially decreasing convolution kernels on x k". This means that

there is a non negative, non increasing and lower semicontinuous function

K0 : fl satisfying 5 K0 (x)x"'dx < oo such that K(x, y) = K0 (Ix - yl). Following

[AH], we will use the notation K(x) = K0 (lxi) from now on.

PROPOSITION 1.4.8. [Ali, pp.38-39] Let l<p<oo and let -+-=1. If K is a

radially decreasing convolution kernel then

(1) 5(K(x)rd <00 implies that CKP({xo})> 0, Vx0 €

5(3) (2) 5 (K(x)Ydx = 00 implies that CKP(E) = 0, VE

IxI>1

(K(x)rdx <00 implies that tEl =0 if C'K P (E) =0. IxI> 1

This proposition explains when a capacity will be useful to reach our goal to

differentiate between sets of null Lebesgue measure. (1) and (2) give us a restriction

on the kernel if we what to measure small sets, while (3) relate capacitarian and

Lebesgue measurements. In particular, combining this with Propositions 1.3.2 and

1.3.5 (6) imply that the Riesz kernel, 'a' defines a meaningful capacity when

o < ap < n, while the Bessel kernel, Ga, does the same when 0< ap !~ n.

1.4.2 Basic Properties and Useful Results

Throughout this text, we will often need to refer to some basic properties and results

about capacities. We will in this subsection try to provide a basic library of these tools.

PROPOSITION 1.4.9. [AH, pp. 25-261 Let CK P be a capacity defined as in the pre-

vious section and let E, lR", i=1, 2, ..., be afamily of sets.

If El E2 then C(E1 )!~ C(E2 ),

If E=UE1 then CKP (E):5:, CK , P (EI ).

This proposition means that a capacity is nearly a measure. The additivity of a measure

11

is missing but it is replaced by a subadditivity property, (2). It should be remarked that

this can not be improved.

A key result to understand the interplay between all the elements defined in Definition

1.4.2 is the following fundamental theorem:

THEOREM 1.4.10. [AH, pp.21-23, 37-381 Let suppose that we are in the setting of

Definition 1.4.2. Suppose also that X is a locally compact topological space and that the kernel K is such that the potential 9 q is continuous on IR" and lim 9 p(x) =0

IxI-

for all T e C, (X). Let 1< p < oo. If E c IR" with CK P (E) < then there is a unique

E ETPb() such that

9(fE)( x)~ 1 C-quasieverywhereon E

9(fE)( x)<1 on Support of/i'

pE() = S (Eidv = 5 9(fE)4E =

where fE_(9pE) and +=1.

E is called the capacitari an measure of E while fE is called the capacitarianfirnction

of E. fE is the only function in If+ (dv) such that the second member in (3) is equal

to CKP (E) [AH, p.28]. We should remark that (1) can not be obtained everywhere

while (2) is valid on the whole of IR" if we restrict ourselves to radially decreasing

convolution kernels by the boundedness principle:

PROPOSITION 1.4.11. [AH, p.391 If K is a radially decreasing convolution kernel

and /1 R + ( IR') then sup(K*p)(x) :!~ C sup (K*p)(x) where C is a constant XE xEsupp(p)

depending only on n.

DEFINITION 1.4.12. Let i e TL(IR"). The function V, = 9((9)q

)

is called the

nonlinear potential of p when I + -- =1.

If K is a Riesz or a Bessel kernel and p = 2 then VK1,P is a linear potential by

Proposition 1.3.5 (2). Equipped with this definition, we can now give the classical

definition of C-capacity (equivalent to Definition 1.4.4):

PROPOSITION 1.4.13.[AH, p.36] Let E be a compact set of IR" and let l<p <00•

Then

12

CKP(E) = max{j1(E): i ¶L 4 (E) and V(x) :!~ 1 Vx E supp(p)}.

Another important aspect for us will be to understand how two capacities interact

especially in the context of the Bessel and Riesz kernels. At a really basic level, we can

see from Definition 1.4.4 that if two kernels K 1 and K2 are such that K 1 :!~ K2 then

CK2P (E) !!~ CKP (E) for E ç 1R'. In particular, this implies that Cap(E) :5 Ca , p (E)

while, to compare Bessel and Riesz capacities between themselves, we have the

following theorem:

THEOREM 1.4.14. [AH, pp. 148-1501 Let O2ap represent either the Riesz or the

Bessel capacity of order (a, p). Suppose also that E 1R is an arbitrary set with

diam(E) :!~ 1 (this restriction is not needed for the Riesz capacities). Then there are

constants c such that

( pq(E))lI(n$) <_

1/(n—ap) if 0<13q< ap<n,

l jq (E) !~ cCap(E)

if f3q=ap<n and p<q,

[log C(E)j <

c(Cap(E)) ' cO</3q< ap=n,

(Cpq (E)) 1 <_ c(Cap (E))'1

if J3q=ap=n and pczq.

We should remark that this is the best which can be done in all cases as there exists

sets E such that C q (E) =0 and ,, p (E) >0.

As mentioned at the beginning of Section 1.4, two tools can be used to study small

sets, capacities and Hausdorff dimension. We will not really study their interplay in

this thesis and so we refer the interested reader to chapter 5 of [AH]. [Ca2] and [KS]

contains also some interesting, but short, surveys about this interaction.

1.4.3 Quasicontinuity

We will now try to make clear why C -capacities are well adapted to study functions

in Z P (R)

DEFINITION 1.4.15. Let CKP be a capacity on IR". A function f defined CK P -

quasieverywhere on an open set of IR" is said to be CK,, -quasicontinuous if for all

e >0 it is possible to find an open set 0 such that CKP(0) < e and f 10 E C(0') .

13

If we restrict our attention to Z(1R") functions, we remark that we can characterise

them in term of their Cap -quasicontinuity:

PROPOSITION 1.4.16. [AM, p.156] Let g E Lf(1R), 1< p< oo and a>0. Then

G*g is (a,p)-quasicontinuous.

In particular, this and Definition 1.3.6 imply that every functions f E Z(1R") has an

(a,p)-quasicontinuous representative. This Cap -quasicontinuity provides us with a

complement to Sobolev's imbedding theorem (Theorem 1.3.9 (3)) when 0 < a ~ .

We will see in the next three chapters that this turns out to be the exact property needed

to explain why spherical partial summations of functions in ZP have better pointwise

convergence than spherical partial summations of general If functions. The following

capacitarian version of Lebesgue's theorem is a good prototype of this phenomenon:

THEOREM 1.4.17. [AM, pp. 156, 157, 1591 Let g€If(1R) with i<p<oo and let

0< a :!~ . Define also f = G*g . Then

lim B(x,r) $ f(y)dy = (G* g)(x )

r-40 B(x,r)

for every XE IR" such that (G* gI)(x )< 0° (this takes place (a, p) -quasieverywhere).

Moreover, the convergence is uniform outside an open set of arbitrarily sinai! (a,p)-

capacity.

This result is a consequence of the following capacitarian analogue of Theorem 1.2.1:

LEMMA 1.4.18. [AH, pp. 159-161] Let i<p<oo, 1:!~ q:!~ n I and 0< a ~ (or Irp

1 :!~ q <00 and a = .). Suppose also that f = G,, * g for g E LP, (R"). Then

op Cap ({X € ii M(f))( x ) > q}) CII , I'°

for every A. ~! IIgII (for all )L >0 if q = 1). C is a constant independent of f.

As most of what we will do is based on quasicontinuous representatives of functions

in Z(1R"), it is important to have uniqueness of such representations to guarantee the

usefulness of the results obtained. The following theorem provides us with this

important insurance.

THEOREM 1.4.19. [AH, pp. 157-1581 Let a >0 and 1< p < oo • Suppose that f and

g are two (a, p) -quasicontinuous functions such that f = g almost everywhere. Then

f = g (cz,p)-quasieverywhere.

14

1.5 Related Classical Results in Fourier Analysis

As mentioned at the beginning, we will study in this thesis the pointwise inversion of

the Fourier transform in dimensions greater than one for some special L"-functions.

This section will present both the analogous study in dimension one and the results

which preceded it to give some insights and background on the problem.

1.5.1 The Historical Results

Some of the preliminary work for our problem was done during the XIXth century and

the beginning of the XX th by mathematicians like Dirichlet (1829, [Kö, pp.56-61]),

Du Bois-Reymond (1873, [Ka2, pp.5 1-52]), etc., but the first steps of importance to

us all came after 1920. One of the first was a negative result of Kolmogorov in 1926:

EXAMPLE 1.5.1. There exists a function in E(T) whose Fourier series diverges

everywhere.

The elaborate argument behind this counter-example can be found in [Kt2, pp.59-61].

The proof is done by building first a family of measures, {p,}, for which

su1SN/1 j(x)l >1 almost everywhere. It then follows from [Ktl] that there is an

f E L' (T) for which SNf(x) diverges on a set of positive measure and, so the same is

true everywhere by Theorem 1.5.2.

THEOREM 1.5.2. [Ktl, p.301] Let l!~ p<oo. Then either all Fourier series of

functions in LP(T) converge almost everywhere or there exists a function in LP(T)

whose Fourier series diverges everywhere.

Kolmogorov's counter-example raised at the time some serious questions about the

validity of Lusin's conjecture even if some positive results were also achieved. The

most important for us is a result of Kolmogorov, Seliverstov and Plessner which

shows that if

:.. .?(k log(2 + ki) <0°

N then SNf(x) = k=_Nf(k)e' - f(x) almost everywhere.

Different progress then followed around related questions like the Bochner-Riesz

Means (See Section 4.3) and the capacitarian extensions of Kolmogorov, Seliverstov

15

and Plessner's result (Theorem 1.5.6 to Example 1.5.11). But, the "pure line" of this

problem stayed relatively unmodified up to 1966 when Carieson, in a giant step

forward, resolved Lusin's conjecture by proving the following weak-type inequality:

THEOREM 1.5.3. [Cal] If f L2 (T) then

E (—pr, 7r) : S,,f(x)> A !! ~ -L 2 IIfII

where C is a constant independent of f and S,f(x) = su$(x -

y)' ef(y)dy.

This result was immediately extended by Hunt to the remaining L"-spaces by

establishing some estimates similar to those made by Carleson's for p = 2.

THEOREM 1.5.4. [Hu] If f e LP(T) and 1< p <oo then

IISJI = il NE su (T) 1 1LP(T)

~

where C,, is a constant independent of f.

In [KeTo], Kenig and Tomas proved a transference method which shows that

Theorem 1.5.4 is in fact valid on IR, i.e. if f E LP(R) with 1< p <00 then

IIS.fII !E~ CIIfII.

In what follows, we will refer to Carleson 's theorem as being this extension of Kenig

and Tomas. The proofs of these theorems are really deep and their techniques are not

directly relevant for this thesis, we will consequently skip them here. What is more

important for us is that, with an argument similar to the proof of Corollary 1.2.2, they

imply that SNf(x)—' f(x) almost everywhere when fE L"(T), l<p:!~ oo.

Moreover, if f e Z(IR) with 1 1), the situation is less clear. There is first of all the

question of how to generalise the partial integral operator Skf because of the infinity

of possible extensions. Let just have a brief look at three natural ways to generalise

Skf: the cubic, rectangular and spherical partial integral operators. The two first

If

operators are not really interesting in higher dimensions. The behaviour of the cubic

partial integral operator on

Sf(x) = jZ{yEiR n : Iyj I<Rforj=l2n}edY

can be reduced to the one dimensional situation. Hence, Sf(x) -4 f(x) almost

everywhere on IR", n ~: 1. On the other hand, the rectangular partial integral operator,

S1..R,,f(x) S forj=l,2,...n}f7)eY

diverge almost everywhere on 1R for some f e L° (1R) when p ~! 1 and n> 1 (For

these two first cases, see [KhN]).

But, the situation is completely different for the spherical partial integral operator,

SRf(x) = SBO.R ye2 y.

In higher dimensions, it is not yet known whether an equivalent of Theorem 1.5.4

holds or not for SRf. But, by Fefferman's counter-example (Theorem 1.5.5), this can

possibly be the case only if p = 2. We will use in this thesis the name Carleson 's

theorem in higher dimensions to refer to this possible extension of Hunt's theorem to

n>l when p2.

THEOREM 1.5.5. (Fefferman's counter-example, [Fl]) The operator

S1 f(x) = SB0.1e 27

defined on 1 2 n L"(IR") can not be extended to a bounded operator from L"(IR") to

L"(IR") when n>l and p#2.

A proof of Theorem 1.5.5 based on Besicovitch's set can be found in [St4, pp.450-

454].

Even if Hunt's theorem does not hold when n >1 and p # 2, we will see in Theorem

3.1.8 that if one can verify that SRf(x) -4 f(x) almost everywhere for f € L! (R")

then the same will also be true for f L" () with 2 :!~ p -Zn—. In particular, Carbery

and Soria [CS 1] showed that this happens when f Z(]R") with 2 !!~ p <-2n— and

a > 0 (Theorem 2.1.1). In this thesis, our principal problem will be to verify if it is

also possible in higher dimensions to get Ca,p -quasieverywhere convergence for these

functions like in dimension one and not only almost everywhere convergence.

17

1.5.2 The One Dimensional Capacitarian Study

In 1939, Beurling initiated the one dimensional "capacitarian study" when he proved

the following theorem:

THEOREM 1.5.6. [B, pp.5-911 If Llk(a +bfl<oo then the points of divergence of

the Fourier series 4a0 + l (ak cos(kx)+bk sin(kx)) form a set of null logarithmic

capacity3 .

In the same article, Beurlmg also stated a more general, but less precise result 4 :

THEOREM 1.5.7. [B, p.9] Let 0< a!!~ 1. If 1 ka(a +b)<oo for all e>O

then the Fourier series 4a0 + I '1(ak cos(kx) + bk sin(kx)) = ° f(k)e' is Abel

surninable5 except on a set of null 0-capacity 6 for all $ >1— a.

A few years later, a sharp extension of Theorem 1.5.7 was given by Salem and

Zygmund (Theorem 1.5.8), while two different constructions of Beurling and

Carleson (See Example 1.5.11) showed the sharpness of Theorem 1.5.6.

THEOREM 1.5.8. [SZ, pp.27-29] Suppose that ' 1 k(a, +b,)<oo and that the

Fourier series 4a0 + k=1 (a, cos(kx) + bk sin(kx)) diverges on a closed set E. Then

E is of null (1— ,B) -capacity if 0< /3<1 and of null logarithmic capacity if /3=1.

We will not discuss here the proofs of Theorems 1.5.6, 1.5.7 and 1.5.8, nor the proof

of their generalisation (Theorem 1.5.9) because the ideas necessary are just really

weak forms of those needed in the higher dimensional setting (Theorem 2.4.7). But,

roughly speaking the argument behind Theorem 1.5.8 combines the idea in Theorem

1.4.10 with the technique used to prove Kolmogorov, Seliverstov and Plessne?s

A set £ has null logarithmic capacity if there is no positive measure supported on E such that the potential

J2f1og(eu1 - re - rl )d/1(t)

is bounded uniformly in x as r—*1. 4 Beurling announced in [B, p.91 that the proof of this theorem was supposed to appear in Arkiv

fOr Matematik, but a remark in [SZ, p.23] lead us to believe that this never happened. Throughout this thesis, we will refer to this result as "Beurling's result".

By this, we mean that lm rf(k)eth = f(x).

( A set E has null /3-capacity, 0 </3 <1, if there is no positive measure supported on E such

that the potential

s:Ieit - reIadIl(t)

is bounded uniformly in x as r - 1.

theorem. Under some more restrictive conditions on the kernel defining our class of -

functions, Theorem 1.5.8 can be generalised to give:

THEOREM 1.5.9. [Ca2, pp.50-53] Let K be a kernel on IR in the sense of Definition

1.4.2 (a) and suppose that K is subharmonic, K(0) =00 and K(x) 0 for x ~! 1. Let

also define

= f K(x)cos(kx)dx and k(r) = r5K(x)dx.

Then the Fourier series (a cos(kx) + bk sin(kx)) converges Q-quasievery-

where7 if 1 )k(ak +bfl< 0°•

DEFINITION 1.5.10. A function f is said to be subharmonic if the Laplacian of f is

always positive (i.e. Af=f+f+...+f ~!0).

When A k = log(k), this result is the sharp generalisation of Kolmogorov, Seliverstov

and Plessner's theorem but, for specific classes of functions covered by it, some better

results can be established (e.g. Theorem 1.5.8). A precise quantification of the

sharpness involved was made by Beurling in the case corresponding to Theorem 1.5.8

[SZ, pp.47-49] and by Carleson for Theorem 1.5.9. They showed the following:

EXAMPLE 1.5.11. [Ca2, pp.53-541 Suppose that K and -2k are defined as in

Theorem 1.5.9. Suppose also that the kernel K satisfies K(x) = O(K(2x)) as x —*0

and let E be a closed set in [—,r, ,r) such that CK (E) = 0. Then there exists a function

in L2 ([—ir, pr)) whose Fourier series, (a cos(kx) + bk sin(kx)), both diverges

everywhere on E and satisfies =

(a + b) <00.

There are two keys ideas behind the classical constructions of Example 1.5.11. They

can be summarised as the absolute need for an everywhere and uniform localisation

principle (Riemann's localisation principle) and the necessity for a version of Theorem

1.4.10 (1) which holds everywhere on the support of PE and not only CK -

quasieverywhere if E is compact. A harmonicity condition on the kernel K which is

generally not satisfied by Ga in higher dimensions is hidden behind this last request.

Before ending this section, we should point out that all the classical results stated

above can be rewritten for the partial integral operator on IR rather than the partial sum

The classical Cg -capacity is similar to Definition 1.4.13 except that 1' ' (X) = (K*ji)(x).

19

operator on T with the modern notion of capacities given in Section 1.4. It is these

equivalent formulations that we will use as starting point for our investigations in

Chapter 2. For example, Theorem 1.5.8 for functions in Z 2 (R) should then read as

follow:

THEOREM 1.5.8a. Let O<a:5-4 and let fEZ(R). Suppose also that g€ L2 () is

such that f = G*g almost everywhere. Then SRf(x) —* (G*g)(x) C a2 -quasievery-

where on

As we will see in the next chapters, the situation is a lot more complicated in higher

dimensions. The main problem is that the Dirichiet kernel does not behave as nicely in

these cases. In particular, it is not possible to build a direct equivalent of Lemma 2 in

[Ca2, pp.5 1-52]. To compare what is happening between the two cases, let us have a

brief look at two simple computations in IR and W:

COMPUTATIONS 1.5.12.

In IR,

KR(x)= F =$R cos(2rxy)dy+iJ sin(27xy)dy =(,rx) sin(27rRx).

While, in IR", we first rewrite K1 in term of the Bessel function of order -,

- J,2 (x) — (,rl/2F(n+1))' £ elxl (1 — t2 )(n_1)/2 dt,

using the classical identity K1 (x) = Ixfl 12 J,2(27rIxI) [St4, p.390]. Then, we replace

f1112 (2 irlxi) by its asymptotic development at infinity [St4, p.1381 and we rescale to get

KR (X) R"11

2 lxi-(n+1)12 2,r,R e + terms of lower order

when lxi —* 00 Hence, there is a term "R"'2 " which appears in KR in higher

dimensions. Because of this term, the KR are not uniformly bounded in R outside a

neighbourhood of the origin as they are in IR.

Chapter 2

The X2 case

In this chapter, we will motivate the problems studied in this thesis. Once the situation

in higher dimensions will have been clarified, we will extend Beurling's result

(Theorem 1.5.7) in dimension one to all dimensions. We will also prove a capacitarian

localisation principle (Theorem 2.3.7) which will enable us in Theorem 2.4.7 to

extend Salem and Zygmund's result (Theorem 1.5.8) to Z 2 (1R") when -- < a :!~ n.

2.1 Carbery and Soria's Result

In [CS1], Carbery and Soria proved a higher dimensional analogue of the classical

Rademacher-Mensov theorem (a proposition similar to the Kolmogorov-Seliverstov-

Plessner theorem seen in Section 1.5 but with a (1og(2+kI)) 2 rather than a

log(2 + IQ). More precisely, they showed the following:

THEOREM 2.1.1. [CS 1, p.3291 Let f e LP(R) with 2 :!~ p < and n ~ 2. If for

each 0 E C the function g = Of satisfies

<c'o

then SRf(x) - f(x) almost everywhere on IR".

PROOF.

Using Carbery and Soria's localisation principle (Theorem 3.1.8)8, we can suppose

without loss of generality that supp(f) ç B(O, 2) and that

1f(

4)12 <00•

8 It makes more sense to us to cover the results in this chapter before those in Chapter 3. Unfortunately, the proofs of Theorem 2.1.1 and its related propositions depend on Theorem 3.1.8 and estimate 3.1.* which will be studied in Chapter 3.

21

We now need to prove that SRf(x) —* f(x) almost everywhere on any ball inside

B(O, 2). We will only do this for B(O, 1), but the proof is similar for any ball.

Let 5R be the operator with multiplier rn = XB(o,R) — 4A) where XB(o,) ~ 'r ~ XB(o,)

and VEC. But,

(sf(x))2 <— — &f(X)J) ' + (SRUT I~"f(X lY - So, using Theorem 1.2.1, we can now suppose that we are working with SR rather

than SR as the first part is controlled by the Hardy-Littlewood maximal function Mf

(i.e.ll5I5Rf() — 5Rf()1112 :!~ C'lf, !!~ Cf2). RT

Let NEiNbe fixed and let R--2'. For lxl<l,

SRf(x) = ( *f)(x) = $ KR (x — y)f(y)dy = J (k )(x — y)f(yly = ((okR )* f)(x)

where 0 E S and 0 1 on B(O,3). The third equality follows from the support

property of f (i.e. lxi :!~ 1 and II !!~ 2 implies that lx — l!5 3). Consequently,

R f(x) = ((0(- /2 N )k ) R * f)(x) = ft0(.)k1 (.)+ (. / 2

' JR *f W

where y(x) = 0(x) — 0(2x). We used here the usual notation g, (x) = Rg(Rx).

