Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann
911 Ole G. Jersboe Leif Mejlbro
The Carleson-Hunt Theorem on Fourier Series
Springer-Verlag Berlin Heidelberg New York 1982
Authors
Ole Greth Jersboe Leif Mejlbro Department of Mathematics, Technical University of Denmark DK-2800 Lyngby, Denmark
AMS Subject Classifications (1980): 43A 50 ISBN 3-540-11198-0 Springer-Verlag Berlin Heidelberg New York ISBN 0-38?-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 w 54 of the German Copyright Law where copies are made for other than private use. a fee is payable to "Verwertungsgesellschaft Wort", Munich. 9 by Springer-Verlag Berrin Heiderberg 1982" Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. t2141/3140-543210
CONTENTS.
PREFACE
3
CHAPTER I
4
i.
Interpolation theorems.
2.
The Hardy-Littlewood maximal operator.
10
3.
The Steln-Welss theorem.
15
4.
Carleson-Hunt's theorem.
17
23
CHAPTER II
Py
and
Qy .
24
5.
The operators
6.
Existence of the Hilbert transform and estimates for the Hilbert transform and the maximal Hilbert transform.
7.
33
Exponential estimates for the Hilbert transform
40
and the maximal Hilbert transform.
45
CHAPTER III 8.
The dyadic intervals and the modified Hilbert transforms.
46
9.
Generalized Fourier coefficients.
51
lO.
The functions
S~(x;f;m*)
and the operator
M* .
69
CHAPTER IV II.
Construction of the sets
12.
Construction of the sets
13.
60
Gk
and
Estimates of the index set
Y*
S*
and
70
~L "
Pk(X;~)-functions and the and
Pk(X;W)
~
and
74
X* .
and introduction of 80
G~ .
14.
Construction of the splitting
15.
Construction of the sets
16.
Estimation for elements
T*
~(p*,r) and
p* ~ G* rL "
V*
of
~* .
87
and
EN .
91 101
IV
17.
Final estimate of
e . o. Sn(X,XF,~_I)
18.
Proof of theorem 4.2.
9
II0 118
REFERENCES
122
INDEX
123
CHAPTER I.
This chapter is composed of four sections. In w 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 Marcinklewicz for the general formulation). mal operator
0
(ef. [9]
In w 2 we introduce the Hardy-Littlewood maxi-
and prove that
@
is of type
p
for all
p ~ ] I,+~[
.
In w 3 another classical interpolation theorem is proved, namely the SteinWeiss theorem, and finally, in w 4 , we prove .the Carleson-Hunt theorem under the assumption that some operator all
p~ ] i,+~[
M
defined below is of type
p
for
.
For technical reasons we shall always consider real-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 generality, since for a complex-valued function we may consider the two real-valued functions shall further assume that
f
and more complicated, full generality.)
and
is integrable , f eLl(1)
finite interval mentioned above. this chapter also hold for
Re f
, where
I
is the
(We may note that most of the results in
f ~LI(R)
f
Im f instead. We
, but their proofs may be different
so we have avoided to prove the theorems in their
w i. InterDola_lion theorems, Let
f
be a real-valued function defined on an interval
pose that
f ~LI([-A,A])
.
The Lebesgue-measure on
R
is denoted by
depending on the function (i.i)
Ey = {xE
understand the function
(1.2)
and sup-
m . We introduce the sets
E
Y
under consideration by [-A,A] I [f(x) l > y} , y e R+
By the distribution function
Definition I.i.
%f : R++ [0,2A]
~f(y) , y c R+ , we shall
defined by
~f(y) = m(Ey) = m({x~ [-A,A] I [f(x) l > y})
Clearly, we have and
f
[-A,A]
%f(y)
0 =
for all
yER+
, %f(y) + 0
is a decreasing function. Furthermore,
%f
as
y++~
,
is continuous from
the right, as
U E = E n=l y+ ~ Y
for all
y ER +
n
From this fact we conclude that
%f
ity points, and especially,
%f
is a measurable function.
Let
LI([-A,A])
T
be an operator from
measurable functions on defined on all of
[-A,A]
LI([-A,A])
has at most countably many discontinu-
into
. The operator
~[([-A,A]) T
, the set of all
will not necessarily be
, but will at least be defined on all simple
functions , i.e. finite linear combinations of indicator functions of measurable subsets of
[-A,A] , and on all continuous fun6tlons, and so
especially the domain of In the following linear set and ~, S~ R(C)] set,
, or
T
T , ~(T)
, is dense in
9
will either be a lineca~ operator [i.e.
T(~f +Sg) = aTf + BTg T
LI([-A,A])
for all
f, g E ~ ( T )
will be a sublinear operator [i.e.
IT(mf)[ = ImI[Tf]
IT(f +g) I ~ [Tfl + [Tgl
for all
f E ~(T)
for all
f, gE ~ ( T ) ]
and all
~(T)
is a
and all
~(T)
is a linear
m ~ R(C) , and
The operator
Definition 1.2.
pc [I,+~])
T
(1.3)
where
for
f r ~(T) .
Usually one considers operators
p,q ~ [i, +~]
. The operator
(p,q) , if there exists a constant
(where
A p c R + , such that
IITfllp ~ ApllfNp
Remark 1.3.
p
is said to be of (strong) type
if there exists a constant
T
T
of (strong)
type
(p,q) ,
is said to be of (strong) type
A,q
~ R+ , such that
IITfllq <= Ap,qnfllp
for
f c ~(T)
.
(Cf. e.g. [9] and [7]). For our purpose it is enough to consider the case, where
p = q . Analogous remarks are applicable
to the definitions
in the
following.
We note that if an operator can be extended to all of dense in of
LP([-A,A])
LP([-A,A])
T
is of type
LP([-A,A])
, and
T
p
with
by continuity,
p e [I, +~[ since
, then
~(T)
T
is
is thus a bounded operator defined on all
.
