Table Of ContentLecture Notes ni
Mathematics
A collection of informal reports and seminars
Edited by .A Dold, Heidelberg and .B Eckmann, Z0rich
991
Charles .J ihcozzoM
Yale University, New Haven, CT/USA
nO eht Pointwise ecnegrevnoC
of Fourier Series
Springer-Verlag
Berlin Heidelbera - New York 1791
AMS Subject Classifications (1970): 43 A 50
ISBN 3-540-05475-8 Springer-Verlag Berlin • Heidelberg • New York
ISBN 0-387-05475-8 Springer-Verlag New York • Heidelberg . Berlin
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Dedicated to the memory ofm y father and mother
Foreword
This monograph is a detailed (essentially) self-contained treatment
of the work of Carleson and Hunt and others needed to establish the Main
Theorem: If f p (-~,~) e L l<p~,~, then Sn(X;f) converges to f(x)
th
almost everywhere where S (x;f) is the n partial sum of the Fourier
n
series for .f
The purpose of the first five chapters of this monograph is to
develop the machinery necessary to reduce the proof of the Main Theorem
to the proof (given in Chapter )6 of a theorem that involves only
characteristic functions in L (-4z,4~).
In Chapter 6 a large number of statements are presented without
proof. Every proof that is omitted in Chapter 6 can be found in Appendix
B.
The reader who is not familiar with the concept of Cauchy principal
value (denoted: P.V.) or the concept of the Hilbert transform is
referred to Appendix A for a summary of the results needed in the sequel.
Appendix C contains the recent results of Kahane and Katznelson on
divergence sets which in a sense are opposite to those of Carleson and
Hunt.
I would like to thank Professor S.A. Gaal, Professor R.E. Edwards, and
Professor .S Kakutani for their encouragement during the preliminary stages
of the writing of this monograph, and Professor R.A. Hunt for his very
generous help during the spring and summer of 1969 at which time he
explained in detail a portion of his original papers to me and made a
number of suggestions for improving a few of the proofs contained in them.
VI
Without his offer of assistance in the spring of 1969 I would not have
seriously considered writing this monograph. Also, I would like to thank
Professor .Y Katznelson for his permission to reproduce in Appendix C a
portion of his text: An Introduction to Harmonic Analysis.
An outline of the proof of Theorem ~2.b2a)s ed on Theorem (1.18) and
Theorem [3.6) was communicated to me by Professor E.M. Stein and my
appreciation for his doing so si to be noted here.
This paper was completed while the author was conducting a seminar
on the contents of a preliminary draft at Yale University during the fall
of 1969. I would like to further acknowledge my indebtedness to the men
who participated in this seminar: James Arthur, Eugene .J Boyer,
G.I. Gaudry, Michael Keane, S.J. Sidney, Joseph .E Sommese, Charles
Stanton, .A Figa-Talamanca and Professors .S Kakutani and C.E. Rickart.
September, 1970 C.J. Mozzochi
TABLE OF CONTENTS
Chapter ;i A Theorem of Stein and Weiss ................... 1
Chapter :2 The Main Theorem ............................. 8
Chapter :3 A Proof of Theorem (2.2) ...................... ii
Chapter 4: A Proof of Theorem (3.6) ...................... 19
Chapter :5 A Proof of Theorem (4.2) ...................... 20
Chapter 6: A Proof of Theorem (5.2) ...................... 24
Appendix A: The Hilbert Transform ........................ 44
Appendix B: Proof of Unproved Statements in Chapter 6 ....... 53
Appendix C: The Results of Kahane and Katznelson ............ 75
Bibliography ......................................... 86
.I A THEOREM OF STEIN AND WEISS
Throughout this monograph we assume each function f is real-valued
and in LI(-~,~) and hence finite almost everywhere in (-v,~). We say f
and g are equivalent iff f(x) = g(x) for almost every x in {-~,~). Hence
we may assume, when necessary, that f is finite everywhere in (-~,~).
The proofs in this chapter have been taken directly with only slight
modification from [14].
m or ~ denotes the Lebesgue measure on (-~,~).