Hence,

~ CMf(x)+ sunI N ((kj)R*f)(X)l

R-I i='

W CMf(x) + )R * f)(x)

where K(x) = K1 (x)1(2'x).

Let assume for the moment the following lemma:

LEMMA 2.1.2. [CSI, p.130] Let (Ki) be defined as above and let f e P. Then

1(sup'(KJ) *fl(x)d ~ 1lfli

J I I i

where C is a constant independent of j.

- Now,

22

1/2

( fB (0 ,1)uRf I (x) ) 2 dx)

\R-2

1/2 N

~ 1f112 )i *fj(x)) dx) j= 1

(* *)

< C(l+N)11f112.

Here, the first inequality is a consequence of Theorem 1.2.1 while the second one is

obtained from Lemma 2.1.2.

Let he L2 be such that Using the fact that the support of k1

is contained in an annulus distant from the origin, we can find a smooth bump function

or with a compact support around the unit annulus and not containing the origin such

that:

(Rf)() = = k1 (1R)h()(1+logj

- 11 = *Rl/R PN'O() = *(k R *(pNh))i

where (pNh)"() = N(1 + iogI) ' a(c / 2'')h().

Consequently,

2 2

N=1 N2 B((,l)(RPL") ~ C 1 pNhj < ChjI $ (su Rf) — ; _L$ B(O,1) ' R~

The second inequality is obtained from the sets of inequalities following the statement

of Lemma 2.1.2 while the third one is a consequence of the support of (PNh) which

is contained in the region where IH 2N Hence, N and (i+logI) 1 cancel each

other and the quasi-disjointness allow us to recover IIhII from the sum.

If we now turn our attention to Lemma 2.1.2, we will find out that it has a classical

proof based on the fundamental theorem of calculus and an estimate for -(k )A•

PROOF OF LEMMA 2.1.2.

(

s.::up ((k ) * f)(x)) ,<2

!~ 2f, I((kj),*f)(x)jI(-.d- (k) * f)(x)dt +((ki ) * f) (x) J1 2

112( 2 2 1/2

~ 2152 ((kJ ) * f)(x)2 rl dt) fIt(f (K ) *f)(x) tldt) + (k' * f) 2 (x)

1

23

The first inequality is obtained using the fundamental theorem of calculus while the

other is a consequence of the Cauchy-Schwarz inequality. Now,

I (1sypI((ki),*f)(x) dx J

2 \112( 2 1/2 ~ c[(512

Jk) dxtdt j f$ ft(f(k3)1*f)(x)dxt1dt) +lIfII] ~ cIIfII ) 1

In the first inequality, the first term is obtained using the previous computation, the

Cauchy-Schwarz inequality and Fubini's Theorem. To get the second inequality, we

note that

(*) (S

U

P J12Iki( / t)12 tdt) (sup5It*(K( / t)12 t 1 dt) ~ c

independently of j because

dtfl I t)!!~ C 2 ( 1 + 2 III — tI)

by a computation analogous to the one in Lemma 3.1.11 (Chapter 3). So, the first term

in (*) is dominated by C2 (A. 1 and 13=0) and the second by C2 (A. =0 and

Theorem 2. 1.1 contains in particular that SRf(x) - f(x) almost everywhere if

f e Z(]R") with 2 :~ P<r and a >0. In Section 2.3, we will show how a careful

study of the ideas in the previous argument and in the proof of Theorem 3.1.8 gives an

equivalent of Beurling's result (Theorem 1.5.7) in higher dimensions as well as some

other more precise capacitarian results. In order to do this, we need to remark that the

proof of 2.1.1 contains the estimate

CrgI < Cr,JIgIj,

21, valid for f Z(1R) with 2 ~ p < ç, a> 0, supp(f) ç B(0,ij), 0 <r < r, <oo, and

f = G*g almost everywhere for a g E L"(]R).

In fact, a combination of this estimate with estimate 3.1.* (Chapter 3) easily implies

the following stronger proposition:

PROPOSITION 2.1.3. Let f (1R") with 0< a and 2 ~ p < . Then

24

yI~r} Is,f (x)12 dx ~ Cr IlgII.

S{x:Ix-

Here f = G* g almost everywhere, g E LP (R ' ), r E ]R, y E JR" and Cr is a constant

independent of y and f but depending on r.

PROOF.

Without loss of generality, we can, by translation invariance and scaling, restrict

ourselves to show

IS*fx)I2f1 < IB(o,l ,2)

Let 0 E c be such that XB(0,1) !!~ 0 5; XB(0.2)- Hence,

I*f( dx 5; 2$B(O.1/2) L(0,1 l(S*Of)(JB(O,l/2) x)12 dx + 2 12) j(S.(1 - Ø)f)(x) 2 dx = '1

+12

by the basic inequality (a + b)2 5; 2(a2 + b 2 ) applied to S(Øf) and S.((1 - Ø)f).

Now, the first term is bounded by C11g112 using 2.1.* while the second is bounded by

CIIflI, using 3. 1.*. And, so the result follows from the L" -boundedness of the Bessel

operator (Section 1.3.2).

2.2 Motivations in Higher Dimensions

if we now compare Theorems 1.5.8a and 2.1.1, we should expect that, in higher

dimensions, a better result than Theorem 2.1.1 holds for functions in Z. If,

moreover, we make the natural supposition that Carleson's theorem is true for all

dimensions, we then realise that we should hope like in dimension one for the

following conjecture:

CONJECTURE 2.2.1. Let f E Z 2 (1R"), 0< a 5; -, and suppose that g E L2 (IR") is

such that f = Ga *g almost everywhere. Then SRf(x) - (G*g)(x) C 2 -quasievery-

where on 1R.

Conjecture 2.2.1 is the heart of this thesis. Our primary goal in the next sections will

be to try to prove it and its ZP analogue (Conjecture 2.2.12). To discuss these results

and many others, we will need an easy but really useful estimate:

25

BASIC ESTIMATE 2.2.2. If f and g are two real functions in some appropriate

function space then S(f* g)(x ) <_ ( If *s. g)(x ) on IR".

PROOF.

Let R>1 be fixed. Then

ISR (f* g)(x) = I(f* Sg)(x) :!~ (111* Sg)(x).

The equality follows by Fubini's theorem while the inequality is a consequence of the

triangle inequality. As the bound obtained is valid for any R> 1, this implies the

desired result.

CONDITIONAL PROOF OF CONJECTURE 2.2.1.

Before starting the proof, let us clarify our hypothesis that an equivalent of Carleson's

theorem is true in higher dimensions. To be precise, we suppose here that

IIs.f 112 !~ C11f112 for all f E L2 (IR") with a constant C independent of f.

Now, define DA = {x E ]R" : S.f(x) > A and D = {x e IR": (Ga * S.g)(x) > Al. The

Basic Estimate 2.2.2 then implies that DA ç DA. Consequently,

Ca2 (Di ) < C (D) < CjSg 2 < Cg 2

- a,2 -

where the second inequality follows from Definition 1.4.4 while the third one is a

consequence of our hypothesis. This, by the standard argument (e.g. Theorem 4.1.5),

implies that SRf(x) —* (G*g)(x ) C 2 -quasieverywhere on IR" as desired.

In fact, with the technique which will be used to prove Theorem 2.3.3, it is possible to

verify that even a local version of Carleson's theorem (IB01 IS*f(x)I 2 C1IfII) in

higher dimensions will imply Conjecture 2.2.1. On the other hand, if only a weak-type

(2,2) version of Carleson's theorem (i.e. I{x € R":S.f(x)I> A}l ~ CA_2 IIfII) was

shown to hold when n> 1 then it is unclear if it would also implies Conjecture 2.2.1.

Nevertheless, it is reasonable to believe that it will be possible to reorganise the ideas

in some way to do so.

But, expecting to prove such a theorem is probably being too optimistic, so two

pill

natural questions are raised: Are there some other reasons to believe the truth of this

conjecture? and What else can we try in order to prove it? We will leave the second

question for Section 2.3 and 2.4 while, in the next pages, we will try to answer the

first one by looking at two particular cases where a "Carleson's theorem" holds in

higher dimensions.

In the lacunary case, Carbery, Rubio de Francia and Vega proved the following

weighted maximal theorem:

THEOREM 2.2.3. [CRdeFV, pp.52 1-522] Let 0 :!~ a <1 and let f L2(Ixradx).

Suppose that (Rk ) l is a lacunary sequence and define Sf(x) = suflISRkf(x)L. Then

$ (sf(x)) 2 Ixra dx !!~ Ca $ lf(x)12 Ixra cix

where Cc, is a constant depending only on a.

DEFINITION 2.2.4. A lacunary sequence {Rk}l is a sequence of positive numbers for

which there exists a constant C> 1 such that Rk+l > CRk for all k.

After dividing the operator into pieces with a partition of the unity similar to the one

seen in Theorem 2.1.1, the proof is done using Littlewood-Paley theory and a classical

result of Hirschman:

LEMMA 2.2.5. [Hi, pp.52-551 Let n>1 and let —1 <a< 1. If fEL2(Ixladx) then

$ (S1 f(x)) 2 IxIa dx !!~ Ca $ If(x)12 Ixia dx

where Ca is a constant depending only on a.

Using Theorem 2.2.3, we can build a maximal inequality implying Conjecture 2.2.1 in

the lacunary case.

COMPUTATION 2.2.6.

More precisely,

JB(O,l) ISf(x)I2 dx

$B(0,1) If(x) 2 Ixi dx

:!~ C1 $ If(x)12 IxI' dx

= C $BO.1 If(X)1 2 IxI'3 dx ~ C1If1IP((B(01)))

for f E LP((B(0 , 1))c) with 2 :!~ p -Ln- because If((B(0, 1))c , dx) L(xdx) for

n(1 - ) <1.3 and, here, 0 can be chosen as we please in [0, 1).

27

Moreover,

.1. ISf(x)d* ~ Lf = C11f11 B(O,1)' 2(B(Ol)) < ClIfIIp(B(Qj))

when fE L(B(0,1)) with 2 ~ p. Hence, 5B(Ol) Lf C1IfII if f L'(1R) with

2<p < and, when p = 2, we have !!~ C1 2

So, for fE Z2 (]R) with 0 < a !!~- , we can extend Salem and Zygmund's result [SZ]

to all dimensions in the lacunary case using an argument identical to the conditional

proof of Conjecture 2.2.1. While, when f E Z(1R") with 2 ~ p < and 0< a <_ AP

a proof similar to the argument behind Theorem 2.3.3 also implies the Ca2

quasieveiywhere convergence of SRf. But, in this last case, we can not get the Cap -

quasieverywhere convergence strictly from Theorem 2.2.3. More precisely, we just

established the following:

COROLLARY 2.2.7. Let {Rk } I be a lacunary sequence and let 2 :!~ p < 2n . Suppose

that f e 0< a !!~ , and suppose also that g E I! is such that f = G*g

almost everywhere. Then SR, f (G * g)(x) ca2 -quasieverywhere on IR".

We should note that, for p # 2, we only build a local capacitarian weak-type

inequality, Cap ({x E B(0,1): S(G*g)(x)> })~ -frg, for A. > c1 ~ 0, to get

Corollary 2.2.7, while, for p = 2, we constructed a global one (i.e. valid for all IF

and not only for B(0,1)). This difference will be of great interest in Section 4.1.

If we are now more restrictive in our choice of functions, it is again possible to prove a

"Carleson's theorem" in higher dimensions without this time any restriction on "R".

Kanjin and Prestini proved independently that for radial functions the following sharp

theorem holds:

THEOREM 2.2.8. [Ka] and [P] Let ç < p < r and let f E L"(IR"). Suppose also

that f is radial. Then

IISfII ~ Cf p p

where C is a constant depending only on n and p.

In joint works 9 with Meaney, Prestini extended this result to different subspaces of

Many references to these joint articles can be found in [MeP].

L2 (IR") where some symmetries exists (e.g. the linear spans of products fg where f is a radial function and g is a spherical harmonic). Theorem 2.2.8 and its extensions

will not be proven here because their proofs would require a long study quite far from

the main line of idea in this thesis. As in the lacunary case, these results can be given a

capacitarian extension. More precisely,

COROLLARY 2.2.9. Let f E (1R") with 2 :!~ p < and 0< a ~ -, and let

g E L" (s") be such that f = Ga * g almost everywhere. Suppose also that f is radial

(or satisfies the conditions of one of Meaney and Prestini's extensions). Then

SRf(x) - (G* g)(x ) ç-quasieverywhere on IR".

In the view of these two examples and with the knowledge that Theorem 2.1.1 has

been proven with a similar proof as its one dimensional analogue, which itself was

already containing the key ideas behind Theorem 1.5.8a, Conjecture 2.2.1 should now

be considered highly plausible. But, these examples raise another natural question: can

we hope to prove Conjecture 2.2.1 without establishing a higher dimensional Carle-

son's theorem?

To do so, we need to be able to build an inequality like IIfIILP (d ) ClIIL for all Bore!

measures 1a satisfying the conditions in one of our definitions of C,,,2

-capacity in

Section 1.4. The results in [CS2] lead us to expect that this estimate is equivalent in

some sense with the "weaker estimate" su?IISRfIILp(d) !!~ Cg; so let have a look at

what can be done in this case:

PROPOSITION 2.2.10. Let fEZ(]R"), O<a ~ -, and let gEL2 (]R") be such that

f = G * g almost everywhere. Then

5,gpI1RfIILI(d) ~ C119 112

if p satisfies the conditions in Definition 1.4.6 (general case) and

su?I1SRfII12(d,) !~ C119 112

if p satisfies the conditions in Definition 1.4.13 (compact case).

PROOF.

(1): IIRfII1.' (dji) = IISR (Ga *g)l (dp

!~ 5(Ga *ISRg)(x)dp(x) by Estimate 2.2.2

= 5sRg (x)j(Ga *p)(x)dx

~

(fISRg(X)12 dX) 112

(5 ((G" *P )(X))2dX ) 112

~ ugh2 (5 ((Ga *p )(x))2 )1!2 ~ 'ugh2

(2): IISRfIIL2(dP) = sup 5sR (G* g)(x )h(x)djJ(x) ! IhIL2(4) ~ l

= sup $sRg(x)(Ga*(hdJl))(x)dx

2 1/2

Ih L2()

~ sup !1, ~ 1

hsRgII 2 (5 ((Ga *(hdiU))(X)) d)

1/2

~ sup 1912 hIhhIL?d (jj (G a *Y )(X))

h , HhL2() 1

1/2

~ IIg lh 2 (Su (G2a * /1 )(x)) < C 9 2 E

The second inequality in (2) follows from labi !~ +(a2 + b2 ) while the last inequalities

in (1) and (2) are consequences of the respective hypothesis on U.

Hence, it is possible to establish the weaker estimate suhISRfjILp(d) for some

p and so it seems plausible that there is a way to extend Salem and Zygmund's results

(Theorem 1.5.8) without using a higher-dimensional Carleson's theorem. We will see

in the next sections how this can be done in some particular cases. But, before doing

so, let us take a moment to ask what should be the analogue of Conjecture 2.2.1 when

p 2.

If we want to make a conjecture valid for all a >0, we know from Herz's work [He]

that it only makes sense to ask this question when r < p < r. But, the two cases

<p <2 and 2 < p < are completely different. When < p < 2, no natural

conjecture can be made because of some recent work of Tao [T, Proposition 4.11 as

we will find out in Section 4.3.2. On the other hand, there seem to be two natural

possibilities when 2< p <

If we think of functions on IR as being semi-radial and if we remember both

COrdoba's work [CM and the way the positive results in [CSI] and [CRdeFV] were

30

proven and Theorem 2.2.810 then we can expect that the optimal result for

f Z(1R") with 2< p < is something like IISafIILP ( L2) :!~ c gI• Here g is a

function such that f = G*g almost everywhere and LP(L2 ) is the mixed norm space

which is LP radial and L2 angular. If this mixed norm inequality is the sharpest result

then, for capacities based on the Bessel kernel (C yq -capacities), the best possible

result takes the following form:

CONJECTURE 2.2.11. Let fE(1R") with 2<p< and 0<a ~ , and suppose

that g E LJ'(IR") satisfies f = G*g almost everywhere. Then SRf(x) - (G* g)( x )

Ca 2 -quasieverywhere on IR".

Conjecture 2.2.11 seems also supported by Corollary 2.2.7. But, if we are more

optimistic and believe that the mixed norm estimate above is not sharp and that a direct

analogue of the one dimensional case should hold even if Hunt's theorem (Theorem

1.5.4) is false in higher dimension, then we should hope for the following:

CONJECTURE 2.2.12. Let f E Z(lR'1 ) with 2 (G*g)(x)

Ca,P -quasieverywhere on IR".

We will see in Chapters 3 and 4 that Conjecture 2.2.12 is the correct one. We can

always do better (at least intuitively) than Conjecture 2.2.11 (Section 4.2) and, when

<a < , we can even verify Conjecture 2.2.12 (Theorem 3.2.11).

2.3 Some Direct Computations

Following the motivations given in the previous section, we will now prove some

direct capacitarian extensions of the work done in [CS 1].

2.3.1 An Extension of Beurling's Result

We will start by extending Beurling's theorem (Theorem 1.5.7) to all dimensions, but,

in order to do this, we need first the following lemma:

LEMMA 2.3.1. Let r,r1 1+ be such that r> ij. Suppose that f is supported in

(B(0, r))c and satisfies

10 For radial functions, II•IIL'(L') = 1•11,, so Theorem 2.2.8 can be rewritten as ItS, ! IL,(L1) <C1IfjI.

31

S f 2 (x)dx<C<oo B(y, r. )

for all y E IR" with C a constant independent of y. Suppose also that iw(x)l is a

radially decreasing function in L' (IR") with exponential decay as lxi - oo (i.e.

ia)(x)i = 0(e"), c1 > 0, for large x). Then

i(w*f)(x)i < <00

for lxi :!~ r with a constant depending only on a, r, ij and C.

REMARKS 2.3.2.

The condition r <r is there to take care of the possible bad behaviour near the

origin of co. Think for example of a) as being a Bessel kernel G a .

The condition about the radial decay of co is not necessary if co is bounded in

(B(0, r -

'j ))'. When reading this proof, just observe that the same proof applies

in this case by replacing the majorisation obtained on the inside border of the

annulus D = B(0,j + 1) \ B(0, j) by the supremum of the values of I col inside the

annulus. The sequence of all such suprema is bounded and decay exponentially

for j large enough because co is bounded and decays exponentially at infinity

The exponential decay is not needed in this lemma but, in practice, that is what we

will be using. In fact, it is sufficient that ii behaves near infinity like a positive

function p on IR" which satisfies q(i)i"1 <00 (by q(i), we mean (x) for i large

x such that lxi = i). This in particular is satisfied by Ga , but not by 'a•


If co is not defined everywhere on IR" \ {0} just redefine it before continuing in a way

that preserves the radial decay of ii (or redefine it by 0 in the case covered by

Remark (2)). This can be done without loss of generality because this set has null

Lebesgue measure.

Throughout this proof, the value of i will be fixed to 1 and the value of r to 2 to

simplify the notation but the same proof can be used for any values of r and ,j.

Let x E B(0,1) be fixed. Then,

(co*f)( x ) = Jw(x - y)f(y)dy = D I (1)(X - y)f(y)dy = 5 U)(x - y)f(y)dy j=2

32

where D = B(O,j + 1) \ B(O,j). The second equality is obtained from the support

property of f.

We now cover D1 with cj' balls b1 of radius 1 where c,, is a constant depending

only on the dimension n (D Uk,). The power of j is n - i here and not n

because the width of the annulus D1 = B(O,j+l) \ B(O,j) is 1. By replacing b11 by

= b. 1 flD, we get D = Ub1,,. Hence,

- y)f(y)dy = L, Iw(x - y)11121c0(x - y)I'f(y)Iv

< )(j11/2J k(x-y)I

1/2 If(y)Idy

1/2 1/2

< w(j _1)11/2(5 Iw(x _Y)tdY) ( i.j

LIf(y)I 2 dy)

< Io(j_1)Ih/2IfrIll2ChI2 by b1 , cb1 and the hypothesis on f.

The first inequality is based on the inequality w(x - y) !~ co(j -1) which is obtained

from the radial decay of w as y E bi ,j ç Dj and x E B(O, 1).

Using this estimate, we have

j=25 w(x—y)f(y)dy ~ 2 I(i_1)Ih/2 IIwII 2 ChI2ci 1 by D =U&

IIwII'2 c112c ( k Io. (j - 1)11/2 in_i + c(l)d:=k jn_1 e_mi )

= C r C

where 0 <rn <°° and 3 !~ k <00. The approximation in the last line follows from the

exponential decay of w.

The combination of Proposition 2.1.3 with Lemma 2.3.1 easily gives an analogue of

Beurling's results (Theorem 1.5.7) in all dimensions:

THEOREM 2.3.3. Let feZ(lR"), 2~ p<and0<a~ , and let geL" (R) be

such that f = G* g almost every where. Then SRf(x)-9 (G,, *g)(x) C, 2 -quasi-

everywhere on JR" for every k >1.