Definition 1.4.
The operator
T
is said to be of tea]< type
p r [I, +~[) , if there exists a constant
p
(where
A p ~ R + , such that
(i. 4) =\Y] for all Note
that
f e ~(T) for each
and for all y eR+
P
y e R+ .
,
IITf IIpP = IITf(x) IPdx > yP %Tf (y) " Hence we conclude that if
i.e.
T
T
is also of weak type
is of type
p , p ~ [i, +~[
, then
p .
We shall later on give an example showing that the converse is not necessarily true.
We shall also need the following concepts, closely related to what we already have introduced.
Definition 1.5.
(where
The operator
p ~ [I, +~[ )
HT•
for all measurable sets
(where
p ~ [I, +~[ )
and if
T
T
Ap ~ R+ , such that
= Ap[m(E)]I/P
T
is said to be of restricted weak tyFe
if there exists a constant
for all measurable sets
p
E ~ [-A,A] .
P <_ (~y)l]•
~T•
Clearly, if
p ~ Ap[l•
The operator
(I.?)
is said to be of restricted type
if there exists a constant
(1.6)
Definition 1.6.
T
p
Ap ~ R+ , such that
~P = ( ~
m(E)
E ~ [-A,A] .
is of type
p
then
T
is also of restricted type
is of weak type
p
then
T
is also of restricted weak type
p . Furthermore,
if
stricted weak type
T
is of restricted type
p
then
T
p ,
is also of re-
p . (This is proved in quite the same way as the asser-
tion following definition 1.4.)
Before we start proving the interpolation theorems we need a result relating the d~stribution function
%f
to the p-norm
IIfllp
fact, we have
Lermna 1.7.
If
(I. 8)
and especially,
(1.9)
i~
f ~LI([-A,A])
ItfIIPp=
then for each
fIt(x)Ip
p E [I, +~[
[+p {-I ~0
f(y)dy ,
of
f . In
Proof.
Using Fubini's
theorem we have
=
P
=
yp-I
yp-lm({x
[-A,A]
If(x) l > y})dy = [§
Note that if
Xf(y)dy
~
O
f ~ Lp
then both sides of (1.8) are
+~
We now prove the first interpolation result.
Assume that
Lemma 1.8.
where
T
is of restricted weak type
I ~ Po < Pl < + ~ " Then
T
is of restricted type
Po
and p
Pl "
for all
p E ]po, Pl [ Proof.
Let
l(y)
denote the distribution
function of
assumptions we know that there exists constants
l(y) <= for all measurable
IIT•
m(E) sets
E
= p
,
and
%(y) ~
and all
. From the
A I , such that
m(E)
y E R+ . Using len~na 1.7 we get
yp-1 X(y)dy + p o
yp-1 X(y)dy 1
< p .m(E)
= p .m(E)
A~
TX E
o y
A
o
A~
9 I
dy +
+ AI
y
A I dy
.
P-Po and thus IITXEH p ~ Ap[m(E)] I/p with
Ap = pl/p 9
pl
A o " P-P---~I+ AI
i IIp
P--~S
for
p E ]po,Pl [
We note that Po
and
Ap
is bounded as long as
is bounded away from
P e]Po' Pl [
Pl "
A closely related result is the following which is a special case of a theorem due to Marcinkiewicz (cf. [9]).
Theorem 1.9.
Let
T
be a sublinear operator of weak type
where I
T
is of type
p
for all
Po
and
Pl "
P ~ ]Po" Pl [ " More
'~ %Tf(y) = m({x I ITf(x) l >y}) <_
%Tf(y) = m({x I ITf(x) l >y}) <= then
where P l -p
Po " Kp = p .2P(p1_po + ~ ) A o
Proof.
We choose
(I.i0)
A = A~
fy
for fixed
y ~ R+ f(x)
-Pl Pl-Po AI
A
P-Po Pl pl-Po 9 AI
defined above we introduce the functions
fY
and
by if
If(x) l J A y
fY(x) =
[ ,
0
and note for later use that
Pl -p Po pl-Po Pl APl -p = AI = A~
o AP~ A p~
Using the constant
Po Pl-Po
P-P o Pl " Pl-P~ " AI Pl-P~
otherwise
fy(X) =
0
i
f(x)
if
If(x) l ! Ay otherwise
Clearly. we have f(x) = fy(X) + fY(x) . and the sublinearity of T gives us that %Tf(2Y ) ~ %Tfy (y) + %Tfy(Y) which by assumption is smaller than A Po o y-P~ I ,fy(x),P~
+ APl 1 y-Pl I IfY(x),Pldx
Using Fubini's theorem to interchange the order of integration, and the constant A introduced above, we get
0~f-~o JO
O
-Po p-I
y I
I
x~
dx) dy
{ If(x) I>Ay}
[~[~ -Pl ~p-i < "O
j ~x ~ dx) dy}
{ If(x) I
= p.2P{A~~
,f(x),P~
yP-p~
+~ yp-pI-I dy)dx} + AIPl ;'f(x)'Pl(;,f(x), A = p.2P{A~~ pO-p .
I P-Po
I
Pl IIf(x) IPdx } if(x) iPdx + AI APl-P. I pl-P
Pl -p P-Po Po" pl-Po PI" pl-Po f i = p . 2 p. A 9 AI 9 o [ P-Po where we have used (i.i0).
D
In a similar way we get the following result.
Pl-P
10
Theorem i.I0. strong type
Let +~
p c ]Po" +~[
T
be a sublinear operator of weak type
, where
P o C [1, +~[
. Then
T
Po
is of type
p
and of for all
" More precisely we have that if
~Tf(y) = m({x I ITfl >y} < ( ~ ) P ~
IIf[l~
and
llTfll < A llfll
,
then IITfllp < p " ~ =
Proof.
