The Carleson-Hunt Theorem on Fourier Series

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911 Ole G. Jorsboe Leif Mejlbro

The Carleson-Hunt Theorem on Fourier Series

Springer-Verlag Berlin Heidelberg New York 1982

Authors Ole Groth Jersboe Leif Mejlbro Department of Mathematics, Technical University of Denmark DK-2800 Lyngby, Denmark

AMS Subject Classifications (1980): 43A50

ISBN 3-540-11198-0 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-11198-0 Springer-Verlag New York Heidelberg Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich.

© by Springer-Verlag Berlin Heidelberg 1982' Printed in Germany

Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210

CONTENTS. PREFACE

1

CHAPTER I

3

1.

Interpolation theorems.

4

2.

The Hardy-Littlewood maximal operator.

10

3.

The Stein-Weiss theorem.

15

4.

Carleson-Hunt's theorem.

17 23

CHAPTER II 5.

The operators

6.

Existence of the Hilbert transform and estimates for

P Y

and

24

the Hilbert transform and the maximal Hilbert transform. 7.

33

Exponential estimates for the Hilbert transform and the maximal Hilbert transform.

40 45

CHAPTER III 8.

The dyadic intervals and the modified Hilbert transforms.

46

9.

Generalized Fourier coefficients.

51

10.

The functions

and the operator

CHAPTER IV

M* •

60

69

'1 .

11.

Construction of the sets

12.

Construction of the Pk(x;oo)-functions and the sets G and y* and and x* k Estimates of Pk(x;oo) and introduction of

74

the index set

80

13.

S* and

70

.

k.

G

14.

Construction of the splitting

15.

Construction of the sets

16.

Estimation for elements

of

00*

87

V* and

EN

91

rl(p*,r)

T* and p* 4 G* rL

101

IV

S*(x·xo· w* ) n 'F'-l

17.

Final estimate of

16.

Proof of theorem 4.2.

110 118

REFERENCES

122

INDEX

123

CHAPTER I. This chapter is composed of four sections. In

§

1 we introduce the concept

of (weak and strong) type of an operator, and we prove an interpolation theorem, which is a special case of a theorem due to Marcinkiewicz (cf. [9] for the general formula tion). In § 2 we introduce the Hardy-Li tt lewood maximal operator In

§

0

and prove that

(')

is of type

p

for all

p e ] I, +

co [



3 another classical interpolation theorem is proved, namely the Stein-

Weiss theorem, and finally, in § 4 , we prove .the Carleson-Hunt theorem under the assumption that some operator all

M defined below is of type

p

for

pEll,+"'[.

For technical reasons we shall always consider reaZ-valued functions defined on a finite interval, although their Fourier expansions will be written by means of the complex exponential functions. This assumption will save us for a lot of trouble in the estimates in the following chapters, and we do not loose any