33

PROOF.

Let first remark that SRf(x) = SR (G,,, *g)(x) for every x in IR". Consequently, we can

replace f by G*g without loss of generality in this argument.

S(G*g)(x) :5 (G a/k *s* (G(k_l)a,k*g))(x)

= (Ga/k * (xB(o.2)S$ (G(k 1)alk * g)))(x) + (Galk * (X(B(0,2)y S (G(k_l)a/k * g)))(x)

The inequality follows from Estimate 2.2.2 and Proposition 1.3.5 (2). This implies

that

E = {x E B(O,1) : S(G* g)(x)> 2L}

jx E=- B(0,1):(Ga/k *(z B(o ,2)S.(G(k _I)/k *g)))(x) >

u {.x e B(O,l):(Ga/k*(x((02))S*(G(k_j)a/k*g)))(x)> +} = E1 u E2 .

Now, Ca1k2 (E) !!~ Ca/k2 (El ) + Ca1k2 (E2 ) by the subadditivity of Ca/12 1 consequently,

we will be done ifwe can show that Ca/k2 (Ej ) for >c>O and j1,2.

But, on one hand, Ca/k 2 (E2 ) = 0 if 2 > 2CIIgII because, by Lemma 2.3.1 with r1 = 1

and r = 2 and by Proposition 2.1.3 with r = 1, we have

(G.1k * (X(B(0,2))c S* (G(k-l)alk * g)))(x) < CJg ,,

The constant C,, 11g lI can be obtained by looking at the proof of Lemma 2.3.1.

On the other hand,

Ca lk 2(E1) < Jig

2 I*I

.2A

(%(o,2)5.(G(k_1)a,k*g))C) dx

- S (S.(G(k_l)a/k*g))2(x)dx ~ -IIgIi. -

B(O,2)

The first inequality follows by Definition 1.4.4 and the last one is a consequence of

Proposition 2.1.3 with r=2.

In fact, a result slightly stronger than Theorem 2.3.3 holds with exactly the same proof

(modulo a small redefinition of Proposition 2.1.3 to fit the new situation).

34

THEOREM 2.3.4. Let f E Z(), 0< a ~ . Suppose also that the function

g e L2 (1R') such that f = G* g almost everywhere satisfies

<00

Then SRf(x) —4 (GOI Ca2 -quasieverywhere.

2.3.2 A C ,2 Localisation Principle

By refining the idea in Lemma 2.3.1 (see Remark 2.3.2 (2)), it is possible to quantify

exactly the contribution coming from the tail of Ga :

LEMMA 2.3.5. Let g E L" (1R) with 2 :5 p < and let 0< a !~ . Suppose also that

0 E C(IR') satisfies the following conditions: 0 is radial, supp(Ø) g B(0,2ij) where

E R, 0!~ 0(x) 15 1 for all x in IR n and 0 1 on B(0,ij). If u E

e 1R, satisfies 11jill < oo then

Js.(((l - Ø)G)* g)(x )d/1( x ) :~ Cg p

where C a is a constant depending only on 0, a and IWIl.

Note that there is no relation between the values of ,j and r2 in this lemma.

PROOF.

We first remark that (1O)G a E ZP if pE[l,2] and 0 <13 < a. This is trivial if p#1

and, for p = 1, it follows from Proposition 1.3.5 (2) combined with Lemma 3 and

Result 6.10 in [SO, Chapter 5] (or essentially by Propositions 1.3.5 (2) and 1.3.10 in

Chapter 1). Consequently, for any such fixed 13, there is a g E L' r L2 such that

((1 0)Ga )(X) = (Gfl almost everywhere; moreover g is radial.

Let usdefme J3=--=$+/3 where p=--, and k>1 isanumber near

1 to be fixed later.

Observe that we can split G,*§ in two parts g and g such that g is supported in

B(0,r) where r is a large parameter to be fixed later and g in its complement. This

implies that and g are both radial and in L' n L2 by the properties of g and G, - We

also suppose for what follow that g is defined everywhere. We can do this because

the set where g is not defined has null measure and so, if this is not the case, we can

give value 0 to every points in this set without losing generality.

35

We now choose a k> 1 near enough to 1 so that (x) is exponentially decreasing

when xl—o°. We can do this because convolving G with essentiallyfl reproduces in Gfl+fl * (i.e. (1— Ø)G,, a.e.) this property of G 0 *g as this function is

nearly as "smooth" as for k small enough.

$ S(((1 - Ø)G)*g)(x)d/1(x) = $ S(G*G**g)(x)d1u(x)

c J( ** çoS (G13*g))(x)dp(x) + $(lgl*(l - (o)S$ (Gfl *g))(x)dp(x)

+ 5(ll*(1_ (p)S* (G* g))( x )dJ1(x ) by Estimate 2.2.2

= 11 +12+13

Here, p has the same properties as 0 but with radius i (rather than ij) to be fixed

later. We will now show that 11, 12 and 13 are correctly bounded.

'1 by Cauchy-Schwarz

~

(ff (lg l*l ,4 g l)(X- y)dji(y)d (x))

1/2 frpS. (G, *

by the L2 -boundedness of the Bessel operator of order 0 (Section 1.3.2)

~ ($5 Gdp(y)d/i (x))1/2 z

by IgI E L2 which implies that is bounded on IR" using Cauchy-

Schwarz

< cx lIgil

by Proposition 2.1.3 and Jdii :!~ C

12 f(XB(O,,)W*X(B(O,t)) 'S*(GP by properties of p and

= $5 Xcj,r (x - Y)X (B(O r))C (y)lèl(x - y)S (G * g)(y)dydi(x)

- y)(S.(G*g))(y)dyd/L(x)

because lx - <r and lxi < r2 imply that lYi < 2r if we choose an r which

satisfies O<i <r<oo.

< $ ugh2 IIZB(o,2r)5'. (G * g) du (x)

by Cauchy-Schwarz

_< C a hlgIIp by Proposition 2.1.3 and g€ L2

'ri

For what follows, we also suppose that we have chosen r large enough to have

bounded. We can do this because g has an exponential decay at infinity and this is not

possible otherwise.

To deal with 13 , we use Remark (2) of Lemma 2.3.1 as the properties of g and q

fulfil every condition of Lemma 2.3.1 in this modified version. We now fix r > r + r2

in order to get a bound of (Ig l* (I - (o)S$ (G,f3 *g))(x) valid for every x in B(0,i). This,

by the support property of p, implies that

13 !~ JC a lIgII p d/J(x) = Cca IIg IIp .

REMARKS 2.3.6.

With an argument similar to the end of the proof of Corollary 3.2.9, we can

deduce from Lemma 2.3.5 that SR (((1 - Ø)G)* g )( x ) converges everywhere on

the support of p.

We could have also deduced the previous remark (with convergence everywhere

on 1R') from the smoothness of ((l_Ø)G)*g, but Lemma 2.3.5 seems more

visual to us.

In Theorem 2.1.1, two steps were needed to prove the almost everywhere

convergence: a localisation principle and an argument dealing with compactly

supported functions. Using Lemma 2.3.5, we can give a capacitarian extension of the

first of these two parts:

THEOREM 2.3.7. Let f€ Z(lR") with 2!5 p< and 0< a~ .ç. Suppose also that

g € LP (R') is such that f = G*g almost everywhere. Then SRf(x) -* (G* g)( x )

Ca2 -quasieverywhere off the support of g.

PROOF.

By translation and dilation invariance, we can assume without loss of generality that supp(g)c(B(0,2))C. We then have to prove that Sf(x)_*(G*g)(x) Ca2

quasieverywhere on any ball inside B(0, 2). We restrict ourselves to show this for

B(0,1), but the proof is similar for any other ball (see remark 2.3.8 (1) after this

proof).

37

Let D represent the set of divergence inside B(0,1). Then D is a Bore! set. This

follows by rewriting D as

D = Ui{x E IRE: lirnsupRe(SRf(x))— liminfRe(S Rf(x))

+ !im sup Im(SRI(x)) - !im inf Im(SRf(x)) > = u: Dk. R—o

But, then each Dk is a Borel set as SRf is a continuous function. Hence, SRf is a

Borel function and Borel functions are closed under lim sup, !im inf and difference.

So. D is a Borel set.

Now, suppose that, for every Borel measure p such that supp(p) g D and

IV a */u112 !!~ 1, we can show that p(D) = 0. Then, we will have C,1 ,2 (D) = 0 by the

definition in Proposition 1.4.6 as D is a Borel set. Hence, the theorem will be proven

if we can show that $ Sf(x)dp(x) :!~ C lIgIl for any Borel measure p supported inside

B(0,1) and such that JjG2a (X—y)dP(y)dP(X)< 1.

Let 0 be defined as in Lemma 2.3.5 with ij chosen to be -i-. From supp(0) and

supp(g), we then have supp(h) c (B(0,1))c where h = (ØG)* g .

$Sh(x)d,u(x) !! ~ 5((ØG)*S.g)(x)dp(x) by Estimate 2.2.2

= 5((ØG)*p)(x )S g (x )dx

= 5((OGa )*P )(x)z(3) (x)s*g (x)dX

because supp((ØG)*p) B(0,.)

II(OGa )*pII2kB(o3)S.g by Cauchy-Schwarz

~ 'IIgIL

To obtain the last inequality, we use

II(0a)*PII ~ IIGa *p11 = 55 G(X - y)dp(y)dp(x) < 1 2

for the first term while the second one is bounded by CIgII using the estimate 3.1.* of

Theorem 3.1.8. Combining this estimate, 5 S.h(x)dp(x) !! ~ Cl Ig Ilp with Lemma 2.3.5

then implies that 5 Sf(x)dp(x) ! ~ ClIgIlp , so SR(f)(x) = S(G* g)(x ) —* (G*g)(x )

W.

Ca2 -quasieverywhere in B(O, 1).

REMARKS 2.3.8.

To see that we can modify this proof for any ball inside B(0,2), observe the

following figure.

Figure 1.

0 is supported in the black ball, supp(g) is contained outside the grey annulus

while supp(f) is in the union of the grey annulus and supp(g). The previous

proof clearly works for any sphere B(O,r) with r <2(1 — ij) (i.e. inside the inner

limit of the grey annulus). To show the same for an r0 satisfying 2(1— ,j) < i <2

define a new function 0 (with the same properties as 0 except that ij is replaced

here by ) with a support small enough to have i <2(1 - TI ). ro is then contained

inside the inner limit of the corresponding new grey annulus, hence the proof

above can be used for r0 .

The localisation provided by Theorem 2.3.7 implies that SRf(x) —* (G* g)(x )

Ca ,2 -quasieverywhere on any region where g satisfies either some "smoothness

condition" (like in Theorem 2.3.4) or some "symmetry condition" (like in

Theorem 2.2.8). This follows directly by applying Theorem 2.3.7 first and then

Theorem 2.3.4 or Theorem 2.2.8 respectively.

Theorem 2.3.7 is in fact a "double" localisation principle. If we want to show that

Sf( x)_(G* g)(x ) C 2 -quasieverywhere for any fEZ(]R"), it is enough,

because of Lemma 2.3.5 and Theorem 2.3.7, to prove that SRh(x) — h(x) Ca2

quasieverywhere on the support of q' for q, E L2 with supp(p) ç B(O,1) and

h = (0Ga )*4. But, the definitions of 0 and q, then imply that supp(h) c B(O,1+2ij)

39

and so both q and h are localised.

2.4 The Sharp Capacitarian Result when p =2 and a >

We remarked in Section 1.5 that the behaviour of the Dirichiet kernel is worse when

the dimension increases (Computation 1.5.12). Because of this, it is impossible to

build in lie, n~!2, a "direct" analogue to the key Lemma 2 [Ca2, pp.51-52] in

Carleson's proof of Theorem 1.5.9. However, a "local" analogue can be built for all

a in the range (Lemma 2.4.5). But, before proving it, we need the following

two lemmas 11 :

LEMMA 2.4.1. If k is a positive integer or half of a positive integer then

x 2 Jk (x) is bounded on

f 12 Jk (x)dx ~ for all y E 1R with k a constant independent of y.

LEMMA 2.4.2. Let N,y,s E 1R. If k is a positive integer then

f - r 2 )' ( 5(k-2)/2 (2 7rNr)J k ,2 (2 irNs) - rfk12 (2 7rNr)J (k2)12 (2 rNs))dr .'!~ C

where Ck is a constant independent of y, s and N.

In these two results, Jk denote the Bessel function of the first kind of order k (see

[St4, p.338]).


(1):

Let first observe that

lim x112 Jk (x) = lirn(2k )- ' x 1/2 (Jk+l(x) + k-1 (x)) = 0

if k >4 and lirn f1"2J112(x) = (1)1/2 The first equality follows from the classical

x-O

Lommel's recurrence formula "tm (x) = 2(tn-1) J 1 () - m-2 (x) [R, p.47], while the

second one uses the fact that lim J0 (x) = 1 and lirn Jk(x) = 0 when k >0 [R, p.48]. X-40

Hence, x112 Jk (x) is bounded on (0,1] by the continuity of Jk and x 112 on 1R. But,

11 The study of Bessel functions has been so complete that these lemmas are almost certainly known, but, rather than having to search them through all the literature, we decided to include their proofs.

trivially, both Jk and x "2 are bounded on (1,00) so (1) follows.

(2):

By [P, p.2081, we have that Jk(x)=-\Jcos(x— 2) )+Ek (x) 12 for x> Ck >0.

Here, Ck is a constant depending only on k and IEk (x)I_< k x 312 . By (1), we can

suppose that y ~! Ck and deal only with x 112 Jk (x)dv. But, by Estimate 2.4.3, we

have

I j y x_h/2Jk(x)dH ~ I

x'cos(x - )dxI k5Yx_2d. C k Ck

The desired result then follows because it is well known that the first integral behaves

like the harmonic series.

REMARK 2.4.4.

Lemma 2.4.1 (2) is also true when k = 0 as J. is bounded and x 112 is integrable

near the origin.


The proof is done in two steps. We will first prove a transference technique which

reduces all the cases to either k = 1, 2,3 or 4 modulo a simple term which will be easily

controlled and then we will prove the four main cases.

The transference method is just a rewriting of the central part of the expression under

study

'k,N,r,s = SJ(k_ 2) ,2 (2 rNr)J,2 (2 irNs) - rJk,2 (2 lrNr)J(k_ 2) ,2 (2 rNs)

using the classical Lommel's recurrence formula jm (x) = 1) ml (x) - jm2 (x) [R,

p.47]. When k> 4, by two applications of this formula, we have

'k,N,r,s = SJ(k_2)12 (2 2rNr)(. "(k-2)/2 (2 ,rNs) - (k-4)I2 (2 JrNs))

k-2 (2irNs) ( (k-2)I2 (2 ,rNr) - (k-4) 12 (2 JrNr))J(k _ 2),2

= —SJ(k_ 2)12 (2)rNr)J(k _4) ,2 (27rNs) + rJ(k_ 4)1 2 (2JrNr)J(k _2)12 (21rNs)

12 We will refer to this approximation of J k as Estimate 2.4.3 in this thesis.

41

-

kdVr -4

- —S(J(k _4)/2 (2irNr) - "(k-6)12 (27:Nr))J(k_ 4),2 (27rNs)

I k-4 +rJ(k_ 4),2 (2 rNr),jj1 (k-4)I2 (2 irNs)

- (k-6)/2 (2 irNs))

(k-4)(r 2 -s2 ) = 'k-4,N,r,s + 2irNrs (k-4)/2 (2 lrNr)J(k_ 4),2 (2 irNs).

Hence, if we can deal with f(rs)_1/2 (k-4)I2 (2 lrNr)J(k _4)/2 (2 irNs)dr, the case k will

follow from the case k —4 and so all the cases will follow from one of the four cases

k = 1, 2,3,4.

But, by Lemma 2.4.1, we have for the necessary k:

If y(rs)-1/2 J (2 irNr) (k-4)I2 (2 rNs)dr

= I(Ns)_112 (k-4)12 (2 rNs)f)Nh/2r_h/2 (k-4)I2 (2 irNr)dr

SoNyr-

1/2 = i (k hI2J4),2(2JrNs)I I J (2irr)dr ~ C. ((Ns (k-4)12

Now, when k = 1, Lemma 2.4.2 is the essence of the proof of Lemma 2 in [Ca2]

written in a different way so we will not do it here (the argument is completely similar

to the case k=3).

When k = 3, we replace J112 (x) and J3/2 (x) by their respective value (i.e. /sin(x)

and I(x' sin(x) - cos(x))). After making all the simplifications, the expression to

evaluate is then

cç'(S2 - r2 ) -' (r cos(2 ,rNr) sin(2 irNs) - s sin(2rNr)cos(2 rNs))dr.

If we now replace sin(a)cos(b) by +(sin(a+b)+sin(a—b)) and group the terms in

sin(2rN(s+ r)) and the terms in sin (2,rN(s - r)) separately, we obtain

cf(s + r) 1 sin (2 ,rN(s + r))dr -

Cj"'(s - r)- 1 sin (2 ,rN(s - r))dr.

But, by Lemma 2.4.1 with k = -, we have J1 'x' sin(x)dx :5 C with a constant C

independent of y E ]R Hence, the desired result follows with a change of variables

when k=3

By the second line of the computation for the transference method above, the case

k = 4 is similar to k = 2 except that r and s are inverted. Like for k = 1 and k = 3,

we will consequently only deal here with the case k =2.

42

By a change of variables, the expression to evaluate is

JN1 ( xZ)hI2 (2 - x2 ) -1 (zJ0 (2,rx)J1 (2rz) - xJ, (2 rx)J0 (27rz))d !!~ C

where z = Ns to simplify the notation. Let f(x,z) denote the integrand in the previous

line and let C = Max(C0 , C1 ) where the Ck are the constants in Estimate 2.4.3 for J 0

and J1 respectively.

z

2C

C

2Cx

Figure 2.

The proof is then done by dividing f in four parts represented in Figure 2. For the

two parts on the right, we will use the idea in Lemma 2.4.1 (2), while, for the two

parts on the left, we will show that f is bounded.

In the region where both x, z and Ny are bigger than C, we have

f(N Y

3,r \ (z - x )' (z cos(2 7rx

- f) cos(2 rz - - x cos(2 rx - ) cos(2 ,rz -

jNy 2 2

by the asymptotic formula seen in Lemma 2.4.1 (2) (Estimate 2.4.3)

= jJNY(2 - x2 )-'(z - x)(cos(2r(x + z) - r) + cos(2r(x - z) -

= c15 (x + z)' (cos(2 ,r(x + z)) + sin(2 ir(x - z)))dx 15 C.

The first equality is based on the identity cos(a) cos(b) = I (cos(a + b) + cos(a - b)),

while the inequality follows like the harmonic series as x + z ~! 2 0.

Now, if x,Ny ~!2C and z!~ C,wehave

NY NY I x"2312 J0 (2 r)J1 (2rz) + x312 z" 2 J1 (2irx)Jo (2 VZ)

12 Jf(x,z) ~ JwI ZX 1f2C

N ~ cI xhI2z2 _x2rl

f2' 3/2 dx+ x (z 2 —x 2 'J1 (2Trx)dx 15

J2C

by z:5 C and the boundedness of J0 and J1

< CSNy

3/2 2 2 - 1 x 312 dx+Cl x (z — x) J1 (2irx)dx ~ C.

2C

In the last line, the first inequality is a consequence of x2 - z 2 ~ x2 -

x 2 - x 2 as

x ~! 2C ~! 2z. The first integral obtained from this is bounded because x ;~: 2C, while

the second one follows like Lemma 2.4.1 (2) as x ~! 20 and z :!~ 0 (for k = 4, this is

replaced by Remark 2.4.4).

When x!!~C and z ~:2C,

1 If(x,z)I ~ Ixz 1112 iz

2 — x2 1 —

'(IzJo(2 x)Ji(2 z)I+k11(2 x)J0( 2 z)I)

2 -1

~ Cz 1/2 ( Z 2 —x

2 ) (lzI± 1 )

by x:5 C and the boundedness of J 0 and J1

Cz3I2 (Izi +1) < C.

In the last line, the first inequality follows from z 2 —x 2 ~z 2 —*z2 =-z2 as

z ~! 2C ~! 2x while the second one is a consequence of z ~! 2C. Hence,

min{Ny,C} -

SIIU{NYC1f(XZ) ~ I If(x,z)Idx ~ C min{Ny,C} ~ C Jo

for z~!2C.

Finally, when x and z are in the white region in Figure 2, we define a modified

function f:

If(x,z) ifx#z J(x,z) I+x \ (jo x 2 +(J1 (x)) 2 if x = z

To obtain this modified function take the limit when x — z in the direction of your

choice and simplify the resulting expression using the classical equations (7) and (9) in

[R, p.47] (i.e. (x'Jk (x))=x"Jkl (x) and *J0 (x)=—J1 (x)). One can now easily

verify that the way x —* z makes no difference i.e. this new function f is continuous

on the compact set

{(x,z) : 0!~ x,z:!~ Cor(0!!~ x!~ CandC:!~ z!~ 2C)or(0:!~ Z!!~ CafldC!5Z:!~ 2C)}.

011

Hence, f is bounded on this set. As f = f almost everywhere, we can replace f by

f in the integral to evaluate and so the desired result follows from the boundedness of

f.