9 A ~
9 -P-Po
llfll
9
We use the same notation as in the proof of theorem 1.9. We choose
the constant
A
as
~
. Then
IlfYll~ =< ~ y
, IITfYII~ =< y , and conse-
.co
quently
;~Tfy(y) = 0 .
This gives
the estimate
~Tf (2y) ~ ~Tf (y) ~ A o Y
y_Pol
fy(X) Ip~ I dx ,
and (exactly as in the proof of theorem 1.9) 2P
=
~
po-p
1 P-Po
9
\A /
proving the theorem.
j
D
w 2. The Hardy-L~ttlewood maximal operatoK. In this section we shall consider the Hardy-Littlewood maximal operator and derive estimates for this operator using theorem i.i0. Let
f ~LI(R)
. We define the maximal operator
G
by
I Ix+t 8f(x) = sup ~--~ If(y) Idy , x E R ts + ~x-t
(2.1)
Clearly, the function
Of
is measurable for each
f e LI(R) , because
@f
is lower semicontinuous as a supremum of continuous functions, and the operator
0
is a sublinear operator, defined on
Theorem 2.1.
The operator
precisely,
satisfies the following estimates
O
0
is of type
+~
LI(R)
and of weak type
I . More
11
(2.~)
llofll. ~ llfll , 4
(~.s)
xoffy) = max IOf(x) >y}) 2 ~ llflll
for all
y e R+
From theorem 2.1 and theorem i.i0 we infer at once that the operator of type
p
for all
p ~ ]I, +~[
g
is
. In the applications later on we only need
functions of compact support~ so let us assume for the proof that
f
has
compact support. It is obvious that (2.2) is satisfied. In the proof of (2.3) we may as well assume that xs
I Of(t) >y}
f
is non-negative. Let
there exists an interval
I
y s R+ . To each
(with center at
x
x ) such
that I
f(t)dt > y .m(l x) . I x
As
f
has compact support, the set
pose that all the intervals
I
x
{Of >y}
are contained in an interval
We claim the existence of a sequence intervals
I
x
is bounded, and we may sup-
(In )
[-B,B]
(extracted from the class of
above) of pairwise disjoint intervals, such that
m
In
U Ix
> [m
n I
x
Assume for the moment that this has been proved. Then the proof is finished as follows:
m({x I Of(x) >y}) __<m
4 =< --Y
U Ix x
< 4m
4 I f(y)dy = ~ n
;
U In n=i
= 4 I m(l n) n= I
4
U I n n
f(y)dy < ~llfll I =
9
Let us prove the assertion above (this is a theorem of Besieovitch-type; results in that direction can be found in e.g. [3]) Let
SI
be the class of all intervals
a I = sup{m(l x) II x r S I} m(ll ) > 3 a I sect
. Let
I I , let
S2
I
considered above. Let x and choose an interval I I from S I with
be the class of intervals
a 2 = sup{m(l x) II x ~ S 2}
Ix
and choose
that do not inter12
from
S2
with
m(l 2) > ~ a 2 . In this way we continue. If the process stops after a finite number of steps, the result is obvious, so we may assume that we obtain a
12
sequence
(a k)
of real numbers
disjoint intervals
(Ik)
Let us consider an interval denote
the first
ak + 0
with I
and a corresponding
m(l k) + 0
for
and let
x
index for which
sequence of
k -+ +
k
I x ~ Sk ; Ik. I
in that case
Ix n l k _ I ~ @
i
and
m(Ik_ I) ~ ~ m ( l x)
. Then we have
where
Ik_ I
Jk-i
and
ter and where
i
i
i
i
Ix c J k _ l
have the same cen-
m(J k i ) = 4m(l k i ) . Using , and the theorem is proved. It
s h o u l d be n o t e d t h a t
possible one.
Note that
the constant
4
appearing in (2.3) is not the best
D 9
is not of type i (if we e.g. let
e
f(x) = X]0,1[(x)
, then
e
is not even integrable')
Corollary 2.2.
The operator
e
is of type
p
for all
p ~ ]i, +|
. More
precisely we have (2.4)
llefllPp __< 4p. 2p . ~
Iifiip
for all
p E ]1, +~[
o
As mentioned above this follows from theorem 2.1 using theorem i.I0 with Po =I
If
, A ~ =4
g(x)
and
A
is a real-valued
g+(x) = max{g(x),0}
Theorem 2.3.
(2.S)
=i
.
~ .
function, we define another function
. A well-known example is
The operator
e
log +
satisfies
[ (of(x)- 2)+~ ~= 8 [ if(x)llog+if(x)j~ J
J
g+(x)
by
13 Proof. We may of course assume that f ~ 0 and that f log+f grable. We define functions fY and r by Y fYtx~/ = ~ f(x)
L
0
0
if f(x) ~y fy(X) =
otherwise
is inte-
if f(x)~y
f ( x ) otherwise
Then clearly Ofy ~ y and so m({x J@f(x) >2y}) ~ m({x JOfy(X) >y}) Using (2.3) on the function f we get Y (2.6) m({x 1Of(x) >2y}) ~ 4 ] fy(x)dx . An integration of (2.6) from I
to
+~
J 0f(x) >2y})dy _<
.
gives
~
my
i = 4
i ( Jy=l ~[,f(x),I dy)dx = 41 f(x)log +f(x)dx {if(x) I>i}
I +~ m({x jef(x) >t})dt = ~If (0f(x) -2)+dx . The left hand side equals 2S2 and thus the result follows.
Later on we shall also need the following two simple lemmata:
Lemma 2.4. Let F
x
on
R
by
f~LI(R)
. For any fixed
we define a function
Fx(t) = S~f(x+y)dy 9 Then
JFx(t)[ ~ 2JtJ@f(x)
Proof.
xER
for all
x~R
, a~l
t ~R
The len~na follows immediately from
irx+t (x+t [x+,t, JFx(t) J = Jx f(y)dy < Ix f(y) dy < Jx-JtJ Jf(y) Jdy ~ 21tJ "Of(x) .
