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
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