By the remark made above, we need to control

f(x, z) = —(xz) 112 (z2 - x 2 )' (xJ0 (2irx)J1 (27rz) - zJ1 (2 rx)J0 (27rz))

to prove the case k = 4. In this setting, the modified function f is:

If(x,z) ifx#z 1(x, Z)

= t(2Jo(x)Ji (x) - x(J0 (x)) 2 - x(Ji (x)) 2 ) if x = z.

To obtain this function, use Lommel's recurrence formula and equation (5) in [R,

p.47] (i.e. f = +(Jk-1 (x) - k+l (x))).

Equipped with the previous lemmas, we can now prove a "local" analogue of Lemma

2 in [Ca2]:

LEMMA 2.4.5. Let 0 be a radial bump function associated with the ball B(O,j) C IR".

Suppose also that 0 satisfies the properties listed in Lemma 2.3.5. If ' < a :!~ then

S*(OGa)(x) :! ~ Ca (1+Ga (x)) with Ca a constant independent of x.

PROOF.

If f is a radial function on IR" then it is well known that f is also a radial function

and that

(t) = 2 ,r J rt J( _ 2),2 (27rtr)f(r)r" 1 dr

(see [St4, p.430]). Here, f(r) means f(lxl) with lxi = r (the same remark applies to

f and all the lines below).

But, (SR (øGa ))() = ZB(O , R) ()(0Ga ) ('). Hence,

(SR (OGa ))(s) = Cs" 212 j. j.R r"'2 (ØG )(r) J( _2) ,2 (2 rts)J( _2) ,2 (2 ,rtr)tdtdr.

o

Now, using the symmetry like in [P, p.2091, we can rewrite this in the form

(SR (OGa ))(s) = CRs"212 5 (2 - r2 )' mfu2 (OGa )(!)'n,R,rA'

where 'n,Rr,s is defined as in the proof of Lemma 2.4.2.

45

To evaluate this integral, we break it in two parts (0, f) and (f , ,j). The second part is

easily controlled by Lemma 2.4.2 and integration by parts (on r''2(ØGa)(r) and

R(rs) 112 (s 2 - r2 )- 1 'nRrs)• While, for the first part, we use some really crude estimates

similar to [P, pp.209-2 10].

Let, first control s

(2.4. *) I = CRs" fo 12

r' (OGa )(r)J( _ 2) ,2 (2 irRr)J,, (2 ,rRs)dr. J

If <— c = Max(q fl_2),2 , C,2 ) (the Ck are the constants associated with J( _ 2)/2 and

J,2 in Estimate 2.4.3) then

I ~

CRS-n12$0

s/2

r12(OGa)(r)dr

< CS — IS —(n-1) 12r12 "112 (OGa )(r)dr Jo

!!~ Ca (1+Ga (S))

by the boundedness of Jk

by -~c and r~ f

by Proposition 1.3.5 (3).

While, if c, we break 1 in two pieces I and 1 corresponding respectively to the

regions of integration (0,-i) and (-,f). Then, we have

CIR

11 :2~ by Estimate 2.4.3

~ CR 112 s 112 (cR1 )1/2 $0

dR

r'''2 (øGa )(r)dr

~ Cs"112 1l2 r" 1 ' 2 (OGa )(r)dr Jo

:!~ Ca (1+Ga (S))

MI

'2 CRS-n12 (Rs)-112

s12 (Rr)"2r''2(ØGa)(r)dr

Ii R

~ Cs'' $S12

r(12 (øGa)(r)dr 0

< Ca (1+Ga (s))

Now, the same estimates are also valid for


by Estimate 2.4.3


cRs-0250

s12

rs'r"12 (ØGa)(r)IJ(fl_2) 12 (2rRr)J 12(2rRs)Jdr

because r:~ f. So, we have the desired control over the part (0,f)

REMARKS 2.4.6.

It is easy to verify that Lemma 2.4.5 is false for a E (0,1] and there is no way to

correct this. The estimate for the part (, ij) still works because we can replace

r"''2(ØGa)(r) by rks_l_22 (Ga )(r) with a k big enough to have integrability

using 2rs 1 > 1. While, when a E (4,] and Ns < C, this is also true for (0,f)

if we replace the estimate I' (X)I C of Prestini [P] by the classical estimate

I Jm (X)I!~ 'xm for x small and use the fact that N < Cs'. But, when Ns> C or

a e (0, 4], there is no solution (all the cases are similar so to be convinced just

study the case n=3).

Hence, a different technique will be needed to prove Theorem 2.4.7 when a is in

(0, 1 ]. One should also expect from the comment above that, if such a technique

is built, the range (0, 4] will probably be more difficult to deal with than (4,z1].

The range (0, 41 often appears as a limit to the actual understanding of both the

Bochner-Riesz problem and the inversion problem for functions in Z (e.g. [Ch,

Theorem 1] and [CS 1, Theorems 6 and 7]).

In [Ka], Kanjin proved that p E (r'r) is best possible when trying to get

SRf(x) —> f(x) almost everywhere for all radial functions f L"(IR"). With this

in mind, it is interesting to note that, when a E (!-,], OGa e L"(IR") for some

p in the range while ØG LP (1R) for any p in the range if

aE(0,].

Lemma 2.4.5 is a refinement of Theorem 2.2.8 for f = OGa . As we will see in

the proof of Theorem 4.1.10, Theorem 2.2.8 is enough to get the "almost

everywhere version" of Theorem 2.4.7, but to get the sharp result (Theorem

2.4.7), we absolutely need a stronger estimate like Lemma 2.4.5.

Combining Lemma 2.4.5 with our localisation principle (Theorem 2.3.7), gives the

following analogue of Salem and Zygmund's result:

THEOREM 2.4.7. Let f Z(IR") with < a ~ and 2 ~ p < . Suppose also

that g E LP(1R) is such that f(x) = (G* g)(x ) almost everywhere. Then

Sf( X )(G*g)(x) C 2 -quasieverywhere on ]R".

PROOF.

By the localisation principle (Theorem 2.3.7), we can suppose without loss of

47

generality that supp(g) c B(O,2) and we must show that

SRf(x) = S(G* g)(x ) (G*)(X)

C,-quasieverywhere on any ball contained inside B(0,2). We will do this for

B(O, 1), but the argument is similar for any other ball.

Now, let p be a Borel measure supported on B(0, 1) which satisfies the condition of

Proposition 1.4.6. By Remark 2.3.6 (1), we can limit ourselves to only consider OG,,

rather than Ga for 0 a smooth radial bump function supported inside B(0,1). Hence,

as in Theorem 2.3.7, we will be done if we can show that

5S. ((ØG)* g)(x )dp(x ) :!~ '1IglI.

But, this follows easily from Lemma 2.4.5:

$ S((OG)*g)(x)dp(x)

:!~ $(Igl*S* (OGa ))(X)dp(X) by Estimate 2.2.2

= 5(gI*(x B(o,3) s.(OGa )))(x )dp(x) by supp(/i) c B(O,1) and supp(g) g B(O,2)

~ IIlL (xB(o.31 5. (OGa ))*p by Cauchy-Schwarz

~ IIII (Ca%B(O 3) (1+ Ga ))*pD by Lemma 2.4.5

1/2

=

1912 (15 ((CaZB(03) (1 + G,, ))* ( CaZB(O 3) (1 + Ga )))(.i - y)dji (y)dp (x))

~

1912 (Cff (1+ G2a )(x - y)dp(y)dp (x)) 1/2

~ C119112 !~ CJJgIJ,, by supp(g)B(O,2).

The penultimate inequality follows from 5$ G2 ,, (x - y)dp(y)dp(x) :!~ 1 and from the

properties of p in Proposition 1.4.6 which imply that

p(B(O,1)) :! ~ (Ca2(B(O, 1)))v2 < c< 00 .

REMARK 2.4.8.

Modulo the proof of a localisation principle similar to Theorem 2.3.7, but valid

C p quaSieve1ywhere, the previous argument can also be used to show that

SRf(x) -* (G*g)(x) C p -quasievrywhere on IR" when f E Z(IR") with

<a !~ and 2 ~ p < . To do this, replace the Cauchy-Schwarz inequality

by Holder's inequality in the previous argument.

Chapter 3

Localisation Principles

In this chapter, we will study different localisation principles for the inversion

problem. This study will start with the classical localisation principles of Riemann,

Sjölin, Carbery and Soria. We will then build a technique based on Fefferman's folk

computation (Folk Lemma 3.2.1) which will enable us to prove some capacitarian

versions of these results. We will in particular obtain an everywhere localisation

principle (Corollary 3.2.9) which will be used in Theorem 3.2.11 to extend Theorem

2.4.7 (the Salem-Zygmund type of result) to 2 <p <. To finish our study on

localisation, we will rapidly present a more general context in which most of the work

done in this thesis can be expected to be generalised.

3.1 On Some Known Localisation Principles

As we have already seen in Chapter 2, a localisation principle for the inversion

problem is a proposition which explains how two functions with the same local

behaviour in a region R react under the partial sum operator inside R. For our

purposes, the main advantage of these principles is that they allow us without loss of

generality to restrict our study to functions with bounded support. This aspect was

made particularly clear in Chapter 2 (Theorems 2.1.1 and 2.4.7). A classical example

is Riemann's localisation principle:

THEOREM 3.1.1. [Kt2, p.54] Let f L! (T) and suppose that f 0 inside an open

interval I. Then SNf(x) -30 for all x E I. Moreover, the convergence is uniform on

any closed subinterval of I.

This theorem covers in particular the L2 -functions as L2 (T) c L' (T). Its proof is

based on two lemmas:

LEMMA 3.1.2. [Kt2, p.53] If f E E(T) andJIf(x)f h ldx < oo then SNf(0) - 0.

50

LEMMA 3.1.3.13 [Kt2, p.13] If f L' (T) then limf(k) = 0. Moreover, this holds

uniformly on compact subsets of L' (7).

An important feature of this localisation principle is the uniformity it provides. In

higher dimension, n :?! 2, we can not have this uniformity even for "nice functions"

(e.g.: f E as the following counter-example of fl'in shows:

ExAMPLE3.1.4. Let Then there is an feAa rCc (IR) such that

f 0 on B(O, 1) and lirnsuplSRf(0)I = 00

DEFINITION 3.1.5. Let k be the smallest integer greater than a. The space of

Lipschitz continuous functions of order a >0 is defined as being

A ,,, (IR " ) = {f E L("): 4-u(x,y) ~ Ay+z}

where u is the Poisson integral of f, —(+1)12

u(x,y) = f( L), n4112 5 y(1t12 + y2 ) n f(x - t)dt.

The norm on the space A a is IIfIL = IIfL + Ak where A k is the smallest constant A

such that the inequality above holds. If 0 < a < 1 the previous definition is equivalent

A,,, (R " ) ={fE L(1R) : IIf(.+t)-f(.)IL :!~ AtI"}.

A more detailed presentation about these spaces and their properties can be found in

[St3, pp.141-15911.

PROOF OF EXAMPLE 3.1.4.

The following argument is an informal proof of A. Carbery based on a classical

approximation of the Dirichiet's kernel. The reader is referred to the original English

translation of Il'in article 14 for a more rigorous proof.

By the uniform boundedness principle (Banach-Steinhaus theorem), if SRf(0) was

convergent for all fE Aa (")flCc (R) with support in B(0,10) such that f O on

B(O,1), we will have ISRf( 0)I~ cljfL for these f.

13 This lemma is known in the literature as the Riemann-Lebesgue lemma. 14 V.A. Win, The Problems of Localization and Convergence of Fourier Series with Respect to the

Fundamental Systems of Functions of the Laplace Operator, Russ. Math. Surv. 23, No.2 (1968), 59-116. An account of this proof can also be found in [KhN, pp.35-36].

EOi4 C:)

Lets now show that this is impossible if a

SRI(0) = JKR(x)f(x)dx = $ KR(x)Ø(x)f(x)dx = 5(KRO) ' ()f(c)dc

2

Ya12 2 a/2

= 5(KRØ)()(1+II ()(l+ ) ()f()dc

Y/2

R5(KRO)'()(l+I , ()f()d

= R 5 KR (x)Ø(x)Daf(x)d

where 0 e C is a bump function such that 0 a l. inside B(0, 10) and Df is the

Bessel derivative of order a of f. The second equality follows from the support

property of f while the approximation is based on the fact that R on support of

(K R O) (this is direct if 0 is compactly supported if it is not the case cut 0 as in the

proof of Theorems 3.1.8 and 2.1.1 and apply the previous argument on each piece).

The computation above combined with the classical approximation (see Computation

1.5.12)

KR (x)= R"e(1+RIxi)-(n+1)/2

implies that ISRf(0)I!~ CIIfIIA for all f Aa (lR)ThCc (1R") supported in B(0,10)

such that f 0 on B(0,1) if and only if

sunR'S (1+ RIxI)'''2dx < oo • txI -1

But, this is true only if a>.

It is also worth mentioning that not only do we lose the uniformity in higher

dimensions, we also do not have everywhere convergence as the previous example

shows. This cannot be improved even for functions in a space of Bessel potentials.

In higher dimensions, the first localisation principle was proved by Sjölin [Sjl] using

ideas in the spirit of his work on spherical means at the beginning of the eighties:

THEOREM 3.1.6. [Sjl, p.4131 If f e L2 (1R) and supp(f) is bounded on 1R then

SRf(x) - 0 almost everywhere off support of f.

The proof of his principle is essentially contained in the next lemma.

52

LEMMA 3.1.7. [Sjl, pA151 Let KR be the Dirichiet kernel on IR" and let (P C= S be

such that (0) = 0. Define the operator TRf =(pKR )*f for f L2 (") and R> 0. If

7f = s,IfI is its associated maximal operator then II2fII2 !~ Cf with C a

constant independent of f.

We will not prove Lemma 3.1.7 here because the argument necessary is essentially a

direct combination of the fundamental theorem of calculus and of Lemma 3.1.11

which will be discussed later in this section.

There is a major problem with Sjölins principle; it only applies to functions with

bounded support. Considering that our main goal in finding such principle is to

transfer the study of a general function to the study of a function with bounded

support, we can only see it as a bench test to try some idea. But, this problem can be

overcome.

Carbery and Soria [CS 1] put together all the ideas in Sjölin's work in such a way that

the proof of Theorem 3.1.6 can be reorganised to prove a complete localisation

principle rather than only a version restricted to boundedly supported functions. Their

key idea is that the good behaviour remarked in Lemma 3.1.7 is in fact nicer that what

was previously observed by Sjölin. We do not only have control on the multiplier

when the function is boundedly supported, but we can in fact split our operator in

small pieces all well behave and such that we can add them all together.

THEOREM 3.1.8. [CS 1, p.324] If f If (]R") and 2:! ~ p<In then SRf(x) —* 0

almost everywhere off supp(f).

PROOF.

Using translation and dilation invariance, we can suppose without loss of generality

that f is supported in (B(0,3))c. To prove the theorem, we need to show that

SRf(x) - 0 almost everywhere in any open ball contained in B(0,3). In what

follows, we will only do this for B(0, 1) but the proof is similar for any other ball

B(0, r) with r <3. More precisely, we willrestrict ourselves to show that

2/p

(3.1.*) Cu (B(O,3))'

If(x)Idx) = CllfII

because this implies as usual the desired pointwise result (see for example Corollary

1.2.2 and Theorem 4.1.5).

53

Let 0 be a radial function such that :!~ 0 !!~ and let i(x) = 0(x) - 0(2x) .

Then,

1= 0(0) = 0(x) + (o() - O(*i-)) = 0(x) +

where y. (x) =

Now, by applying this partition of the unity to both KR and f, we get

SRf= KR*f=(OKR)*f+lK*(Of)+JlKj*fk

where Ki = KR V1J and fk = flIfk. But supp(Ø) ç B(0,2) and supp(f) c (B(O,3))c, so

Of 0 which implies that the first summation is 0. If x e B(0, 1), we also have that

((OKR )*f)(x) = 0 for the same reason while (K*fk )(x) = 0 under the additional

condition 1 ~! 3 as supp( Vim) B(0, 2m+1) \ B(0, 2m1). Consequently, we only

need to show that

2

(*) IB(O,1)SU K*f1+ f (x)dx CIIfII R>I i=

for Isi < 3 to prove 3.1. *. We will only do here the case s =0 because the five

possibilities are similar.

When s = 0 and p = 2, (*) is the particular case 0 < a = y< 4 (a arbitrary) and

gj = f = fyfj of the following lemma 15 :

LEMMA 3.1.9. [CS1, p.325] If 0!!~ a<-4, y>O and {91 } ° 1 is an arbitrary sequence

of test functions then

2

f sup1jns Kj*g' (x)-- < Cay 22 S g1 (x)J

Ixl2a 1R R>i IXI2 - i> 1

where K is defined as in the proof of Theorem 3.1.8.

Accepting for the moment the truth of this lemma, we then have as desired

.IB(0,1) s.f (x)12 dx ~ Ca7 5(B(0,3))"

If (x)1 2 Ixr dx !~ CIIfII

for f supported in (B(0, 3))C• Here, 17 = 2 a -2y < 1 (q arbitrary), hence, the last

inequality above follows from the inclusion L((B(0,3))c,dx) c L2(Ixrd..i) valid

15 When p;& 2, we replace O<a=y<4by n(1- 2 )<2a-2y<z1 with O:~ a<+ and y>O.

54

when n(1 - <11 (i.e. 2 ~ p <

Lemma 3.1.9 is based on three lemmas (3.1.10, 3.1.11 and 3.1.12). Of these three

results, Lemma 3. 1.11 is the most important for us. This is particularly clear if we

remember Lemmas 2.1.2 and 3.1.7.

LEMMA 3.1.10.[CS 1 , p.325] 16 Let fEZ where 0:!~ a<--. For t>0, there is

constant C,, independent of t and S such that

.{{E1R'1I—:Iss} If( )l d < CaS2a If II.

LEMMA 3.1.11. [CS 1, p.3261 Let m()=(KI)"() where K,' is defined as in the

proof of Theorem 3.1.8. Suppose that 0 E N, 2 ~: 0 and t >0. Then

dfl I

in C (i + 2 i 114 1 -

dtfl

with a constant independent of t and j.

LEMMA 3.1.12. [CS1, p.326] Let y:IR ---> IR be afixed smooth function such that

y(x) 0 when x !! ~ -

and y(x) 1 when x ~: 1. Let also L = y(t)K/ for t E IR with

K defined as in the proof of Theorem 3.1.8. If 8 ~! 0 and 0 !!~ a < then

2

lxl -2a 2

5 fl(D1 L*fXx) dtdx ~ C222 f5lf(x)l lxi

where DO acts with respect to the t-variable.

We used here Dfl to represent the fractional differentiation operator of order /3 on R.

This operator is defined by (Dh)') = Ih(), e R. Equipped with these three

lemmas, we can now return to the proof of Lemma 3.1.9.


We will use directly the notation and assumptions built in Lemmas 3. 1.11 and 3.1.12

throughout this argument. Now,

2 2

SuD"V Kj,*g ~ sunl0 L* g4 ~ C I ' D1 L1 *g4dt J= 1

'3JIRI j=1 '

R~'l V'i'

16 This lemma is closely related with Lemma 3 in [CRdeFV].

55

if 8 > 1/ 2. The first inequality follows from the definition of y in L as the first

supremum is taken on R ~: 1 while the second inequality follows from the inclusion

(1R)c C(IR), /3>1/2 (Sobolev's imbedding theorem, Theorem 1.3.9 (3)). Hence,

1/2 . I I (K* g )(x )xj -2a

dx) i~ I j=1

1/2

5C (L ~

(DL*g 1 )(x)2

dtlXi 2a dX)i

2 -2a \1/2

~ c i"" (I i (DA3L*g1)(x) dtlxl dx) by Minkowski's inequality

/3 =i g"

1/2

~ C T'1 2_22ih3(L.

gj(x)x_2(L dx) by Lemma 3.1.12 if 0 <_ a <4

c$ (_122lh2_2.h7)1/2

(1-12 2ir f g. (x)12 1/2

ixl_2a dx) by Cauchy-Schwarz

2 1/2 (' li

-2a dx) if y>0-4 i.e. y>0.

-

We will now prove the crucial Lemma 3.1.11 while Lemmas 3.1.10 and 3.1.12 are

left to the reader (A proof of Lemma 3.1.12 can be found in [CS 1, pp.327-328], while

an argument similar to the proof of Lemma 3.1.10 is contained in [CRdeFV, Lemma

3].).


By a change of variables, it is enough to show the case t = 1. In the following lines,

we will only consider the case /3 =0 as all the others are done in a similar way by re-

expressing -m() in terms of j-m() and a convolution involving the

derivatives of order ,(3 of (for the details on how to do this see for example [Sj 1,

pp.417-418]). The proof is done by splitting the 4 in two classes: < 1 and > 1.

Let first suppose that 141 < 1. Then

ki (4)1 = I(XB (0 , 1) * ( if), )( ~ )j

= L12 1)(XB(0,1) - ii) - 1)d77 by 5 = i(0) = 0

!~ I (i)2 ())d

56

I ' I( O2i (ij)d

J iI~ '-II

~ I Ju1(u)Idu byII< 1 JIUI~2 III II

!~ c(1+2uIjI-1I) by ieC°

for any 2 ~!O. Here (qi) (77)_.2'I(if)(2i )

When 141 >1, we get in a similar way that

= (7i)2(rj)di7 ~ $ I()21()k' ~ CA( 1 + 2 II 1 ) I'iI ~III - 'I

for any /1 ~! 0. In this case, the differences are that we compute directly m()

without using the trick in the second equality above and that the first inequality is

obtained from the inverse triangle inequality and > I rather than the triangle

inequality.