Le~ma 2.5. If
feLl(R)
, then
if(x) I <=Of(x)
for almost every
x rR .
14
Proof.
The lemma follows from the following two facts x+t Of(x) > ~ t
If(y) Idy
for all
t9
Jx-t and
[x+t
I -2t ~ - t
If(y) Idy -> If(x)]
where we have used Lebesgue's (cf. e.g. [3]).
for
t~0 +
differentiation theorem for
x 9R ,
Ll-functions
D
From lenm~a 2.5 and theorem 2.1 follows that if get
for almost every
f 9
nL=(R)
we even
[[fll= = ll@fll~
Finally we shall prove an exponential estimate for f(x)
Of , when
f ~L ~
and
is zero outside a compact set.
Theorem 2.6.
Let
c 9 R+
be any positive constant and let
f
be any es-
sentially bounded function, the support of which is contained in an interval
I
of length
A . Then for
y 9
(2.7)
XOf(y) = m({x Igf(x) >y}) <_ 2A . e x p c 9 ~ Y
Proof.
It follows from theorem 2.1 that
even have
l~Gfll = llfI[~ ) , so if
.exp
(
-c.
llfll~
)
lI0fll~ ~ IlflI~ (in this case we
y ~ llfll
the left hand side of (2.7)
is zero, and the estimate is trivial. Let
y 9 ]0,11fll [ 9 If
x 9 {x I@f(x) >y)
and
dist(x,l) = d > 0 , then an
application of (2.1) gives
y < Of(x) < ~
If(T) IdT __<~--~ llfll
,
from which we conclude that
m({x I Of(x) >y}) j A + 2
Let
t
=
9
A llfIl~ (llfll~) Y A. i + - -y -
~-F77F-l Y e ]0,I[ . Then it is enough to prove that
15
1 1 i + ~ =< 2exp c. ~ 9 exp(-ct)
,
tE ]0,I[
,
which is equivalent to the trivial estimate (l+t) exp(ct) ~ 2. expc and the theorem is proved.
for
t E ]0,I[
,
D
w 3, The Stein-Weiss theorem. We shall now introduce an operator
T~
associated with the operator
T .
In the following we assume that all functions considered are defined on a fixed interval p , where
[-A,A]
p e ]I, +=[
. Let
Let
q
E ~[-A,A]
be the number conjugate to
A p ~ R+
such that
,
p , i.e.
. We define a set function
contained in
[-A,A]
y
! + ! = 1 , and let P q on the class of Borel sets
by
(3.2)
y(E) = I (T•
H~Ider's inequality implies that that
operator of restricted type
IITXEIIp =< AplIXEII p = Ap[m(E)] I/p
f eLq([-A,A]) E
linear
be a
, i.e. there exists a constant
for every measurable set (3.1)
T
X
y
is well-defined,
and it is easy to see
is a countably additive set function which is absolutely continu-
ous with respect to Lebesgue measure. Using Radon-Nikodym's theorem we then get an (up to nullsets) uniquely determined function
(3.3)
~(E) = ~ h ( x ) d x
for all Borel sets
E S[-A,A]
Because of this relation we
of
such that
= IXE. hdx = ITXE" fdx ,
.
define
(3.4) Clearly,
h
an operator
T*
on
Lq([-A,A]
by
T*f = h . T*
is linear, and
T . We have e.g. that if
(3.5)
formally g
Tg 9 f dx = A
T~
behaves as the adjoint operator
is a simple function, then
Iig9 T * f d x
9
16
Lemma 3.1.
Assume
p ~ ]I, +~[
where
Proof.
that the linear operator
. Then
T~
is of restricted q , where
By assumption there exists a constant
(3.6)
type
P ,
~ + ~ = I . P q
A oR+ , such that P
][TXEH p =< ApI[XEI[p = Ap[m(E)] I/p
We shall prove the existence of a constant
(3.7)
XT, f(Y) ~ and let
[Ifll
Let
f EL q
E
(x I lh(x) l >y} 9 We put
Y
T
is of weak type
h=T*f
for all
and E
Bq~ R+ , such that
f
and all
y~R+
.
%(y) = %h(y) = m(Ey) , where as usual,
= E + uE- , where Y Y Y
E + = (x lh(x) >y} , Y
E- = {x [h(x) < - y } Y
Finally, we define X+(y) = m(E~) , Clearly,
A-(y) = m(Ey) .
X(y) = X+(y) +X-(y) , and for
%+(y)
we get the following esti-
mate
Y
Y
ffi I TEE+y" f dx
Y
< IIT• p y
= Ap[m(E~)] I/p
tlfllq
where we have used the definition of
IIfllq ~ ApIIXE~IIp [IfI[q
= Aq[X+(y)] l-(I/q)
B
= 2 I/q .A q
X+(y) = 0
X (y) , and hence
X(y) = X+(y) + X-(y) ~ 2 which is (3.7) with
0 P
Hft[q
,
T* , H~Ider's inequality and (3.6).
From the inequality above we deduce that either [X+(y)] I/q ~ (Ap/y) 9 Ilf]lq We have a similar estimate for
Y
IlfI[
or
17
If the operator
Theorem 3.2. (Stein-Weiss).
ed weak type
Po
p
p c ]po,Pl [ .
for every
Proof.
Let
Pl
p c ]po,Pl [
stricted type T*
and
p'
is of weak type
q'
for all
Tf = T**f T
for
T .