(i.i) Definition. For each y > O the function
Xf(y) = m{x ¢(-~,v) I l > If(x) y}
is called the distribution function of .f
(1.2) Remark. Since kf(y) < oc for each y > 0 and f is finite almost
everywhere Limit %f(y) = O. Clearly, %f is non-negative and non-increasing.
y-~
Using the fact
co
U (x c (-~,~) 1 Ifcx)1 > Yo + 1 } : n ~×~ (_~,~) I I f(x)l > yo }
n=l
we have that Xf is continuous from the right. Since Xf is monotonic, it
has a countable number of discontinuities; so that it is measurable.
Let T be a mapping from a subset of the integrable real-valued
functions defined on (-~,~) that contains the simple functions into the
set of measurable real-valued functions defined on (-~,~). In this
chapter we assume 1 < p < ~, 1 < q <
(1.3) Definition. T is of type (p,q) iff there exists A > O such
that ";l"~ f~lq~ "" A-- p~[f~I- for every simple function .f
(1.4) Definition. T si of weak type (p,q) iff there exists A 0 >
such that for each simple function f and y O >
~Tf(Y) < [y q ['f~(P~
(1.5) Definition. T si of restricted type (p,q) iff there exists
A 0 > such that for each measurable tes ~ E )~,~-(
A IJXEU
llv~"EI! q - p , where ~E si the characteristic function
of E.
(1.6) Definition. T si of restricted weak type (p,q) iff there
exists A > O such that for each measurable set E~ (-~,~)
q
X E
(1.7) Lemma. fI T si of restricted type (p,q), then T si of
restricted weak type (p,q).
Proof. Let Ey = (x E(-~,~) 1 ITXE(X) I > .}y teL E be any
measurable set contained in (-~,~).
-Tf
(1.8) Lemma. For 1 < p < - and f ~ LI(-~,~) we have
Proof.
[f[Pdp: pyp-ldy d~= o o[ If (x)])
. ~T .'IT .TT '
so that by Fubini's theorem since the set {(x,y) I x )~,rT-( and
f(x)] I > y} is product measurable
j (-oJ )
]f]Pdu= pyp-I X )y( ul~ dy. But
T~ T~. ,o[ f(x)|)
~[ (y) = %E (x) where E : {x e xIifCx) I > y}.
[o, lfCx) l ) Y Y
This completes the proof of (1.8).
In the rest of this chapter we assume:
1 < p £ q < ~ k( = 0,1), p 4: P , q ~'q and for 0 < t < i
k k o 1 o 1
1 : (l-t) + t ; 1 = (l-t) + t
Pt Po Pl qt qo ql
If s ~ ,i then 's si that number (including ~ )
satisfying ( 1 + 1 i _ )=
S S v
(1.91 Lemma. Let T be of restricted weak type (Po' qo )
(pl,ql), then it is of restricted type (pt,qt) for 0 < t < .i
Proof. Let p = p q , = qt for a fixed t between 0 and .i
t
Suppose E ~ (-~,z) is a measurable set, ~E its
characteristic function, h = T~ and l(y) the distribution function
E
of .h We can assume, without loss of generality, that q < q . Then,
o 1
using the restricted weak type relations (with constants A and AI),
o
4
ew obtain by (1.8) yna rof 0 > C
=~dq[h[ q yq-lx(y)dy = q yq-lX(y)dy + q yq-lx(y)dy
~. JC
fC ( Ao ]l/Po)qo l;yq- ( ~l )lP/1 q
< q I-qy )7E(-~ [ yd q + 1 ])E(u[ 1 yd
0
:["q [q Alql ]
°qA 7 +
(q-q°)-[ [v(E)]q°/P° cq-q° (~l---q)'] [v(E) ]ql/Plc q-qt
gnitteL = C ])E(~[ s , where
)oq-q( P )lq-q( P lP
ew have
(q-qo) q/p
qo/Po
] C = [~(E)] = [~(E)] .ql/pl cq-ql
[~(E)
Thus ew have shown that
. (cid:127)
d~< A [~(E)] ,
Ihl q q q/p
erehw
q )
1/q
A =
q-q _ + q -q 1
o 1
completes This eht of proof )9.1(
esoppuS T si linear dna fo restricted type (p,q) dna tel
'q
f s L (-~,~). Let~be eht set-function, defined no eht measurable
stesbus C E (-~,~), such that
(1.10) ~ )E( = (T~)f ud
E
(p,q),~is
ecniS T is of restricted type countably additive dna
absolutely continuous with respect to ~ . Thus yb the mydokiN-nodaR