DEFINITION 3.1.13. E c B(0, 1) is a set of divergence for the localisation problem if

it is possible to find an f E L2 (IR) such that f 0 on B(0,1) and SRf(x) diverges

for all xEE.

Comparing Carbery and Soria's localisation principle (Theorem 3.1.8) with Il'in's

example (Example 3.1.4) raises the question of trying to understand which sets on

B(0, 1) can be sets of divergence for the localisation principle. This question will not

be studied in this thesis but some of the ideas developed to answer it in [CS2] are of

interest for us (See Theorem 3.2.2). The proofs containing these ideas cover in

particular the following partial answers:

THEOREM 3.1.14. [CS2, p.3] If E ç B(O,1) has Hausdorff dimension less than n —1

then E is a set of divergence for the localisation problem.

THEOREM 3.1.15. [CS2, p.3] If n ~! 2, there is a set of divergence for the localisation

problem which has full Hausdorff dimension (i.e. dimension n).

THEOREM 3.1.16. [CS2, p.41 If F(0,1) has Hausdorff dimension greater than

and if n ~! 2 then E(F) = {x E IR" :IxI E F} is not a set of divergence for the localisation

problem.

57

3.2 An Approach to Localisation Based on Fefferman's

Folk Computation

Following a line of ideas started in [CS2], we will develop in this section an approach

to localisation based of Fefferman's folk computation (Folk Lemma 3.2.1). We will in

particular prove an everywhere localisation principle (Corollary 3.2.9) for functions in Zp (lR'7 ) with 2 :!~ p < fr and a > as well as some other localisation results.

Nevertheless, it seems that the interest of this approach lies not only in these results,

but also in its possibilities to be used in the future to prove a C analogue of the

capacitarian localisation theorem (Theorem 2.3.7) for functions in (]R") with

2:!~ p<and a>O.

3.2.1 Fefferman's Folk Computation

In [F2, p.451, Fefferman introduced a heuristic approximation for the partial sum

operator. Even though his idea is just beginning to be formalised ([CS3] and some

recent work of T. Tao), it is often a good starting tool to understand the behaviour of

SR.

FOLK LEMMA 3.2.1. (Fefferman's Folk Computation) Let f e LP (R") be compactly

supported. Then

SRf(x) = R"112ixi

—(a+1)12 2,riR e kI](ff)

for large values of x.

FOLK PROOF.

Using the classical approximation for the Dirichlet's kernel (Computation 1.5.12), we

have

SRf(x) = ((R (n-1)/2 1-1— (n+l)/2 e 2 riRlol )*f)(X)

= R '"1'2 $ ix - y''2 e f(y)dy 2,r,

R "'" - e fix yi

-(n+1)/2 2iriR(IxI- (x .y)14') f(y)dy

2 riRIxI R"'2 IxI - (n+1)12 e $e_2xf(y)dy by ii << ixi

= R "''2ixi e -(n+I)12 216RlxI kIxI /

W.

The second approximation follows from the MacLaurin expansion of g(y) = lx - yl (i.e. g(y) = lxi - (X - y)lxr').

Fefferman's computation is closely related with the restriction theorems for the Fourier

transform. These are results stating that the Fourier transform seen as an operator is

If - If bounded when restricted to a manifold satisfying an appropriate curvature

property. For example, when S is a compact subset of a submanifold S c: IR" of

dimension n —1 with non vanishing Gaussian curvature, we have

(fS . 1 1 q

da(c)) !~ '11fiI

if 1 :!~ p :!~ and q = (n1) for f S and dc the induced Lebesgue measure with

respect to S. In particular, using this result, the Fourier transform of any f e U can

be defined on S as S is dense in U. This result will not be proven in this thesis, but a

proof can be found in [St4, pp.386-388.

Using Fefferman's computation, Carbery and Soria were able to build different

heuristic estimates in [CS2]. They proved in particular the case p = 2 of the next

theorem [CS2, CS3]. A minor modification of their argument shows that in fact the

same holds for 2 < p <

THEOREM 3.2.2. Let n ~! 2. Suppose that f E L"(]R"), 2 :!~ p < , is such that

supp(f) c (B(O, r))c, r E 1R. Suppose also that /2 is a finite positive Borel measure

supported on B(O, r) satisfying

(*) (gdii ) (R•) IL2 (S'') _ C2 R 1 "'2 11g112 (dz)

for all g. Then JJSJ 1IL2 (dp) !~ C' where C is a constant independent of f. If l lp

PROOF.

By dilation invariance, we can without loss of generality restrict ourselves to the case

r = 3. We also suppose that supp(p) ç B(0,1) to keep our notations consistent with

Theorem 3.1.8 but the same argument work for any closed set inside B(0,3). The

proof will now be done in four steps. We will first study heuristically the L2 case

using Fefferman's folk computation to obtain a weaker estimate under the theorem's

hypothesis, namely,

59

(3.2.a) su II SRfIIL2(d 4) !!~ C, 11f112

for f e L! (R " ) with supp(f) g {x:px } and i e IR. This estimate will then be

used to prove another heuristic L2 estimates implying Theorem 3.2.2 and finally we

will explain how to formalise the two first steps.

So, we want to show that su?IISRfIILZ(dp) ~ for f supported in

(B(O,1))c but this is equivalent by duality with

(3.2 .a) IISR (9d1u )11L2 ((fi(oF))) !!~ CT - 1 IIgIIL2 (dp)

And, using Fefferman's computation, 3.2.5 can be rewritten as

R(T_ 1 )' 2h (gdt)(.ft) ~

L2 ((B(O,i))')

which is true by hypothcis (*) and 5 x2dx = F1.

Now, the proof of Theorem 3.2.2 closely follows the proof of Theorem 3.1.8 but

K1 (x) will be approximated here by t' (1 + tIxI)"12 eUN (see Computation 1.5.12).

To avoid repetition, we keep here the notations made in that proof. After rewriting

everything with the same partition of the unity as in Theorem 3.1.8, we are left to

show that

(3.2.b) J sup (K * (f~ ))(x) d (x) ~ C( 1 If(x)I )2/P

= IfII2

for Isk3.

Let suppose for the moment that we have

(3.2.c) 5sup(K*g)(x)2d/J(x) :! ~ C25 Ig (x)I2 dx IxI -2

for g supported in {x:IxI - 2'}. Then, by Holder's inequality, we will have

(3.2.d) 1su I(K/*g)(x)2d(x) ~ C22(fixi-2i

Ix(x) 2/p

)

< C2"" 1i !I

-

g .1

with I +2=1 But 2 ~ p< r ,hence P q

Consequently,

5 (K/ * f+ )(x) dji (x) <— $ (;i pI('7 * f+ )(x)) d1a (x)

60

:!~_ 1 sup(K1*

2 1/2

d1(x)) )2

j=1 j 5 )(x)

by Minkowskits inequality

1/p\ 2

~ C22(f+V fj ~ (x)I dx) by 3.2.d j= 1

~ diIfII2 V_2)

2

=

,,i 2 p(2

j=3

So, Theorem 3.2.2 will be proven if we can show 3.2.c and formalise 3.2.a. But,

3.2.c follows from the cases /3 = 0,1 of

(3.2.e) J J°'(- K,* g)(x) 2 dtdp(x) < C2 -2j(1-0) f Ig (x)I 2 dx d,fi fix[-2i

by the usual majorisation based on the fundamental theorem of calculus and Cauchy-

Schwarz inequality. In this case, the boundedness of the adjustment term generated by

the fundamental theorem of calculus is a consequence of 3.2.a. The two inequalities

(with 8 = 0, 1), are similar, consequently, we will only deal here will the case 8 = 0.

By duality, 3.2.e is then equivalent to 1/2

(3.2.0 f (iq * (f(., t)dp ))(x)dt[?Qx:JxJ-2iJ)

5 C2 (f 5 If(x, t)12 dt (x)dt)

And, this inequality is true because

2k*l

))(x)dtD : 52k

(K *(f(.,t)d

D * IIL? ((x:lxl-2i )) with 0 E satisfying 0

(3.2 .g) 2'' 5 e'fl'N (f(., t)dp )A

(j)o(*)dtD 2 ({x:Ixi-2)) k=1 2j(2

by Fefferman's folk computation, fri 2k and lxi -. 2

2j+l

= (5s 1 52) I 2k(lI)/2

5 2,nzr 2 1/2

• k=1 2j(+ 1 ) 1 2 e (f(., t)dp )A

(tw)O(*)dt r1drda(o4)

1/2 t 2zizr

< - 2j12 (s- s 212

J e (f(., t)dp )A

(to)O(*)dt drda(w))

61

1/2

= C2- $ f(.t)d/2)(to)Ø(*)dtd(0))) I 2k(n-1)/2( JR -

by Plancherel's theorem 1/2

(3.2.h) C2- 1$ by 0 X[ 1 ,2] J1 S''

1/2 1/2

~ C2 (5: t $ If (x, t) 2 dji (x)t 1 dt) = C2 (5 L If(x, t) 2 d/1(x)dt)

The last inequality follows from the hypothesis (*) for each fixed t.

Finally, 3.2.a and 3.2.g are formalised in a similar way using Plancherel's theorem,

consequently we will limit ourselves to prove 3.2.a. Trivially, the hypothesis (*)

implies that

(3.2.i) .1{:III_RI ~ l} (gdp

)A

( )12 d ~ (d/L)

Hence, we will have finished if we can show that 3.2.i implies the dual form of 3.2.a,

3.2. a.

Now, let KR(x) = co(x)K R (x) + (1— co(x))K R (x) = KR1 (x) + KR2(x) where e

with support inside B(O, 4). Let us also choose a 11 E S such that V(x) = 1 on

B(O,). Then 3.2.ã follows from 11(1— ip')S(gd/i)j 2 ~ which itself follows

from

(3. 2 .jk) Do - v1)(KRk*(gdi1))2 15 091IL2(d,)

for k=1,2.

But, supp(gdj.i) g B(O,1) and supp(K 1 *(gd/2)) g B(O,) by our hypothesis on the

support of q'. Consequently, (1— K*(gdJ1))(x) 0, so 3.2.j 1 is true. While,

11 lfIII—Rl ~ 2

((1— )*%B(oR))()

r(III - RI) fIII - RI>2

because q has been chosen such that qx (o,) is a smoothed version of ZB(OR) on

scale 1. In the previous inequality, y is a rapidly decreasing function (almost 0 when

1141 - RI> 2). Hence, by Plancherel's theorem and 3.2.i, we have

II (I -V ) (KR ,2*(gdjU)) 11 2 :5 IK(' -

row

~ CgIJ3(J/) .

If we add some smoothness to the function f, we can let the Fourier transform of

gdu on S' have less decay than in Theorem 3.2.2.

THEOREM 3.2.3. Let he L"(lR) with 2 !!~ p <r and n ~ 2. Suppose that

f = (ØG)*h = Ga *h on with 0 defined as in Lemma 2.3.5 and a e (o,..].

Suppose also that supp(h) (B(O,r+ e))c (so supp(f) c (B(O,r))c), r e , and that

t is a finite positive Borel measure supported on B(O, r) which satisfies

(**) (gdp)(R.) < L2 (S'') -

for all g. Then IIS*fIIL2(d,) !!~ Cjj hjjP where C is a constant independent of f.

PROOF.

By dilation invariance, we can without loss of generality restrict ourselves to the case

r = 3. We also suppose that supp(ji) B(0,1) to keep our notations consistent with

Theorems 3.1.8 and 2.3.2, but the same argument works for any closed set inside

B(0,3). The proof of Theorem 3.2.3 is then done by mimicking the proof of Theorem

3.2.2. We will consequently do the analogue of 3.2.a as an example and then we will

only describe the main modifications to do to the previous argument to prove Theorem

3.2.3.

We want to show

(3.2.a') sIISRfIIL2(d) ~ CFhj

for he L2 (IR") and f = Ga *h almost everywhere with supp(f),supp(h)c {x:IxI -

and 0< s << i. By duality, this is equivalent with proving that

I II81IL2(d).

Using Fefferman's computation, this last inequality can be rewritten as

R"''2 I.' V -(n+1)/2 ä,, (R)(gdi) ()D ~ ci:-! IIgIIL2 (du) II

which is true by hypothesis (**), G,, (R)R a and =

Now, rather than applying the partition of the unity directly to f like in Theorems

3.1.8 and 3.2.2, we apply it here to h. The small increase in the size of the support

induced by Ga to the different pieces (Ga * (hvf)) can easily be seen to be negligible.

Formally, we can keep track of these perturbations in 3.2.b' or, in a slightly less

formal way, we can just replace the dilation of order 2i by dilations of order ci for

c> 1 large enough to "make them negligible" (We will keep the notation 2i for

simplicity in this argument.). This lead us to replace 3.2.b by 2/p

(3.2.b') fsu Iy (Kj*(Oa *(hyuj+s)))(X 2 dii(X) < qf Ih(x)Idx) = C1IhII. t~ PIL_Jj= 1 'l~3

As for Theorem 3.2.2, we will be done if we can show that

(3.2.c') $ spl(Ki* (X) 12

di (x) < C2 11 g(x) 2 dx

for g supported in {x:IxI - 2i}. But this follows as in the previous proof from the

fundamental theorem of calculus with estimates analogous to 3.2.e and its dual 3.21, 1/2

(3.2.f') 1 k=1 f2, (K i * C. * (h(-,t )dy ))(x)dt

2 C2_1(JJIh(x,t)12d11(x)dt)

L ((x:IxI.. 2 ))

In the equivalent of 3.2.h as well as in the formalisation of 3.2.a' and the equivalent of

3.2.g, the lack of decay in hypothesis (**) is compensated as in 3.2.a' above by the

decay coming from Ga .

REMARK 3.2.4.

It is easy to verify that the hypothesis supp(h) g (B(O,r + E))c is not necessary in

Theorem 3.2.3 (supp(f) g (B(O,r))c is enough), but we prefer to keep it here as

it is the form of this theorem that we will normally use in this thesis.

Theorem 3.2.3 will be our main tool to build capacitarian localisation principles in the

next two sections. Under its actual form, it will allow us to prove directly localisation

"off support of g" (f = G* g almost everywhere) while the "natural" localisation

takes place "off support f" when Theorem 3.2.3 is in the form of covered in the

previous remark.

3.2.2 On a Closed Sets Condition

We observed in the proof of Theorem 2.3.7 that the sets of divergence generated by

ZI

SR are Borel sets. Consequently, we can always use Definition 1.4.6 to compute

them. But, Definition 1.4.6 gives us a condition on the energy, while it will sometimes

be easier to have a stronger condition on the potential like in Definition 1.4.13. In this

section and in Section 3.3, we will study two results obtained using a combination of

Theorem 3.2.3 and such a potential condition. Because of Definition 1.4.13, they will

both contain a restriction to closed sets similar to the condition in [SZ] 17 .

From the one dimensional case ([SZ] and [Ca2]), we should expect to be able to

remove this restriction, but possibly at the cost of finding a completely different proof.

This was done for example in the localisation Theorem 2.3.7 where a stronger version

of Theorem 3.2.6 was obtained. What is more interesting is that we will be able in

Section 3.2.3 to improve on this work while still using the part of our argument

coming from Section 3.2.1 and not only by finding an alternative technique of proof

like in Theorem 2.3.7.

Before proving our first "restricted' localisation principle, we need a lemma relating

the energy associated with a Ca2 -capacity and the Fourier transform on S 1 of a

measure p satisfying the conditions of Theorem 3.2.3.

LEMMA 3.2.5. Let p be a finite positive Borel measure on IR" with compact

support. For 0 !!~ a <n, there is a constant C independent of p (but depending on the

size of its support) such that

P (r) y)dp (y)dp(x)

for r>1.

PROOF.

Let first prove this lemma for 0< a <n.

We approximate p with functions in C (IR"). Hence, its is enough to show that

5,1f(r4 f da(4 ) < Cr' 55 G,,_ (x - y)f(y)f(x)dydx

for f e C,- (R') such that f is non-negative and supp(f) ç B(0,1).

Let ii e ç(IR") be a radial function supported in {x E IR": 4< IxI < 21 such that

17 This restriction to closed sets is the only real similarity between this section and [SZ].

65

I- k=-2 v,(2kx) = L=-2 V'k(X) 1

when ixi ( 2 and x # 0. To simplify the notation, let also q(x) = k=-2

Now,

= 5s'' (f..

f(x)eb)(5, f(y)e''dy)dcY()

= I fR. (fs.- , e

= 5 5, - y))f(x)f(y)dxdy

= 5, J, &(r(x - y))cp(x - y)f(x)f(y)dxdy

by supp(f)B(0,1) and pl in B(0,2)

= 5,5 &(r(x - y))(_ 2 41k(X - y))f(x)f(y)dxdy

= k=-2 15 ô(r(x - y)) I'k (x - y)f(x)f(y)dxdy = k=-2 Lk (r)

where ô- is the Fourier transform of the measure o on S". The majorisation of

k=_2I (r) is now made in two steps. We will first bound L and then, using this

estimate, we will bound the others Lk . Now,

L0 (r) = $5 &(r(x - y)) 4(x - y)f(x)f(y)dxdy = $5 k,. (x - y)f(x)f(y)dxdy

= 55k(X_y)f(X)f(Y)ddY = 5(kr*f)(X)f(X)dX = j2

where kr (X) = &(rx)y(x) = &(X)vI(X). The last equality follows by Plancherel's

theorem in its polarised version and (k*f)' = kf. Let Vr be the surface measure on

Ix e IR" : lxi = r} (i.e. v = T"'cy).With these notations, k() = r '" (v* ')() as

kr (X) = r ' '(x)IV(x).

But, i Ce" implies ii e S and so we have the estimates

Ic ifII!~ 10r 1C ifIl!!~ 10r

cIcl-N if lOr

- tc(1 + II2 )_ if lOr

where r >1 and N is a large integer. The last inequality follows from > lOr > 10.

Consequently,

I (1 r * ,i)():!~ crn_a(1+IcI), (a-n)/2

for r> 1 and so, using the computation above for krI we get that

! Cr(1

Hence, using this estimate and the identity built for L above, we get as desired that

(**) II.(r !!~ cr1_a$(1+lI2

)(a-n)/2

f( 2 d4 = Cr"'$$Gn_a (x—y)f(y)f(x)dydX.

Now, to deal with the other Lk , we divide them in two classes based on the value of

k, 2'>—r and 2k <r. For 2 k >—r,weget

I y C $Jf(x)f(y)dxdy 2511r, k*O Ix—yI<21r

~ C $5 (rx - yj) f(x)f(y)dxdy Ix— yI<21 r

!~ Cr 5$ (x - y)f(y)f(x)dyth

~ Cr_c $5 Gna (x - y)f(y)f(x)dydx.

The first inequality follows from the support of the Lk and the boundedness of ô and

tVk on IR' while the third one is based on Proposition 1.3.5 (3).

On the other hand, if k <r, we have

'i (r)l = Iff ô(r(x - y)) V'k (x - y)f(x)f(y)dxdy

= 2 -21 55 &(2 r(x

- y )) i(x - y)f(2 x)f(2 y)dxdyl

= 2 2 (f)(2 k r)

~ C2 2 c7 (2" r)_a $5 Gn_a (x - Y)fk (Y)fk (x)dydx by (* *)

= Cr' 2kr_1) 55 G,,-a (2 k n_a (2" (x - y))f(y)f(x)dydx

~ C 41 2 k(a-1)2 -ka $5 Gn_a (x - y)f(y)f(x)dydx.

where LO(fk)(r) is the L0 (r) corresponding to the function fk(r) = f(2_kr) rather than

f. The last inequality is based on

67

Gna (2 k X):!~ C2Gna (X)

which follows from the decay of Gn_a on IR". This finishes the proof because the

majorisation just obtained implies that we can sum the Lk such that 2k <r and get the

desired bound.

To get the case a = 0, replace (rlx - y by log( 1 ) (Using Proposition 1.3.5 (4)

rather than 1.3.5 (3)) when dealing with the case 2k >— r.

We should point out that this argument is just a rewriting for the Bessel kernel of an

argument done by Sjölin for the Riesz kernel [Sj2, pp.324-3261. We only modify here

some estimates to adapt them to the Bessel kernel but nothing deep is involved in these

modifications. Now, by combining Theorem 3.2.3 and Lemma 3.2.5, we obtain the

following localisation principle 18 :

THEOREM 3.2.6. Let f E X P (R') with n ~! 2, 2 ~ p < 0< a ~ . Suppose also

that f = G*g almost everywhere and let E be the set of divergence for SRf off

support of g, (i.e. E={xsupp(g): SRf(x) diverges}). Denote by E xr the

restriction of E to the closed ball B(x,r), x IR" and r e IR. If Exr is closed then

Ca2 (Exr )=0• In particular, SRf(x)—* (G* g)(x ) C 2 -quasieverywhere off support

of g if E is closed.

PROOF.

As SRf(x) = S(G* g )(x), we can suppose without loss of generality that f = G*g

everywhere on IR". By translation and dilation invariance, we can also suppose

without loss of generality that g is supported in (B(0,3))c. Hence, Theorem 3.2.6 will

be proven if we can show that we have Ca2 (Exr )=0 for any closed Exr inside

(B(0,3))'. To fix the ideas, we suppose that E01 is closed and we will show that

SRf(x) - (G*g)(x) C 2 -quasieverywhere on B(0,1) (the proof is identical for any

closed Exr inside (B(0,3))c).