" then
T
T*
T** = (T*)*
" Then
T*
is of weak type
ql'
is a bounded linear operator on is bounded on
T**
T
is of re-
are linear and
T**
and and
T*
Pl' qo' '
is of type
L q , which
L p . From (3.5) we conclude that
for all simple functions, i.e. on a dense set in and
is of type
Po'
q' r ]ql,qo [ . We choose
theorem (theorem 1.9) we infer that
q , and especially that implies that
i8 linear and of restrict-
p' e ]po,Pl [ , and then lemma 3.1 tells us that
Po < p o < p
such that
T
I
be given. From Lemma 1.8 we get that
for all
and from Marcinkiewicz'
both
(where
is of type
L p , and since
p , the same is true
Q
4 , Carleson-Hun~'s theorgm. In this section we shall prove the main theorem under the assumption that an operator
M
defined below is of (strong) type
p
for all
p ~]I, + ~ [
The rest of the text is devoted to the proof of this assertion.
We shall only consider functions
f
defined on the interval
[-~,~]
. We
mention (without proofs) the following well-known facts about the spaces LP([-~,~])
.
(4.1)
If
I =< q =< p -_ <+~
(4.2)
Let
e ~ R+
9
then
LP([-~,~]) c Lq([-%~])
be given. To any
find a polynomial
fcLP([-~,~])
P , such that
, p E [1, +~[ , one can
llf-PHp < c .
.
18
Let
(4.3)
p c [1, +~] , let
from
fELPc[-~,n])
LP([-~,~]) , such that
ces~ extract a subsequence
fnk(x) + f(x)
, and let
Ilfn-fllp+ 0
(fnk)
of
(fn)
as
be a sequence
n§
. Then one
(fn) , such that
a.e. x E [-~,~]
as
k §174 .
From these properties it is easy to derive the following lemma.
Lemma 4.1.
Let
To any given
p E [I, +~[ , and let
~ ~ ]0,1[
f E LP([-~,~])
one can find a sequence
be a given function.
(PE,k)
of polynomials,
such that Pe, k(X) + f(x)
(4.4)
a.e. x E [-~,~]
p p < 2k ilf- e,kilp E
(4.5)
Proof.
as
for all
Using (4.2) we can find a polynomial
k§
ken
Qk ' such that
Iif-QkiI~ < c2k By (4.3) we can find an increasing sequence nk~k)
e
2k
is decreasing in
both (4.4) and (4.5).
k
(note that
N
f ELI([-~,~])
and
a.e. x E [-~,~] .
n k ~ k , the polynomials
LP([-~,~]) ~ LI([-~,~])
. Then the Fourier coefficients
satisfy
c
n
=
~
i [z f(t) e-int dt ,
we denote the partial sum
for
p E [i, +~]
9 Let
e n , ns Z , are all well-
defined by
Sn(x;f)
(Pe,k)
D
It follows from (4.1) that
By
from
, such that Pe,k(X) = Qnk(x) + f(x)
As
(nk)
nEZ
19
n
Sn(x;f) =
[ ckeikx k=_n
of the Fourier series for
,
xE[-~,~]
,
n~N o
f .
In the following we shall only consider the case, where Let
~7 denote the class of functions with values in
an operator
M:LP([_~,~])
(4.6)
§
M
[0, +=]
.
. We define
~7 by
Mf(x) = sup{[Sn(x;f) l I n E N
It is obvious that
p E ]I, +=[
o
} .
is suhlinear, M(f +g) ~ Mf + Mg .
The following theorem is a very deep result, which shall be proved in a later section.
Theorem 4.2. The operator M is of type p for all p E ]I, +~[ . More precisely, to every p r ]1, +~[ there exists a constant CpC R+ , such that
llMfllp ~ % I l f H p
(4.7)
for all
f ELP([-~,~]) .
~ere we shall take theorem 4.2 for granted. Then it is easy to prove the following lemma.
Lenm~a 4.3.
Let
pc ]1, +~[
and let f ~LP([-~,~])
Let % be the constant in (4.7). For nomials defined in len~na 4.1. For each
be a given function.
~ c ]0,1[ let Pe, k k c N we define
Ee, k = {xr [-~,~] IM( f-Pc,k)(X) > Ek/P 1 9 Then
m(E , k) < ~p Ek
be the poly-
20 Proof.
From lenm~a 4.1 we get
IIf-Pe,k[l~F < e
k
I) k
<
Theorem 4.4.
~ {M(f-ee, k)(x)} p dx =
Cp . P
e
2k
k = Cp e P
(Carleson-Hunt,
f9
, so
<
m(Ee'k)= (1)k~ ekm(Ee'k)=(1)kIEe,k <
2k
e,k
k 1[M(f-ee,k) llp P <=
C IIf-Pe,kll
D
[i], [4], [7]).
Let
p E ]1, +~]
. For every
we have
a.e. x 9 [-%~]
Sn(X; f) + f(x)
Proof.
As
L=([-~,~]) S LP([-w,z])
p 9 ]i, +~[ . Let
e 9 ]0,I[
as
n§
it is enough to consider the case
be an arbitrary constant, and let
the polynomials defined in lemma 4.1. As
Sn(x;f)
Pe,k
be
is a partial sum of
the Fourier series it is trivial that Sn(x;f ) = Sn(x;f-Pe, k) + Sn(X;Pe, k) for all all
x 9
k 9
and all
[-~,~]
that
x 9 [-~,~] . This implies for all
k ,n 9 N
and
ISn(x;f) -f(x) l ~ ISn(x;f-Pe,k) l + ISn(X;Pe, k) -f(x) I Since
Pc,k(X)
is a lim
C -functlon on
]-~,~[ , we have
Sn(X;Pe, k) = Pc,k(X)
for all
xE ]-~,~[ .