Now, let 0 be as in Lemma 2.3.5 with ij = - where E is a small number to be chosen

later. Then S(((1_O)G)* g)(x) converge everywhere on IR" by Lemma 2.3.5,

18 The restriction to closed sets required in Theorem 3.2.6 can be removed as we have already seen in Theorem 2.3.7. Our goal here is to illustrate a new technique rather than to prove a new result.

.re1

hence we can limit ourselves to prove that S((ØG)* g)(x) converges Ca 2

quasieverywhere on B(O, 1). From now on, g will be replaced by h to harmonise the

end of this argument with the statement of Theorem 3.2.3.

Using the compactness of E01 , we can compute its Ca2 -capacity with the definition

given in Proposition 1.4.13. Thus, we will be done if we can show that u(E01 ) = 0

for any pE111(E0 ,1 ) satisfying sup VG. By Proposition 1.4.11 and x€supp(p)

because V 2 =Ga*(Ga */1)=G2a */1 by Proposition 1.3.5 (2), we then have

SU)(G2a *P)(X) ~ C < oo • We also have that u is finite 19 .

Now, by Lemma 3.2.5,

!~ Cr12a 55 G c (x - y)dp(y)d/1(x).

Hence, 1/2

(7)L (s') Cr2'2 (55 G2 (x - y)g(y)g(x)dp. (y)di (x))

Cr2a_2 (IS G2(X - y)(9(x))2d/J(y)dp(x)) 1/2

1/2

~ Cr 2 " IIgII2(d) (XE1

sm) (G2a */1 )(x)) < Cr2a_2 1gIl2 (djz)

The second inequality follows from 2IabI !!~ a2 + b2 applied to g while the last one is a

consequence of Thus, IIS* ((OGa )*hZ)11 L2 (d/1) !~ Ch p by

Theorem 3.2.3 provided that e < 2. Consequently, i(E01 ) =0 as usual.

REMARKS 3.2.7.

The ideas used in Theorem 3.2.6 can also probably be used as a starting point to

study some of the kernels covered by the remark following the proof of Theorem

3.4.1 in Section 3.4.

The ideas in Lemma 3.2.5 were originally developed in [Ml] to study distance

sets. A recent improvement of these results in R 2 and 1R 3 by Bourgain [Bo]

shows that these estimates are closely related with the restriction problem on the

19 From definition 1.4.13, we want a " au" such that u(E01 ) :5 C0 ,2(40 !5 CO2 ((o, 1)) < 00.

sphere. Hence, Theorem 3.2.6 provides us in this line of thinking with another

link between the restriction and inversion problems.

3.2.3 An Everywhere Localisation Principle

To prove an analogue of the Salem-Zygmund type result (Theorem 2.4.7) for

functions in Z ' (R') with 2< p < we need to build an equivalent of Theorem

2.3.7 (our localisation principle for Z 2 ) valid C p -quasieverywhere. But dealing

with Ca , p -capacities is not an easy task if only because of the non-linearity involved.

In this section, we will avoid this problem by using the ideas initiated in Section 3.2.1

to prove a localisation principle valid everywhere.

When a the hypothesis (**) in Theorem 3.2.3 can be removed because it is

already contained in our conditions on U. In this case, Theorem 3.2.3 should read:

THEOREM 3.2.3a. Let h e If() with 2 p < ç and n ~: 2. Suppose that

f = (ØG)*h = Ga */1 on with 0 defined as in Lemma 2.3.5 and a €

Suppose also that supp(h) (B(O,r+ e))c (so supp(f) c(B(O,r))c), r E , and that

u is a finite positive Borel measure supported on B(O, r). Then IIS.f Ile (d) :!~ Cjjh jj P

where C is a constant independent of f.

PROOF.

If R -::~! 1, from the finiteness of u, we have

j(gdp)A (R•)11L2 (S'')

C(gd/.i )" IL :!~ Cllgdp ll l

!!~ cpII 112 hg 11L2 (dp)

:2~ cR-0-1- 2a) 12 by the hypothesis on u and a.

Theorem 3.2.3a then follows from Theorem 3.2.3.

REMARK 3.2.8.

As for Theorem 3.2.3, the hypothesis supp(h) (B(0, r + e))c is not necessary in

Theorem 3.2.3a.

By combining Lemma 2.3.5 and Theorem 3.2.3a, we are easily lead to an everywhere

70

localisation principle for functions in (IR"):

COROLLARY 3.2.9. Let feZ(R") with 2 ~ p<, ae[,.1 and n~ 2.

Suppose also that f = G*g almost everywhere. Then SRf(x) -* (G* g)(x )

everywhere off support of g.

PROOF.

Without loss of generality, we can suppose that f = G* g everywhere because

SRf(x) = S(G* g)( x ). We can also suppose without loss of generality that g is

supported in (B(O,3))". So we must show that SRf(x) -p (G* g)(x ) everywhere on

any ball contained in B(0,3). For simplicity, we limit ourselves to prove this for

B(O, 1) but the argument is similar for any other ball.

Now, let 0 be as in Lemma 2.3.5 with r =L where i is a small number to be chosen

later. Then S(((1_)G)* g)(x) converges everywhere on IR" by Lemma 2.3.5,

hence we can limit ourselves to prove that S((ØG)* g)(x ) converges everywhere on

B(0,1). But, the set where S((ØG)* g)(x ) diverges outside the support of g is a

Borel set by an argument made in the proof of Theorem 2.3.7. So, we can use

Definition 1.4.6 to compute it.

Let E be this set of divergence inside BA I) and let p be a fixed number such that

> 2- 1 . We will now compute C(+!)/ (E). Let u Efl(B(O,1)) be a measure

satisfying G(fl+l) ,*/4 ~ 1 whereI + = 1. It is then easy with weighted characteristic

functions of balls to verify that

((E)) <_ C(+l)/ (E) :!~ fl+1)1,(B(0,1)) <00•

Hence, p(E) =0 by Theorem 3.2.3a provided that e <2. So, this implies that

C("+I)/P,P (E) = 0 by Definition 1.4.13. But, by Proposition 1.3.5 (6) and Proposition

1.4.8 (1), we have that the only set Ac ]R" such that q +l), , (A) = 0 is the empty

set.

REMARKS 3.2. 10.

(1) Using the C(1),-capacity is perfectly legitimate as all the propositions seen in

Section 1.4 remain true for this capacity, but it is really unorthodox because it is

71

not a "meaningful" way to measure small sets. To avoid this, the last part of our

proof can also be replaced by a limiting argument involving the verification that,

for any meaningful capacity (not only the Cap ), we can prove that the set of

divergence is of null capacity.

Corollary 3.2.9 (in the form "off support of f") can also be proven with an

argument similar to the classical proof of Riemann's localisation principle

(Theorem 3.1.1).

The restriction of a to the range a ~!is sharp when looking for an

everywhere localisation principle because of Il'in's counter-example (Example

3.1.4).

It does not seem to have been observed in the one dimensional case that the every-

where convergence takes place not only in the complement of the support of f,

but also outside the support of g.

We noted in Remark 2.4.8 that a C version of Theorem 2.4.7 will be true for all p

in [2, r ) if a C analogue of Theorem 2.3.7 for functions in (R') was proved.

But, we just did this. In fact, we showed that a stronger result holds, namely an

everywhere localisation principle (Corollary 3.2.9). So, we have proven the

following:

THEOREM 3.2.11. Let f E Z (]R") with 2 !!~ p < and a E (, ].

Suppose also

that f = G*g almost everywhere. Then SRf(x) -* (G* g)(x) C p *quasieverywhere

on lR.

3.3 Some Improved Localisation Results

From Win's counterexample (Example 3.1.4), we know that there is no chance that an

everywhere localisation principle holds below the critical index .!j1 . We also know that

we should at least expect a minimal Ca2 result when p =2 if we believe in Conjecture

2.2.1. So what kind of localisation should we hope for a function in a space of Bessel

potentials (when p =2) below the critical index?

We cannot unfortunately answer this question. Nevertheless, by replacing the idea

coming from [5j2] by those analogue in [Ml] in our proof of Theorem 3.2.6, we can

do a first step toward such understand ing of the localisation of functions in when

72

p = 2 and a is in More precisely, we have the following:

THEOREM 3.3. 1. Let fE(1R") with n ~ 2, 2 ~ P<, a>Oand aE{',).

Suppose also that f = G*g almost everywhere and let E be the set of divergence for

SRf(x) off support of g (i.e. E = {x 0 supp(g) : SRf(x) diverges}). If E c E is

closed then ca+(112)2 (E) = 0. In particular, SRf(X) - (Ga * g)(x) Ca+(1/2)2 -quasievery-

where off support of g if E is closed.

We omit here the proof of Theorem 3.3.1 because, like Theorem 3.2.6, it is a direct consequence of Theorem 3.2.3 and of an estimate for

LEMMA 3.3.2. Let p be a finite positive Borel measure on IR" with compact support.

Then, there is a constant C independent of p (but depending on the size of its

support) such that

j 1/2 (rc )12 da() ~ Cr if G2 ,, (x - y)dp (y)dp(x)

where $=n-2a if aE[,) and J3= if a€ \4'41

PROOF.

In [Ml, Theorem 8.8], the same result is proved with G2a replaced by 12, It is well

know that G2 a (X) :!~ 12 ,,, (x) for all X IR". Hence, this lemma would be an improve-

ment on Mattila's result if we did not restrict the support of p. But, by doing so, this

lemma is just a rewriting of Mattila's theorem because the restriction on supp(p)

implies that

5512a(X - y)dp(y)d/i(x) = 55 (Xa)(' - y)dp(y)dp(x)

and it is easy to verify that ZB'2a !~ CG2 111 as a consequence of Proposition 1.3.5 (3).

Here B is a ball determined by the support of p.

REMARKS 3.3.3.

The improvement by-1 in Theorem 3.3.1 is not really surprising. Carbery and

Soria already realised implicitely in [CS 11 that any function in H' (IR") with n ~ 2

and 2 !!~ p <1, has essentially half a derivative outside of its support.

For a E a counter-example in [Ml] shows that /3 = n - 2a is optimal.

73

While another counter-example in [Ml] implies that, for a e (0,z), we can not

have a P as good as n — 2a, but it is still unclear in this case what should be the

best possible result.

It seems reasonable to believe that the approach to localisation used in this section and

in Section 3.2 can also be used to prove a Cap localisation principle restricted to

closed sets of divergence for functions in Z P (1R") with 2 !! ~ p < and a> 0. In

fact, such a result is already contained in Theorem 3.3.1 when n ~!2, 2!!~ p<,

a >0 and a e [-, )

(This follows from Proposition 1.4.14.). In the view of this,

Theorem 3.3.1 is not only a first step toward a real understanding of I ocalisation when

p=2, but itis also one for all the other pin 2!~ p<.

Even better, it is not perhaps too much to hope to obtain from this approach a Ca p

localisation principle without any restriction. if we had an estimate like

(3.3.*) II/(rs)II2(S_I) <_ Cr2 an1p(IRn)

for any finite and boundedly supported Borel measure p satisfying IVa *911q:5 1,

this will be the case. But, 3.3.* is a plausible estimate because it is

essentially expressing that, in average, the Fourier transform of a "nice" measure

behaves like an absolutely continuous measure does pointwise. In fact, an estimate

even weaker than 3.3.* is needed, but, in any case, the key idea to obtain such a

localisation principle seems to require a better understanding of Ifr(r.)II(SI) when

IIGa *PIIq <1.

3.4 A Capacitarian Study for a General Class of Functions

Even though Sjölin's result (Lemma 3.1.7) is a rougher version of Carbery and

Soria's work [CSI], it is of great interest if we want to generalise the results contained

in this thesis. This interest is double. It allow us on one hand to study generalisations

of SR while it can, at the same time, be used to handle different class of functions with

a nice local behaviour. We will not follow in this thesis either of these two possibilities

to generalise our work, but we nevertheless want to show why one can hope to do so

by proving two easy first steps in this direction. We need first a stronger version of

Lemma 3.1.7:

LEMMA 3.1.7a. [Sjl, p.415] Let 0€ C1 (R' \ 101) be positive and homogenous of

74

degree 1. Suppose also that 0(0) = 0. For DR = {x E R:Ø(R'x) < i}, define the

kernel KR (x) = JD,, e2 d and then, with 9 E=- S such that 9(0) = O, define the

operator TRf =((OKR )*f for fEL 2 (IR") and R>0. If T.f=supITRfI is the maximal

operator associated with TR then II4I2 !!~ CIf2 with C a constant independent of f.

In the spirit of what we did in Theorem 2.3.1, we can now easily deduce from Lemma

3.1.7a the following:

THEOREM 3.4.1. Let K be a radially decreasing convolution kernel on W xF"

satisfying 1IRJK(r)I2 dx = 00 and L>11K(x)12d1 < 00• Let SRf = I< R *f where 'R is

defined as in Lemma 3.1.7a. If f c L2 () has a bounded support and satisfies

$1f(4)1 2

I()I d4 <00 then SRf(x) - 0 C,-quaszeve?ywhere off support of f.

We assume implicitly in Theorem 3.4.1 that our kernel K has a well defined Fourier

transform which is non zero except possibly on a set having null Lebesgue measure.

The last two conditions on the kernel K are there to guarantee that the capacity defined

is meaningful as seen in Proposition 1.4.8 of Chapter 1. In a sense, Theorem 3.4.1

can be viewed as a partial analogue to Theorem 1.5.9. It should be remarked that it

covers in particular the homogeneous Sobolev functions (i.e. spaces of Riesz

potentials) evaluated with respect to their associated C 2 -capacity.

For the spaces of Bessel potentials, Theorem 3.4.1 can be rewritten as follow:

THEOREM 3.4.1 a. Let f Z (1R"), 0< a ~ . Suppose also that f has bounded

support. Then SRf(x) - 0 C 2 -quasicverywhere off support of f where 5R is

defined as in Theorem 3.4.1.

We will limit ourselves to prove only this particular case of Theorem 3.4.1 to keep the

notation simple, but the general result follows in exactly the same way.

PROOF OFTIIEOREM 3.4.1a.

By dilation and translation invariance, we may suppose without loss of generality that

supp(f) ç {x E : a <lxi < b}. We then choose a function p E C° such that q3 1 on

{xElR:a—r<IxI<b+r} and q0 near 0. Here 0<r<a<b<°°. For IxI<r,

what precedes and the restriction on the support of f imply that

SRf Cx) = $ KR (x - y)f(y)dy

75

= f co(x—y)RR(x—y)f(y)dv = R)* f)(x) = TRf(x).

Define now E, = Ix c= B(O, r): Sj(x) > ;LI, E = E B(O, r):(G * Tg)(x) > A.} and

k. = IX E R":(G,, *T.g)(x) >A j where S is the maximal function associated with S R .

The Basic Estimate 2.2.2 and the previous computation then imply that E c k, c: E.

Consequently,

< cI g2 Ca,(EA) ~ Ca2(E;3<

-

The second inequality is obtained using Definition 1.4.4, while the third one is a

consequence of Lemma 3.1.7.

Hence, following the standard argument, we get SRf(x) 0 Ca 2 -quasieverywhere in

B(0, r). By translation and dilation, we then extend this to all the region outside the

support off.

REMARKS 3.4.2.

A different argument similar to the proof of Theorem 3.1.8 can also be given for

Theorem 3.4.1. In this case, the series remaining to be evaluated is finite because

of the restricted support. So, using Lemma 3.1.11, we can easily compute

directly the L1 (d,4)-norm of its supremum. For any measure 1 1 satisfying the - 1/2

conditions in Proposition 1.4.6, this gives ~ C(IG.

(u)) 119112. But, we

have seen in the proof of Theorem 2.3.7 that the set of divergence is a Borel set,

hence SRf(x) - 0 C 2 -quasieverywhere outside the support of f.

The localisation "off support of g" obtained throughout this thesis for the Bessel

kernel, Ga , is more precise than the localisation "off support of f" obtained here

because of its fast decay. But, for general kernels (e.g. 'a)' this is normally false

(see Remark 2.3.2 (3) for a better understanding of this).

The ideas seen in Theorem 3.4.1 to deal with general classes of functions are not

limited to localisation principles. For example, if we turn our attention to the closely

related spaces of Riesz potentials, we realise that the idea in [CS 1] contains in fact the

following result:

PROPOSITION 3.4.3. Let f Z(1R'1 ) with a> 0 and 2 !~ p < . Suppose also that

g E L1'(1R') has bounded support and is such that f = I*g almost everywhere. Then

SRf(x) -* f(x) almost everywhere.

PROOF.

Let 0 be a smooth cut off function supported inside B(O, r), r E 1R. For r small

enough, we have ((Øl)* g)(x ) = ((ØG)*g)(x ) by Proposition 1.3.5 (3). Thus,

SR((OIU)*g)(x)-_* ((I)* g)(x ) almost everywhere on R'1 by Theorems 2. 1.1 and

3.1.8.

Now, by translation and dilation invariance, we can suppose without loss of generality

that supp(g)cB(O,). So, for xEB(O,*), we have (((1_Ø)I)*g)(x)=0 by the

support of 0 and g. Hence, SR(((1 - Ø)I)* g )(x ) --> 0 almost everywhere on B(O,*)

by Theorem 3.1.8. And, consequently,

SRf(x) = S(I*g)(x) --> (I*g)(x) = f(x)

almost everywhere on B(0,-). The same then follows on all IR' 1 by dilation invariance

(Think of supp(g) as being supp(g) u V where V is a compact set in the previous

part of this argument and this will be clear.).

REMARK 3.4.4.

If both f and g are boundedly supported then, by replacing Theorems 3.1.8 and

2.1.1 by Theorems 3.4.1 and 2.4.7 in the last argument, we immediately obtain

SRf(x) ._ (I*g)(x) C 2 -quasieverywhere on IR" when p = 2 and' < a <.

We will not push further our study of general function spaces, but we want to note that

"good" localisation results are the only missing pieces in trying to understand the

spaces of Riesz potentials as well as we understand the related spaces of Bessel

potentials.

77

Chapter 4

A Collection of Related Results

In this chapter, we will examine what kind of uniformity is present in all the results

obtained in Chapters 2 and 3. We will also concentrate our attention on what happens

to the inversion problem for functions in ZP when p # 2. In particular, we will see

how interpolation should enable some improvements on Theorem 2.3.3 when

2 < p < for some a in (O,1 and we will justify why there is no equivalent of

Conjecture 2.2.12 when I1 < p < 2. The idea behind this chapter is mainly to raise

and motivate some questions closely related to what was done in Chapters 2 and 3, so

the style of presentation will be slightly less polished.

4.1 On a Question. Related with Uniformity

In all the results covered up to now, we were only interested in the pointwise conver-

gence itself and not by the precise qualitative way in which this convergence was

taking place. If we now require our functions to satisfy a more restrictive smoothness

condition than in Theorems 2.4.7 and 3.2.11, the convergence will take place every-

where on IR". Moreover, this will happen uniformly. More precisely, the following

classical result holds:

THEOREM 4.1.1.20 Let a> . If f E Z '(]R1) then SRf(x)—* f(x) uniformly on

IR". f is the continuous representative of f (i.e. the function G*g such that

f = G * g almost everywhere).

PROOF.

ISRf(x) - STf(x)I = JfR<1 ~15T j(4)e"'1d41

~ R<IlI ~T j ?()(l+II2)(1+II 2

)-aI2

d

20 This result has been in the literature for a long time. We do not know who originally proved it.

78

1/2 1/2 < (c 2(II2)aJ (J(1+Il2rd) - JR<j1 ~T

a 1/2

= c(J ~T

<lf((1+II 2 ) d) since a>.

t. R

Now, the right hand side tends to 0 as R and T tend to infinity because f Z.

In [CSI], Carbery and Soria extended this result to a larger class of functions:

THEOREM 4.1.2. [CS 1, p.332] Let n ~ 2. If f L%7" (IR") then SRf(x) - f(x)

uniformly on 1R" where f is the continuous representative of f.

DEFINITION 4.1.3. Two functions f1 and f2 are said to be equimeasurable if they

have the same distribution function (i.e. where 2f(a)I{x:lfj(x)I> a}). f2

DEFINITION 4.1.4. A function f is in the Lorentz space L'(1R") if

11f11,q =

(.,

f (t11i ()) 11q

t)

where f* is a nonincreasing function on (O,00) equimeasurable with Ifi. In the special

case where p = q, L" is equal to L", while L" corresponds to the standard weak-

!]' space. if q !!~ q2 1 we have the inclusion L"

By Il'in's counter-example (Example 3.1.4), the convergence does not always hold if

f A (IR") r C, (R"), a < '-, consequently, the index in Theorem 4.1.2 is

sharp. This completely answers the question of when the convergence takes place

uniformly, but comparing Theorem 2.1.1 with Theorem 4.1.1 raises another natural

question. How "far from being uniform" is the convergence when fE Z 2 (1R") and

0< a !~ ? We will not solve this problem in this thesis, but we will try to explain

what the natural conjecture should be. To achieve this, we will start by studying what

happens in the lacunary case:

THEOREM 4.1.5. Let {Rk } l bealacunary sequence and let fEZ(R"), 0< a ~ .

Suppose also that g E L2 (]R") is such that f = G* g almost everywhere. Then

SRf(x) - (G*g)(x) C 2 quasieverywhere on IR". Moreover, the convergence

takes place uniformly outside an open set of arbitrarily small Ca2 -capacity.

79

The first part of this theorem is just a restatement of Corollary 2.2.7. The new part

here is the uniformity.

PROOF OF THEOREM 4.1.5.