n + + ~
Hence, for every
k 9N
and every
x 9 ]-~,~[
limn++=supISn(x;f) -f(x) l <= limn++~supISn(x;f-Pe,k) l + If(x) -Pe,k.rX) l Using that
lim sup ISn(x;f -P ,k) l < M(f -Pe,k)(X) s
n + + ~
x 9
]-~,~[
(4.8)
for all
k 9N
and all
=
we infer that lim sup ISn(x;f ) -f(x) I < M ( f - P n.+
+ =o
-
,k )(x) + If(x) -P ,k(X) l 9 e
21
Let
Ee, k
be defined as in lemma 4.3. Then
by the definition of
E
m(Ee, k) =< C pp ek
9 Furthermore,
e,k '
M(f-Pe,k)(X)
~ ek/p
for
x ~ Ee, k ,
SO +co
M(f-Pe,k)(X ) + 0
for all
x~A e =
U E n= I
as
k-~+~
.
e,n +~
Hence, we only have to estimate the measure of the set Since
e e ]0,I[
Ae =
U E n=l e , n
we get
m(Ae) ~ n=l~ m(Ee'n) ~
cp ~ en = cp 9 e P n=l p l-e
According to len=na 4.1 there exists a null-set If(x) -ee,k(X) l + 0
for
B e ,m(B e) = 0 , such that
x c ]-n,~[kB
e
Hence it follows from (4.8) that lim Sn(x;f) = f(x)
where
m(A e UBe) < C p 9 e = p l-c small the theorem is proved.
. As
for all
e E ]0,i[
x E ]-~,~[\(A e u B e) ,
can be chosen arbitrarily
CHAPTER II.
In this chapter we shall define the Hilbert transform and the maximal Hilbert transform. These two transforms will play a crucial role later on. We shall follow [2]. The classical exposition may be found in [8]. The two transforms are formally defined in w 5 . It follows immediately from the definition that the maximal Hilbert transform is well-defined. This is not, however, so obvious in the case of the Hilbert transform itself. The proof of this assertion is postponed to w 6 . We use two auxiliary transforms
Py
and
Qy
combined with the Hardy-Littlewood maximal operator in
order to derive the properties of the Hilbert transform. As mentioned above we prove in w 6 that the Hilbert transform is welldefined. Furthermore we show that both the Hilbert transform and the maximal Hilbert transform are operators of type
p
for all
p ~ ]I,+~[
.
Finally, in w 7 , we prove some exponential estimates for the Hilbert transform and the maximal Hilbert transform. These results are similar to the exponential estimate for the Hardy-Littlewood maximal operator proved in theorem 2.6.
24
w 5. The operators
Let
f E Ll(R)
. For each
(5.1)
Hyf(X)
y c R+
let
= ~i
[j
Py
Hy f
and
Qy .
be the function defined by
f(t) x-t dt '
xe R "
{ Ix-tl>y} The Hilbert transform Hf
(5.2)
Hf(x) =
where
(pv)
however, that
lim y§
f
is then defined as
H f(x) = ! (pv) Y #
If(t) ~
dt
stands for "principal value". At this stage it is not clear,
that
Hf(x)
of
Hf(x)
is a well-defined
expression. We shall in w 6
show
does exist almost everywhere.
We shall also consider the maximal Hilbert transform H~f
of
f , which
is defined by (5.3)
H~f(x) = sup{IHyf(X) l I y E R + }
In the applications we shall only consider functions nite interval,
. f
defined on a fi-
so without loss of generality we may in the following assume
that all functions
f,g, ...
This assumption simplifies
under consideration have compact support.
some of the proofs. It may be noted, however,
that the results hold in general.
We introduce two linear operators
(5.4)
Pyf(X) = ~
Py
Qyf(X) = T
Pyf(X)
and
and formally we have
Qyf(X)
related to
Hy
~" dt , (x-t)2+y 2
y E R+ ,
f(t) 9
x-t dt , (x-t) 2+y2
y E R+
-~ Clearly,
Qy
f(t) 9 -~
(5.5)
and
, yER+
Qof(X) = Hf(x)
, are well-defined .
for all
as follows
x(R
,
25
In the following Pyf(X) ky(X)
= ~i
Qyf(X) s
y
will always denote a positive number. We note that
is the convolution of 9
~ = - ~v Im(~) x2+y 2
f
with the integrable function
, z = x + iy , i.e.
is similarly the convolution of = ~i 9 x2+y 2x
~IRe(~) , i.e.
f
we note that k (x) > 0 Y Y even, and decreasing on [0,+ ~[
If
f(x)
k
is a continuous
and
and
with the function
Qyf(X) = (f*Zy)(X)
not integrable. Here and in the following, Concerning
Pyf(X) = (f*ky)(X)
. However,
s
is
k
is
z = x +iy . I+~ ky(X)dx = i _~
and
Y
~
L -functlon, it is easy to see that
~(x,y) = Pyf(X) = (f *ky)(X)
is the only bonded solution of the Dirichlet
problem in the upper half plane for the Laplace operator { A~(x,y) = 0
for
x s R , yE R+ ,
for
x~R
(5.6) ~(x,0) = f(x)
.
The formula (5.7), (5.8) and (5.9) below, which we shall need later on, may be proved in different ways. The most boring method uses a partial fraction expansion of the integrands and freshman's calculus. It is easier, though still tedious, to use residue calculus. A third method is to use that
ky
in probability theory is the density function for a Cauchy di-
stribution and that the Cauchy distribution is reproductive in the sense given below in (5.7). A fourth way is to use the remark above concerning the Dirichlet problem (5.6), guessing a bounded harmonic function in the upper half plane satisfying the boundary condition
~(x,y)
~(x,0) = f(x) .
Due to the uniqueness of the bounded solution of (5.6) this function ~(x,y) is equal to the integral in (5.4). In the proofs below we shall select the shortest method.
28 First we prove i I+~ Y2 ~ -~ (t-x2)2+Y~
(5.7) Let
z = x+iy
Yl YI+Y2 (Xl-t)2+Y~ dt = (xl-x2)2+(yl+Y2)2
, y ~ 0 . Choose Im <~(x,y) = -
i
\
z-x2+iY2)
Y+Y2 (x-x2)2+(Y+y2)2
Then ~(x,y)
is bounded and harmonic in the upper half plane, and Y2 ~(x,0) = f(x) is bounded and continuous, so ~(x,y) is the (x-x2)2+y ~ bounded solution of (5.6). Hence, actly (5.7).
~(xl,Y I) = (f*k
)(Xl) , which is exYl
Similarly one proves
1 Ii (5.8)
~
In this case
t-x2 (t-x2)2+Y~
~(x,y)
"
Yl
Xl-X2
(xl-t)2+Y~ dt = (Xl-X2)2+(YI+Y2)2
is chosen as
qO(x,y) = Re(x2+--~y2) = x-x2 z(x-x2)2+(y+y2) 2
Now, consider the function
f(z) = -----!---I 9 1 xl-iY!-Z z-x2-iY 2
in the upper half plane except for the simple pole calculus (cf. e.g. [6]) we get
~ x l _ i Y l _ t ~ t _ x 2 _ i Y 2 d t = 2~ i R e s
=
..
, which is analytic
x2+iY 2 . Using residue
x l _ i Y i _ z ~ z_x2_iY 2 ; x2+iY 2
2~ i
Xl-X2-i(Yl+y 2) Taking the real part of this equation we get
27 t-x 2
Xl-t
dt
+~
-
Yl
I
Y2
9
-~ (Xl-t)2+y ~
-dr (t-x2)2+y ~
YI+Y2 =
-
2~
(Xl-X2)2+(yl+Y2)2 Comparing this result with (5.7) we finally get the formula
(5.9)
i I +~ t-Xl 9 t-x2 dt = YI+Y2 ~ -~ (Xl-t)2+Y~ (t-x2)2+Y~ (Xl-X2)2+(Yl+Y2)2
We shall now prove a few results concerning the operators
Py
and
Qy .
LenTna 5.1. For every integrable function we have
(5.10)
Qy1(QY2 f) = -Pyl+y 2 f ,
(5.11)
Pyl(QY2 f) = Qyl+y2 f
Proof. tion
Applying Fubini's theorem and (5.9) we have the following computa-
x-t dt Qyl(QY2 f) (x) = ifJ-~ += QY2 f(t) " (x-t)2+y~
t=
-- co
U =
-- oo
t-u } x-t f(u) " (t-U)2+Y~ du (x-t)2+Y~ dt
I~]
~ = -=
9
t-u
-(YI+Y2 )
r = !J +~
x-t
f(u)
9
du
(x-u)2+(yl+Y2)2
=
dt} du
f(x) .
-e
YI+Y2
By another application of Fubini's theorem and (5.8) we have the computation
28
P
Yl
(QY2 f) (x)
=
Yl
f(t)
~ -~
~I,~ {~I~~ -oo
I +~ = ~ u=-~
f (u)
=
f(u)
Proof.
For every
~-~
~-U= - ~
t=-~
9
i I+~ u= - ~
Lemma 5.2.
dt
QY2
'" (t-u) 2+yp2
dt
(x-t) 2+y~
. 9 (t-u)2+yp2 (x-t)2+Y~ at
x-u du (x-u)2+(yl+Y2)2
f, g e L2(R)
du} ~l
=
Qyl+y 2
du
f (x)
we have
By Fubini's theorem we have
_ f(x) Qyg(x)dx = f(x) g(t) 9 x-t dt dx I" I" I~f~~ x= - = t= - = (x-t)2+y 2 }
= -
I~t= - ~ g(t) {~I~x= - cof(x)"
t-x dx (t-x)2+y 2
}dt = - II~,~Qyf(t)dt .
To prove (5.13) we compute
45.14)
~
Qyl+y 2 f 9 Qyl+y 2 f dx
I
+~
i +~ (x-t) 2+ (yl+y 2) 2
at='<~
~ )u =-~
\~Jx=-~ (X-t)2+(yl+Y2)2
(x-U) 2+ (yl+Y2) 2
(X-U)2+(yl+Y2)2
29
I'+~
f(t) { I I +~
2YI+2Y2
f(u)
du}dt = ; i f (t)P2Yl+2Y2 f(t)dt ,
(t-u) 2 + (2Yl+2Y 2) 2 where we again have used Fubini's theorem and the identity (5.9). The other terms are handled in the same way.
D
Our next aim will be to relate the operator maximal operator
P
to the Hardy-Littlewood
Y
0 . We have +~
+~
ky(t)
Y
Jut0
{Itt~kyl(u)}
~162
I Y
<_ II0+2~k y
where k -I Y
~
{ It I
(u) ef(x)du = 8 f(x)
,
is the Hardy-Littlewood maximal operator introduced in w 2 , and
is the inverse function of the restriction of
(5.15)
IPyf(X) l ~ (Pylf[)(x) ~ e f(x) for all
xE R
and for all
It was earlier remarked that formally
yER+
k
Y
(5. ~8)
.
Qof(X) = Hf(x) . More precisely we
For every integrable function
f
lim {Hyf(X) - Qyf(X)} = 0
y§
[0,+~[ . Thus
[=(@Ifl)(x)]
have Lermma 5.3.
to
a.e.
30 Proof.
We have i
Hyf(X) -Qyf(X) =
!
+
r ! 1
f(t)
from which we get (using that
x--t
9
dt ,
(x-t)2+y2
{Ix-tl
Ix-tl
x-t ] dt (X-t) 2+y2
I f(t) [ Ix-t { iX-tl>y}
Ix-tl ~ y
in the former integral and that
in the latter one)
! [ If(t)ly2 f dt +--1 iHyf (x)-Qyf (x) I < = ~{lx-tlty}; Ix-ti{(x-t)2+y 2 } ~ ] {Ix-tI
If(t)ly dt (x-t)2+y 2
-
_< ~i (
I + I ) If(t) 17 dt = fl(x) { Ix-tI_->y} { Ix-tl
,
so we have proved that (5.17)
lHyf(X) -Qyf(X) l __< (Pylfl)(x) .