For £ >0 fixed, choose a g0 ES such that ll —g0112 <e. Define f0 = G* g0 then

S and so lim(K r*fo )(x) = 10 (x) for all x E 1R' where 10 is the continuous

representative of f0 (by Theorem 4.1.1).

For öE{Rk}l, c5>0,we define

Qf(x) = sup Re((K r *f)(x))_ inf Re((K r*f)(x)) 8<r 83<r

rE(Rk)I retRkJI

+ sup Im((Kr *f)(x))_ inf Im((Kr *f)(x)). 5<r 5<r

rE(RkI re(R t )

Hence, Q S f(x) :! ~ Q (f - f0 )(x) + Q ,5fo (x). Using the uniform convergence (Theorem

4.1.1), it is possible to find a S E{Rk}l large enough to have 9f0 (x) :!~ £ for all

xE1R. Thus, If(x):! ~ 4S(f—f 0 )(x)+E for that S.

Let £ <4. From the previous computation, it then follows that

= {x IR" f(x)> } c: IR": S(f - f0)(x) >41

and, consequently,

Ca2 (Os ;) < C (IX .,2 E= S(f—f0)(x)>4}) - a,2

< Ca,2 (irx e IR": (G,, * S (g - g0 ))(x) > '71) -

~ (f)2si(g_g0)2 ~ frIIg - goII

< Ce 2 -

The third inequality follows from Definition 1.4.4 while the fourth one is a

consequence of Theorem 2.2.3 (with a = 0).

Now, for k E IN fixed, let A = 2", 4-k and 5k be the associated S. Define also

the sets Ek ={x€IR" : 0 5k f(x)>2_k } and U m 1j Then, it follows by the

previous computation that Ca2 (Ek ) ~ This implies that Ca2 (F;n ) ~ C:2m_2k by

Proposition 1.4.9 (2) and, consequently, as the right hand side tends to 0 when

m —* 0O we have Ca,2 (nm-=. F- )

= 0. Hence, Q ,5. f(x) 2" if x F for 45 m

for all k~!m because F =fl m {xE : 8 f(x)! ~ 2} and )8f(x)! ~ f(x)

for such 5k• Thus, Sf( x)4(G* g )(x ) if x 0 nM-=1Fm and the convergence is

taking place uniformly outside F,, for any m.

REMARKS 4.1.6.

In different sections of this thesis, we established maximal inequalities and wrote

that they implied some pointwise results. The missing steps, in all these cases, to

pass from the maximal inequality to the pointwise result are in fact similar to the

proof of Theorem 4.1.5.

The previous extension (i.e. uniformity on all IR") for p = 2 cannot be achieved

for p # 2 with the estimates established in Theorem 2.2.3 because they only

provide us with a local capacitarian weak-type inequality rather than a global one

as we already observed in page 28.

Like in Section 2.2, the computation in the lacunary case can also be done for radial

functions in Z, (IR"), 0 < a ~ -, or for any extension of Meaney and Prestini (see

Section 2.2). Hence, it is natural to conjecture that the general result should be:

CONJECTURE 4.1.7. Let 0< a !~ . Suppose that f E Z 2 (IR") and suppose also that

g E L! (R') is such that f =G*g almost everywhere. Then Sf( x)._*(G* g)(x )

Ca 2 -quasieverywhere on IR". Moreover, the convergence takes place uniformly

outside an open set of arbitrarily small Ca2 -capacity.

In the spirit of Theorem 4.1.5, proving this conjecture will require a weak or strong

type inequality valid on all IR" and not only on a bounded part of it. Consequently,

even to get some partial results (Lebesgue measure, Ca/k 2 -capacities, etc.), it will be

necessary to extend the estimates seen up to now to all IR".

When p # 2, it is not as easy to understand what the conjecture should be. By Hunt's

theorem (Theorem 1.5.4), the situation is the same 21 in dimension one when p = 2 or

p#2:

THEOREM 4.1.8. Let 0< a !~ - and let 1< p < oo. Suppose that f Z(IR) and

suppose also that g e L"(IR) is such that f = G*g almost everywhere. Then

21 This follows from Theorem 1.5.4 (version of Kenig and Tomas) with an argument similar to Theorem 4.1.5.

["I [I1

SRf(x) (Ga * g)(x) Cap -quasieverywhere on R. Moreover, the convergence takes

place uniformly outside an open set of arbitrarily small Cap -capacity.

But, in higher dimensions, Fefferman's counter-example (Theorem 1.5.9) shows that

there is no strong or weak type p - p inequality and so no equivalent of Hunt's

theorem. Moreover, all the positive results obtained up to now when 2 < p <r are

based on local estimates which essentially rely at some stage on the fact that L" c L2

on bounded sets. It seems consequently plausible that no global estimate can be built

on the whole of IR". If this was the case, then we could only expect the following:

CONJECTURE 4.1.9. Let 0< a :!-~ & and let 2<p < . Suppose that f

and suppose also that g E L"(") is such that f = G* g almost everywhere. Then

SRf(x) —* (G* g)(x) C p -quasieverywhere on IR". Moreover, on any compact set,

the convergence takes place uniformly outside an open set of arbitrarily small Cap -

capacity.

But, when - < a !~ we can easily obtain an "uniform" result with a simple idea

(11 5*GaI1, :~ CIGaIIs by Theorem 2.2.8 as Ga is a radial function and Ga E L(1R") for

an sE(f,r)1f a>).

THEOREM 4.1.10.22 Let f€(]R") with 1<p< and < a~ . Suppose also

that g E L" is such that f = G*g almost everywhere. Then SRf(x) —3 (G* g)(x )

Cyq -quasieveiywhere. Moreover, the convergence takes place uniformly outside an

open set of arbitrarily small Cy ,q -capacity. Here, y = a — (' + e), q = 2n-np-24'+p and

0< e < a—v.

Consequently, it seems reasonable to believe that the "correct" conjecture, even when

p#2, is:

CONJECTURE 4.1.11. Let 0< a !~ -- and let 2< p < . Suppose that f e Z, (R")

and suppose also that g E L"(IR") is such that f = G*g almost everywhere. Then

SRf(x) _(G* g)(x) C p quasieverywhere on IR". Moreover, the convergence takes

place uniformly outside an open set of arbitrarily small Cap -capacity.


By Young's inequality, Estimate 2.2.2 and Theorem 2.2.8,

22 The new elements in Theorem 4.1.10 are the uniformity and the region 1 <p <2. The quasieve-

rywhere result in it was already contained in Theorem 3.2.11 when 2 :5 p <r.

It s. (G(fl_1)12+E * :!~ II&IL Its. (G(fl_1)12+C )IL :!~ CI jgIIpG(n_l),2+e II,, where -=+..-1 and a,b>1.

Now, we want to choose a as big as possible (because this will imply a better

pointwise convergence of SRPI so we need to pick the biggest b in( 2n , 2n ) for

which G(fl _ 1)12+6 E Lb(1Rl) . Hence, using Proposition 1.3.5 (3), b = n+1-2c and, conse-

quently, a= 2n-,T-26p+p

Using the technique seen in the proof of Theorem 2.3.3, the previous estimate then

implies that SRf(x) (Ga * g)(x) C_((fl _1)12ç),2flp ,(2fl _flp_2+p)_quaSieVerywhere on IR".

Moreover, with a proof similar to Theorem 4.1.5, it can be shown that the

convergence takes place uniformly outside an open set of arbitrarily small

Ca_((n_I)12+e),2npI(2n_np_24,+p) -capacity.

REMARKS 4.1.12.

(1) Theorem 4.1.10 is true for any e in (0, a -

U2 I ), but the best result is when

In the limiting case (i.e. a = ), the values obtained in Theorem 4.1.10 are quite

good, but this rapidly changes when a decreases towards . In fact, when

a = , we have y q = n improving consequently even when p =2 on Theorem

2.3.3 (see Theorem 1.4.14 for an explanation on how to compare different Cyq

capacities). But Theorem 4.1.10 is still a weaker proposition than the sharp result

(Theorem 2.4.7) because y < . Nevertheless, it is a good first step toward

getting the full uniform result (Conjecture 4.1.7).

We want to point out that, with Theorem 4.1.10, we areas near as we wish to a

C112 2 -result for f E Z,2 (1R"). So the n-dimensional results are "near" the

optimal one dimensional result (C1122 ). This shows, even if it was already clear,

how much Theorem 4.1.10 is a one-dimensional idea and not truly a higher

dimensional one.

It is not clear yet how one could prove Conjecture 4. 1.11 even when !!L < a !!~ , but,

with the results in [CRdeFV] in mind, an interesting possibility is certainly to try to

use some "smooth" approximations of the ball multiplier, MR = This should at

least help to extend Theorem 4.1.10 below the critical index n2' .

Another possibility to obtain a partial result (e.g. uniformity except on an open set of

arbitrarily small Lebesgue measure) when 0 < a :!~ n2 l and p :t- 2 will be to show that

the estimate in Proposition 2.1.3 is global when p = 2 (i.e. that it holds for all IR and

not only for B(y,r) with r <oo). Combining such an estimate with Theorem 4.1.10

will then imply the desired partial result using Stein's complex interpolation theorem

[StW2, p. 2051.

4.2 Interpolation (The ZP Case when 2<p<and O<a<j-') OL n- 1

Another question left in a rough state in Chapters 2 and 3 is what happens below the

critical indexwhen 2 < p < . Our only answer is actually Theorem 2.3.3 which,

for p = 2, is as near as we wish to the optimal conjecture, but which is less and less

precise for large p in 2 <p <. If we now use this solid base with a rougher frame

of mind like in the last section, can we obtain a more precise result or at least motivate

one?

The answer is yes, but we need first to return for a moment to the proof of Theorem

2.3.3. When doing so, we rapidly remark that we could have also proven it by

showing that IS.! IIL'(dp) !~ CIigII for any boundedly supported Borel measure u

satisfying 11G.1k *All 2 !!~ 1. To do this just use Lemma 2.3.5 to handle the tail of Ga/k

and a combination of Estimate 2.2.2, the Cauchy-Schwarz inequality and Proposition

2.1.3 to handle its central part. This new estimate is really similar to the estimates built

in Corollary 3.2.9 and Theorems 2.4.7 and 3.3.1. So, by interpolation, it should be

possible to obtain some new results when 2 5 p <

By taking the estimates behind Theorems 2.3.3 and 3.2. 11 as a basis for interpolation,

we are lead to the following:

CONJECTURE 4.2.1. Let f e Z(1R) with 2 ~ p < r and 1X,2 2p <a ~ . Sup-

pose also that g E L" (]R) is such that f = G*g almost everywhere. Then

SRf(x) - (G*g)(x) C,quasieverywhere for every k > 1.

MOTIVATION FOR CONJECTURE 4.2.1.

Using translation invariance, we can without loss of generality restrict ourselves to

prove Conjecture 4.2.1 in B(0, 1). By the remark made above, we have

(4.2.a) i*('a, *g1)j1

when g1 EL2 (lR), a1 >0 and i1 EI1V(B(0,1)) is such that Ga,k*plO :!~ 1, k>l.

On the other hand, by the estimates behind Theorem 3.2.11, we have

(4.2.b) JI S. (Ga, *g2 )IL'(d2) :!~ C1192 11 ~

when 92 EL"(1R), 2:!~ <r, za2 ~ t and /2 E11(B(0,1)) is such that

I 4 4 lGa *92 I :~ 1, *+.= 1. By our restriction to B(0,1), we can suppose that

(4.2.c) u(B(O,l))!~ C<

for j = 1,2 (like in the proof of Theorem 2.4.7). For what follows, we also replace the

condition Ka2 */1211 :~ 1 by the stronger condition IIGa21k*2II !!~- 1.

Now that our proposed plan of interpolation has been clarified let return to the "proof'.

The argument is similar to the classical interpolation built by Stein for the Bochner-

Riesz operators in [5th] (a detailed version of this is also contained in [StW2, pp. 279-

28 11). As in [StW2], we complexify the a 3 in 4.2.a and 4.2.b and we replace the

maximal operator, S(G* gj )(x), by a linear one, S()(G*gj)(x). Here, R(.) is a

nonegative measurable function taking only a finite number of values.

Then, we apply the strong form of Stein's complex interpolation theorem which

allows interpolation of measures (see [Sti] and [StW1]). For all 0 !! ~ t < 1, this gives

us 1S* (Ga * g )II 0(/1) !~ CIgII where a=ta1 +(1—t)a2 , - = t4+(1—t)* and p

is the measure defined by p(E) = f ftf201d(p1 + 92). The fj are the Radon-

Nikodym derivative of 1u, with respect to /- + /2' J = 1,2.

If we could now verify that

(4.2.d) 11G.1k *11 q = 11f G,,,,( — y)f( y )f_l)( y )d(p +/i2 )(Y) q !~ 1,23

for I + -- =1, we will be done and Conjecture 4.2.1 will be a theorem. Unfortunately,

we are unable to prove this last result. There& are some indications both for and against

such an estimate.

On the positive side, there is the fact that we expect the interpolation to work like in the

23 The 1 here is not really necessary. Any positive constant will also prove the desired result.

Bochner-Riesz situation and that some particular cases can even be proven (e.g. f and

f2 both simple functions (but the bounds obtained depend on the functions)). On the

negative side, we can see just by taking f1 f2 CZB(Or) (C, r >0 chosen to respect

condition 4.2.c) that 4.2.d is false when Ga is replaced by 'a• In fact, some problems

in dealing with simple functions are apparent even with the natural approximation

XB(O,e)'a ( E > 0 fixed) when taking f1 f2

REMARK 4.2.2.

Estimates 4.2.a and 4.2.b can be rewritten as a "kind of Holder's inequality"

which can be useful to help think of Conjecture 4.2.1 as a convexity question

rather than an interpolation problem. If we suppose that G gj = 1 when

j = 1,2 (this can be done without loss of generality), we then have

II S. (Ga ) *g.IIL' (dJL)

< -

C gfIr JIG.j 1k * P i 11

Conjecture 4.2.1 will be an improvement on what we know (Theorems 2.3.3 and

(n-1)(p-')n 1 3.2.11) only when -2P <a ~ -. But, the new estimate for Theorem 2.3.3

constructed at the beginning of this section is also valid when 2 < p < 11 and

0 < a :5

(n- 1)(p-2)n So, at least intuitively, this suggests that we could have also2p .

"interpolated" with these results and not only with those for f EZ2 to get some

improvement when 0 < a !~ (n-1)p-2)n The interest of this point of view is that it

suggests at least informally that looking for some mixed norm results (Conjecture

2.2. 11) is never optimal when 2:! ~ p< r and a>0.

If we now repeat the same process with Theorem 3.2.11 replaced by Corollary 3.2.9

and Theorem 2.3.3 replaced by Theorem 2.3.7, we should obtain the following:

CONJECTURE 4.2.3. Let f € Z(1R") with 2 :~ p < and 1)2),i 2p <a~ . Sup-

a n-1

pose also that g E L''(1R) is such that f = Q,, *g almost everywhere. Then, for

y = a + 2pa-(p-2)(n-1)n ~ a, SRf(x) - (G* g)(x) C7 -quasieverywhere off support of

g.

Conjecture 4.2.3 contains in particular a Cap localisation principle for functions in

Z (IR") when 2:5 p < and (n-l)(p-2)n <a !~ A remark identical to the obser-

vation made after Conjecture 4.2.1 can also be passed here when a is smaller than

(n-1)(p-2)a It should consequently be expected that this conjecture can be extended

through interpolation to a full localisation principle for some other a below (n-l)2(p-2)n

In a similar way, one can also hope to interpolate between Theorems 3.3.1 and

2 . 3.724, but we prefer to leave this case for the Appendix because of the technical

details involved in it.

REMARK 4.2.4.

After returning this thesis to the Faculty of Sciences and Engineering of the

University of Edinburgh for its evaluation, we were able to improve on Theorem

3.3.1. At the cost of an 6>0 (as small as we wish) in the index a++, Theorem

3.3.1 can be proven without the restriction to closed sets. If we now interpolate as

above between the new estimates built and those behind Corollary 3.2.9, we are

lead to a result contradicting Il'in's example (Example 3.1.4). This proves that

(4.2.d) and the interpolation proposed above are wrong. The full details for this

result and its surprising consequence will appear in a forthcoming article.

4.3 Some Connections with the Bochner-Riesz Operators

4.3.1 Some "Forecasts" Based on Bochner-Riesz

By interpolating between our estimates of Chapters 2 and 3, we just saw that we can

hope to improve on Theorem 2.3.3 when 2 <p < r and 0< a:5 ', but is this the

only "accessible way" to achieve this? The answer to this question is almost certainly

no. One possibility is to study what was done for the closely related Bochner-Riesz

means.

DEFINITION 4.3.1. The Bochner-Riesz mean of order S is defined by

45

Sf(x) SB(O,R!( _ R2

1C e 2 d.

Its associated maximal operator is as usual S.f(x) = suISjf(x WI. (A brief survey of

the known result for these operators can be found in [St4, Chapter IX].)

Roughly speaking, the Bochner-Riesz mean of order S and the convolution operator

with kernel KR *G can be expected to behave in the same way 25 . Consequently, we

24 We will refer to the result of this interpolation as Conjecture App-2. 25 This is not always true (e.g. [CS 1, Theorem 5]), but it can often be used as a guiding principle.

RE

can use the extensive study which was done for the Bochner-Riesz operators to

understand better our problem 26 .

In particular, if we have a look at what can be gained from extending [C] and [Ch] to

our setting, we realise that in the most ideal case where both articles can be extended

perfectly we should be able to show that II5.fII !~ ClIg1l, for f E Z(IR) when,

respectively, n =2, 2 :!~ p <4 and a> 0 or n ~! 3, 2 !!~ p < ç and a > 1) Here,

g E L (1R) is such that f = Ga * g almost everywhere. With the technique used to

prove Theorem 2.3.3, this will then imply the following:

HYPOTHETICAL RESULT 4.3.2. Let f € Z( 2 ) with 2:5 p<4 and 0< a< -!

Suppose also that g € L" (1R 2 ) is such that f = G* g almost everywhere. Then

SRf(x) —3 (G*g)(x) C, p quasieverywhere for every k> 1.

HYPOTHETICAL RESULT 4.3.3. Let f€Z() with 2!~ p<, f<a~ and

n ~! 3. Suppose also that g € L' (E") is such that f = G* g almost everywhere. Then

SRf(x) —* (G* g)(x) c( _((fl _1)12(fl1)))1,p quaSieVerYWhere for every k> 1.

Result 4.3.3 would be an improvement on what we know (Conjecture 4.2.1 and

Theorems 2.3.3 and 3.2.11) only for a small region (when 1) -Ln- and

2(p-2Xn±1) <a :!~ ). Moreover, this improvement will always be far from the optimal

result (Conjecture 2.2.12). On the other hand, the Hypothetical Result 4.3.2 is always

near the optimal conjecture, but it has the disadvantages of only working in dimension

2 and of relying on a technique of proof difficult to extend to our setting. So, both

results would represent a progress of our knowledge, but this progress would be made

at a really high cost and with only a limited improvement. We will consequently not

push further this investigation here, but we will see in the next section a case where we

will transfer directly some information from the Bochner-Riesz problem.

4.3.2 The ZP Case when p< 2

We studied up to now what happens when 2 !! ~ p < r For fr < p <2, the situation

is completely different. Up to a few months ago, the only known results were

Theorems 5, 6 and 7 in [CS 1] whose estimates easily imply the following theorems

with a technique similar to Theorem 2.3.3:

26 In a really rough sense, one can think of the work done in this thesis as a sharpening of the estimates built for the maximal Bochner-Riesz operator in [Ch) and the other related papers.

THEOREM 4•3•4•27 Let f Z(lR") with n ~! 2 and 1< p!~ 2. Suppose also that

g E L(]R) is such that f = G* g almost everywhere. Then SRf(x) —* (G* g)(x )

Cyq -quasieveiywhere for all 0< e <e0 , where

(2pa+pn-2n)(n-1+2E)

= a — 2pn q

=

- ((n_1+2e)(2Pa+Pn_2n)_nPa)(2Pa+Pn_2n)

Y. )'

2,2n 2 2pa+n(n-1)(2-p) and = 2(2pa+pn-2n) when (2-p)n <a < 2-1

:7-- (2-p)(n-1+2c) I + and = 2pa-(n-1)(2-p) a — ,

q — — 2np o

(2-p)(n-I) < (2-p)n when 2 <a_

THEOREM 4.3.5. Let f Z(lR"). Suppose also that g e lf(IR") is such that

f = G*g almost everywhere. Then SRkf(x) —* (G* g )(x ) C,, ,,,,,-quasieverywhere

for all k, 1> 1 and for all lacunary sequences {Rk }

with

Iy=czandq=ptfn=2,zp<2 and0<a 1 , n+ I

_________ y = a —

(2 -p)n-p and (2- and q = r if n 2, 1< p ~ r <a ~

or if n ~ 3, 1< p < and (2-p)n-p 2p

7=a— (n- 1 and q= 3p3p,f6fl2 ifn ~ 3, ~ p<2 and 7l) <a ~ ---,

y=a—randq--Pn wherer=(l ifn ~ 3, 'zp<2 n pr

and (n-i)(2-p) <a ~

It was consequently expected that SRf(x) --4 f(x) almost everywhere for f e Z (IR")

with 2nI 0, but this is wrong. Last summer, Tao surprisingly showed

in [T] that the Bochner-Riesz operators fail to converge almost everywhere when

p e (1,2) if 0 !!~ a < (2 2P)n1 One can verify (at least heuristically) that a positive almost

everywhere result in our setting for p < 2 implies the same positive result with a loss

of e > 0 in the index a for the Bochner-Riesz problem (See Computation 4.3.11).

Consequently, Tao's result implies (at least heuristically) the following:

THEOREM 4.3.6. If 1 :5 p <2— I and 0 !!~ a < (2-p)n-1 then there is an f Z(lR")

such that SRf(x) does not converge almost everywhere.

Formally, Theorem 4.3.6. can be proven in exactly the same way as Tao's counter-

example if we slightly modify the point of view in Tao's proof [T, pp. 11-12]. So let

27 We do not include the proofs of Theorems 4.3.4 and 4.3.5 because they follow like Theorem 2.3.3 except that Proposition 2.1.3 is replaced here by the estimates built in Theorem 4.1.10 for Theorem 4.3.4 and in Theorem 7 of [CS 1} for Theorem 4.3.5. The indices y and q obtained in Theorems 4.3.4 and 4.3.5 are not optimal. (This lack of optimality is explained in the Appendix.)

me

have a look at this argument before proving Theorem 4.3.6.

PROOF OF TAO'S COUNTER-EXAMPLE.

The proof is done by contradiction. If Sf(x) converges almost everywhere for all

f E L"(lR), by Stein's maximal principle [St2, p.142], we should have

(*) E 1R: Sf f(x)> 2L} ~ CAT"llfll'

for all A >0 with C a constant independent of f and X.

Let f(x) =e 2 'Ø(f"2 x1 ,). Here, 0 is a smooth bump function associated with the

ball B(0, k), k is a small parameter to be fixed later, t>> 1 and I =(x2 ,x3 , ... ,x).

From the definition of f, we then have 11flIp = Ckt "2'. Let suppose that 0 was also

chosen in such a way that Ck !~ C < 00 when k <1 (this can always easily be done).

Now, suppose that we can show the estimate Sf(x) when 0< x1 - lxi - t and R Lt . By choosing ;L = t -(W2 1-6 in (*), we will then have

t :!~ C{x E IR":Sff(x)> t_2)_8 }:!~ CtPil2llfil < Ct2t'

as JJX

E 1R':0 <x1 - lxi - t}I - t. As desired, this is a contradiction if 8< (2-p),-1 -

To prove the estimate Sf(x) - t"'5 , we will use the classical expansion [St4,

p.391] for the kernel of S:

K (x) = C+ (x)R 282 lxi (7f1f25)/2 e2 N + C (x)R'252 I e 2''1 '1 + ER (x)

where IC±(x)I - 1 and ER (x) is the error term.

For y small and x satisfying lxi- t, so the error term

contributes like in Sf(x). Hence, we need to show that the main

contribution is coming from

(**+) t1+'2 5 e _YIe221 0(f1I2 y1 , y)dy.

The phase in the "-" case is not stationary so (**) is rapidly decreasing in t. While,

by doing a Taylor expansion in y, we have Ix — vI = lxi- (x - y)lxr', thus the phase in

the (**+) is withinof 2,rxIxI2 if we choose k small enough. So, there is not a lot

of cancellation in this case as the phase is nearly constant on the region of integration.

Consequently,

ME

Sf(x) t123 5 0(ty1 , = Ct128.

Now, we can return to the proof of Theorem 4.3.6:


If we let k(X)=(K R *Ga )(X), we found that K~ (x) R'(1+Rixi)_(fl+1)/2

e2'" . So,

Tao's argument will directly work to prove Theorem 4.3.6 if we did not transfer the

large parameter from R to x by setting 0 < x1 lxi - t and R = 1. killing in the process

the only place where a appears in K(x). But, this can easily be corrected by

rescaling everything. Let 0 and t be defined as in the proof of Tao's counter-example

and let f =e2 '0(t1/2 x1 ,t). Then SR(Ga*f)(X) — t_(h12)_a when 0< x1 — lxi - 1

and R = Z t. The loss of t on the size of the set where this estimate is valid is

compensated by an equivalent loss on llfIL

REMARK 4.3.7.

Theorem 4.3.6 goes against the intuition provided by the lacunary case [CSl,

Theorem 7] when n 2, < p ~ 2 and O< a< as well as when n ~ 3,

2n P - — < and (1 < v < (2-p)n-1

n+1+2a 2(n+1)

In fact, a result slightly stronger than Theorem 4.3.6 holds. More precisely, there are

functions f in Z (IR") with 1:5 p <2- and 0 ~ a < (2_2_1 such that SRf(x)

diverges everywhere on IR". We only give a sketch below of this result, but the full

details will appear in a forthcoming article.

COMPUTATION 4.3.8.

Let = f(x - y) where f(x) = e2 ' 0(t112x1 , t) as in Theorem 4.3.6. By the pre-

vious computation, we then have = Ct' 22" and SR(Ga*fty)(X) —

when 0<x1 —y1 —Ix--yl—land R-t.

Now, let {ak}"l be an increasing sequence of positive numbers satisfying both

akk "si 0 and ak k_ (2)_a 7 oo when 0 !!~ a < (2,n1 (e.g. ak =

with e > 0 small enough). Then, it is possible to choose a sequence 1k, 1 such that

91

Tk.

for all j N, akkJ2 '21)-a

00 and V° a k (l-2n)I(2p) <00 _1= k j

Figure 3.

When y=O, the region A =(xEIR 0<x1 —y1 _ix—yI-1} look like the shaded

part in Figure 3. So, by taking the union of all AY such that y has integers or half

integers coordinates (i.e. yel={y=(y i ,y2 , ... ,y) 2y Eforj=1,2,...,n}), we

can cover all IR".

Now, let {f3 } be a sequence of functions f,,y such that t = k. ye I and all y0 E I

are assigned to an infinity of different f (i.e. #{f :f, =fkY}=oo for all y(, El).

Hence,

I 'f1Ip ~ a, Dfi I = akkJ <00 I I

for f = afJ (so Ga *f E Z). We also know that SR (Ga *fj )(X) k7 12 and

ak 7 oo when 0 < -

y1 - Ix - y 1 and R = E- k 1 . Consequently, we will

be done if we can show for every jeN that SR (Ga *f)(X) is essentially

SR (Ga *fj )(X) when R — k.

But, this follows from standard estimates on oscillatory integrals in a way similar to

the counter-example in [CS2]. The main difference between this case and [CS2] is that

we want here some estimates valid for all IR" and not only for a compact set small

enough. When > R> 1, this factor is not important and we get, for all k >0,

I SRfM (x)I :!~ Ck M2M'

on IR".

For 2M < R, the loss of "smallness" forces the weaker estimate ISRfM(x)I ~ Ck M1R'

valid on ]R" for all k > 0, nevertheless the analysis stays similar to what was done in

[CS2]. This is enough as we want to use this estimate for R essentially fixed and

M< 1 T 2

N.B.: In fact, there is an additional factor of R"'' 2 in the denominator of all the

92

previous estimates when lxi> C> 0.

REMARK 4.3.9.

To get the two last estimates in the previous argument, we follow the proof of

Lemma 1 in [CS2] and use 2

= 7,41/2 - R(xi_M"2y1) I ___

M I12 1x_(M_ 1I2y, ,M')i I + M 2 1x_(M 112 yl .M)J j=2 lxj -

M -, Y 12

which is bigger than CM and C-M612 for the corresponding cases. (For 2M < R,

we split in the three cases lxi _M_h/2yiI>>tx__M_151, lxi

and lxi - M12y1I << I - M-'yl to get lower bounds C--, C-s- and C& which

are all bigger than Cr as M ~! 1.)

Because of Theorem 4.3.6, it is unclear what should be the correct analogue of

Conjecture 2.2.12 when <p <2. But, hoping for a Cap -result seems impossible

when we think of the ideas contained in arguments like the proof of Theorem 2.3.3.

Even when < a !~ -, it is unclear if one can truly hope for a C -result.

In fact, this also appears to be impossible when we look at what happens to the

convergence when a tends toin Theorem 4.1.10. In this result, the convergence

is almost optimal when a = , but this becomes less and less true as a tends toward

As there is no natural obstacle when 2 !! ~ p < 2nl- which generates this problem and

there is one (Theorem 4.3.6) when 1< p < 2, it seems plausible that this increasingly

"bad behaviour" is a consequence of Tao's phenomenon (Theorem 4.3.6) making

impossible any kind of C,, -result even when < a :!~

We want to point out that if this is true for some *1< p0 <2 and < a0 ~ - (i.e. if

there is an f (IR") for which SRf(x) does not converge C,, -quasievery-

where) then Il'in's example (Example 3.1.4) can be extended to Z0(lRn). More

precisely, the following result will then be true:

HYPOTHETICAL RESULT 4.3.10. There is an f Z(lR") such that f 0 on

B(0, 1) and lirnsuplSRf(0)I =00

Hypothetical Result 4.3. 10 is an immediate consequence of the fact that we only lack a -theorem when n C -localisation principle to get a C ~ 2, < p < 2 and

<a <iL28 . Nevertheless, extending IPin's example to some Z,° (IR") will not rule

out a Ca0 ,p:reSult as the reverse of the previous implication can be false.

For the convenience of the reader, we will now sketch rapidly how one can prove

heuristically that a positive result in our setting for p < 2 implies the same positive

result with a loss of £ > 0 in the index a for the Bochner-Riesz problem.

COMPUTATION 4.3.11.

One may easily show (see for example [CRdeFV, p. 514]) that there are m E C(1R)

supported in [1— 3,1] and satisfying both 0 ~ ,n 8 (x) < 1 and f-rn(x)I ~ C8 for all

k IN such that

(i. - 2-kA m2 ().

Consequently, we only need to understand smooth bump functions of the type

represented in Figure 4 to prove a positive result for the Bochner-Riesz operator.

-1 • ° 1

Figure 4.

On the other hand, by an argument similar to the beginning of the proof of Theorem

2.1.1, we need to control multipliers looking like in Figure 5 to obtain a positive result

for the inversion problem of ZP functions.

Figure 5.

- 28 This is contained in the proof of Theorem 2.4.7 in the modified version of Remark 2.4.8. Even if the remark was made for 2!!~ p<, it also applies for <p<2.

But, the function in Figure 4 is roughly speaking the difference of two functions (at

scale Rand R-fl) looking like in Figure 5ifwelet ö=and )a. So, if we give

up an e > 0 in 2 (i.e. A. = a - ) then the desired result follows with a little bit of

work.

4.4 Retrospective

In Section 2.2, we raised several different questions. This thesis answered some,

partly answered others and also ;ised new ones. So, to conclude this work, let us

rapidly recall the main open problems.

Our primary concern was to determine whether an analogue of Salem and Zygmund's

theorem (Theorem 1.5.8) was true or not in higher dimensions. We showed in

Chapters 2 and 3 that there is such a result when 2 !! ~ p < and 1!1. < a :!~ A. We also

showed that, for 0 < a !!~ I, it is possible to obtain an equivalent of Beurling's partial

result (Theorem 1.5.7) in higher dimensions when p = 2 and that it may be possible to

extend this to 2 < p < ç under some additional conditions on a using interpolation

(Conjecture 4.2.1). Nevertheless, Conjectures 2.2.1 and 2.2.12 are still open when

0 < a :!~ and moreover the uniformity part in their strengthenings (Conjectures

4.1.7 and 4.1.11 respectively) is open for all a E (o,].

When working on Conjecture 2.2.1 for 0< a :! ~ 1 , the obvious first step to try is the

classical idea of bounding S (Ga ) like for a > , but this does not make sense as we

realised in Remark 2.4.6 (1). A natural alternative is to replace the estimate made in

line (*) of the proof of Theorem 2. 1.1 by a better estimate which will ultimately lead to

an estimate without an N in line (**). Unfortunately, one can verify that this does not

work when we try to use the standard tricks (Fundamental theorem of calculus,

Sobolev's imbedding theorem, etc.). Consequently, we are left with three choices.

The result is true and a new set of ideas with more flexibility can be found to prove

Theorem 2.1.1, or the result is true, but it requires a higher dimensional Carleson's

theorem (this absolute need for a so strong result is not really plausible). Finally, the

result could be false in which case there is no strong type (2,2) estimate analogue to

Hunt's version of Carleson's theorem in higher dimensions. But, we should note that

the extensive search for such a counter-example has given nothing in thirty years.

95

Beside this principal framework, we were interested throughout this thesis in problems

which can be divided in two groups: localisation and uniformity. For the latter of

these, uniformity, nearly everything remains to he done. We only motivated in this

work some interesting conjectures which were not raised before (Section 4.1). In

contrast, some serious progress was made for the first one, localisation, in Section

2.3.2 and Chapter 3. Nevertheless, four problems of interest remain open.

When 0 < a :!~ , the work is at three really different stages depending on the value

of p. For p = 2, we have a Ca , 2 localisation principle (Theorem 2.3.7), but we know

from Theorem 3.3.1 that a better result holds. So, the question is to determine and

then prove the optimal localisation principle. For 2 < p <2n , a Cap principle has still

to be established when 0 < a :!~ - (we only have a C,, , 2 result for the moment). As

for the pointwise convergence above it should be possible to do a first step for larger

a using interpolation, but a completely new set of ideas is needed to deal with the

smaller ones. Finally, when < p < 2, we still need to determine when it is possible

to prove a C localisation principle and what kind of localisation takes place if any

when it is not possible to prove one. From this, it will then hopefully be possible to

determine what is the correct analogue to Conjecture 2.2.12 when < p < 2.

In comparison, for ' < a ~ -i-, everything is known when 2 ~ p < The only open

problems in this case are to determine if there is a C principle when < p <2 and,

in the affirmative, to verify if it holds everywhere as when 2 ! ~ p < .

It should also

be remarked that this range, '- < a <-, is also a good starting place in trying to ex-

tend the work done in this thesis for the Bessel kernel, Ga, to the closely related Riesz

kernel, 1.

In planning such an extension, one realises that the region near the origin of both

kernels is similar, so the study done in this work also applies to the central section of

'a But, outside a small neighbourhood of the origin, both kernels behave in a

completely different way. The Bessel kernel can be thought of as being zero and the

inversion operator converges everywhere on this part (Remark 2.3.6), while the Riesz

kernel has not enough decay for this (Remark 2.3.2 (3)). In fact, this is the crucial

region to establish the sharp convergence of the homogenous Sobolev functions. We

should expect that the convergence takes place precisely Cap -quasieverywhere on this

part.

061

Appendix

In this appendix, we will explain with more details the footnote 27 in Chapter 4 and

the last sentence of Section 4.2. We will also sketch rapidly how were obtained the

indices in Conjecture 4.2.3 and in Theorems 4.3.4 and 4.3.5. These results are

studied separately from the main body of the thesis because they are all based on some

interpolation involving an optimisation process (a technicality induced by the lack of

sharpness of the estimates used for the interpolation).

Normally, there is no question of optimality involved with the interpolation, so what is

the problem in these cases? The answer is that we do not only want an estimate like

(App.*) 11S. (G,, *g), !~ CIIg I,,G *1111,

where I + * 1 for Conjectures App.2 and 4.2.3 or like

(App.**)

:!~ CjG8 *g

for Theorems 4.3.4 and 4.3.5. But, to get the best possible capacitarian result, we

want in fact the estimate of these types which both maximise first the product pJ3 and

then the value p. In Conjecture 4.2.1, this was not a problem because the value a

was equal to the value 0 (modulo an arbitrarily small e >0) in the estimates (App.*)

used as basis for the interpolation. But, for the results studied in this appendix, the

distributions of a and 8 (as well as of p and r for Theorems 4.3.4 and 4.3.5) "do

not match". Consequently, different interpolation lines give different values.

aA i n-i I

P I

1 1

Figure 6 (Example of two different interpolation lines for a pair (-k, a)).

97

But, the optimisation problems are not really difficult to pose or to solve, so why were

the optimal answers not computed in Chapter 4? Even if these problems are trivial to

pose and if their solutions do not require a lot of thinking, they do require long

computations with really "nasty" equations. As the three results are quite far anyway

from what should be the optimal results in these three cases, we decided to only

include a "restricted" version of them which is easily computed, but which neverthe-

less achieves what we wanted (i.e. motivates that one can improve the localisation

below the critical index -- when 2 <p < (Conjectures App.2 and 4.2.3) and also

show that it is possible to do a capacitarian study when p < 2 (Theorems 4.3.4 and

4.3.5)).

In most cases, the restriction chosen is that the interpolation line has to pass through

(4,0~ ). The exact restriction for each case and the kind of computation "forced" by

this condition can be seen below.

COMPUTATION APP. 1.

In all the computations in this appendix, the point (4,o) means (4 ,$) with e >0

tending toward 0. In the same way, (*') means a point above the line joining

(* ) and (-- ,

a) which is slightly over the horizontal line 1 .

To get Conjecture 4.2.3, we use a weak form of the estimates built in Corollary

3.2.929. More precisely, we worked with the estimate IIS*(Ga2 *g2)1

!~

which is valid when 92 EL"(1R), 2!~ 3<ç, L<a2~ and /i 2 Efl(B(0,1)) is

such that IG,p*au2Ilq !~ ' p 1 = 1.

,a)I C

IN

n-I 2pa-(p-2)(n-1) -. 4pa p

Figure 7.

29 A similar remark applies for Conjecture App.2 in the region A of Figure 8.

If we start by doing an interpolation similar to Conjecture 4.2.1 "along" the line

joining (+ ,o) and (i.e. D in Figure 7), we then have three different kinds

of possible interpolation lines for the points (.,a) satisfying 2<P<1 and

(n-1) ( p-2)n <a :!~ These are a line passing through (-,a) and joining either A to C,

B toDorCtoDin Figure 7.

With a technique similar to Conjecture 4.2.1, any of these interpolations lines gives an

estimates of the type S. (Ga *g)11 I(p) :!~ CIg II , G,,* /4 with -+- = 1 (e.g. for the line

passing by (-,a) and (+ ,o) in Figure 7, we have ,

n(2Pa_(P2)(n-1))) The best esti-

mate is the one for which the product py is maximal (i.e. we want to maximise y).

An easy but long verification shows that the optimal interpolation lines for the three

cases are the two lines passing through (-,a) in Figure 7. As both lines gives exactly

the same value, the optimal line of interpolation in this case corresponds to our simpli-

fication (i.e. the line passing through (4,O)). We want to remark that the optimisation

was necessary as any other line passing through (.,a) and joining A to C gives a

smaller y and consequently a worse result.

If we now do in a similar way the interpolation planned in the last line of Section 4.2,

we should obtain the following:

CONJECTURE APP.2. Let f E Z(1R") with 2 ~ p < . Suppose also that f = G*g

almost everywhere and let E be the set of divergence for SRf off support of g (i.e.

E = JxO supp(g) : SRf(x) diverges}). If É c E is closed then cfl (E) = 0 where

=a+4- n(p-2) > azfn 2 .ir1 ' 2

- /3 - (a +.L)( 4pa_n(n_1)(p_2)

n(p-2) (n-l)p / — > a if n ~ 2 and a E

( n(n-l)(p-2) '

_________ Pn 2 _p_2n 2 +2n1 4p

In particular, SRf(x) (Ga *g)(x) C -quasieverywhere off support of g if E is

closed.

For Conjecture App.2 30, we used a slightly different simplification which is better

adapted to our estimates. To get the values in this "conjecture", we divided the region

30 Conjecture App.2 is not optimal. The values of /3 given here are strictly smaller than the best possible interpolation results because of our choice of interpolation line. (The indices are just there as an indication.)

real theorems and not only Conjectural results.

I I I a

n-i

I •

Figure (case n=2).

of interpolation in three two parts A and B (see Figure 8 below). When the point was

in A, we took the interpolation line passing through (-, ) (arbitrarily close above

(-.,o+) when n = 2 or 3) and the point we wanted up to the vertical line . While,

when the point was in B, we took an interpolation line passing arbitrarily close above

(+ ,o) and by the point we wanted up to the line limiting B. In choosing in such a

way our interpolation lines, we are simplifying the computations, but we are also

always clearly underestimating the size of the set of convergence.

Q n-i -T

n-i -r

n—I I -

-. 7 p

Figure 8.

For the last two results, Theorems 4.3.4 and 4.3.5, we used an interpolation similar to

the motivation of Conjecture 4.2.3 in Chapter 4. The only difference is that we

replaced the theorem of complex interpolation which allows changes in measures by

the "normal" theorem of complex interpolation of Stein [StW2, p.205]. With the esti-

mates obtained from this interpolation, we then proceeded like in the proof of Theorem

2.3.3 as explained in footnote 27. As a consequence, Theorems 4.3.4 and 4.3.5 are

jI iiii 1 ni n-s-3 7

Figure 10 (case n~!3).

Nevertheless, to ease the computations, we also choose some simplifying restrictions

in the interpolation leading to these results. For Theorem 4.3.4, we proceeded like in

Conjecture 4.2.3 (i.e. the interpolation line was forced to pass by (+,0)), while, for

100

Theorem 4.3.5, we followed a two steps argument. We first maximised the estimates

obtained by.Carbery and Soria in A of Figures 9 and 10 and then we interpolate in the

usual way between these values and the case p = 2 (Theorem 2.1.1). This standard

computation was done using an interpolation line passing arbitrarily close to n+3 n—I (slightly above and on the right of this point) in region C of Figure 10

and with an horizontal interpolation line in B of Figures 9 and 10.

N.B.: Theorems 4.3.4 and 4.3.5 are only included in this work by spirit of comple-

teness, but both are clearly far from optimal (except when n = 2 in Theorem 4.3.5).

As it was explained in Section 4.3, the sharp results in these cases are unclear because

of Theorem 4.3.6.

101

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