From the integral representation of operator
Hy-Qy
(Hy-Qy)a = 0 . we replace
f
by
above it follows that the a
and that
Thus, the left hand side of (5.17) remains unaffected, if f-a
. Choosing
hand side of (5.17) can replace (5.18)
llyf-Qyf
may be extended to all constants a = f(x)
f(t)
IHyf(X) -qyf(X) I < ~
by
we see that we in the right f(t)-f(x)
If(t) -f(x) i "
, giving y
(x-t)2+y 2
at .
Thus (5.16) follows, if we can prove that (5.19)
Clearly,
lim ! [ + ~ If(t) -f(x)[ y+0+ ~ J-~ (5.19) holds if
f
9
7 dt = 0 (x-t)2+y 2
for
a.e.
x~ R .
is continuous. To prove (5.19) in general we
put f(x) = limsup y+0+ From (5.17) and (5.15) follows that in w 2 we get
IHyf(X) -Qyf(x) l 9 ~ f(x) j 0f(x)
, so from theorem 2.1
31
m({xl~
(5.2o)
f(x) > y}) ~ ~ -~ If(x) Idx
Now, the left hand side of (5.20) is unaffected, if we from any continuous function
fo
f
subtract
of compact support. Thus,
m({x la f(x)> y}) ~ ~ -= If(x) -fo(x) Idx
9
Using the fact that the class of continuous functions of compact support is dense in and so
LI(R)
~ f(x)= 0
we deduce that
m({x I~ f(x)>y}) = 0
a.e., proving the lena.
We have the following result concerning
If
Theorem 5.4.
f~L p
for some
(5.21)
y>0
,
D
Y
p ~ ]1,+~]
, then also
Pyf~ Lp, and
llPyfllp=< llfl[p ,
i.e. the operator Proof.
P
for all
P
Y
is of strong type
p
Using H~ider's inequality we get for
for all
p ~ ]I,+ ~]
p ~ ]I,+~[
IPyf(x)]P< ([ +~k (x-t). If(t) Idt)P = {I+:(ky (x-t) I/p 9 If(t) l) 9 ky(X-t) i/qdt} p : \J-~o Y
(Ck
<
:
-
\ J._=,
Integration with respect to
llPyfllp =<
For
p =+ ~
5
x
Y
(x-t)dt / p/q
gives
lf(t) lp
{;: ky(X-t)dx} dt
=
llfllp p
we ge t
IPyf(X) l =<
" If(t) Idt =< llfll~
ky(X-t)dt = llfl[~ 9
32
Remark 5.5.
We may get a shorter proof from (5.15), if we apply theorem
2.1 and corollary 2.2. In fact, llPyfllp ~ II0 flip ~ Cpllfllp
,
p E ]I,+~]
However~ we do not get that the constant may be chosen equal I .
As a corollary we have
Corollar 7 5.6. (5.
If
fE
for some
~)
p e ]I, + ~[ , then
lim llPyf - fIlp= 0 y~O+
and (5. ~S)
Proof.
lim P f(x) = f(x) y+O+ Y
for almost every
First we notice that (5.22) and (5.23) are obvious if
tinuous function of compact support. Let Then there exists a continuous function [If- fr
x ER .
f E Lp fs
and let
f
e e R+
of compact support such that
<e , so we have by Minkowski's inequality llPyf-fllp ~ llPy(f-fe) llp + llPyf e -fEIlp + life-fli p _< 2]If-fell p + llPyfe-fEll p ,
and thus limsup lleyf-fll p=< 2e
for all
c > 0 ,
y~O+
hence (5.22) holds. To prove (5.23) we let f(x) = limsup IPy f(x) -f(x) l y+0+ We note that f(x)
~ Of(x)
is a conbe given.
+ If(x)]
9
Now the proof follows the same line as the proof of (5.19).
D
33
Unfortunately, for
we cannot in the same way as in theorem 5.4 get estimates
IIQyfIIp
it possible
, because the function
s
is not integrable. Neither is
in that way to get an estimate for
6.
IIHyfIlp
Existence of She Hilbert Sransform and estimate~
for the Hilbert transfor[n and the ~aximal Hil~ert transform, The purpose of this section is to show that the limit (6.1)
Hf(x) =
i lim ~
[
f(t) x-t dt
y+O+ (Ix-tl ~y) exists almost everywhere the Hilbert transform
for
H
f
integrable and to prove estimates for both
and the maximal Hilbert transform
H* , defined
by (5.3). We start with a len=na due to Loomis [5] .
Lena
6.1.
Let
x I < x 2 <... <x
n
oi, 02, ... , e n
be positive constants and let
be given real numbers.
We define a function
n
g
by
o.
g(=) = j=l Z x-xj ~,ER+
Then for each
n
m(Tx I gCx) >x}) = m(Tx 1g(x) <- X})= yI JZ c j .
(S.2)
Proof.
We shall in detail prove the statement for
trivial modifications
the proof for
m({x Ig(x) >%})
m({x ] g(x) < - % } )
. With
follows the same
line. In each of the intervals decreases from from if
0
-~
where
to
]xj, Xj+l[ , j =1,2, ..., n-I -~
, and in
al(%), ..., an(%)
(6.3)
have
to
+~
. In
]-~, Xl[
]Xn, +~[
, the function
the function
it decreases
from
g(x) +~
to
g(x)
decreases 0 . Thus,
are the roots of the equation g(x) = X ,
aj(%) ~ ]xj, Xj+l[ , j= 1,2 .... , n-I , and
an(% ) ~ ]Xn, +~[
, we must
:34 n
m({x I g(x) >x}) =
(6.4)
n i [ j~l(X-Xj)
we get that
of the normalized
polynomial
Multiplying both sides of (6.3) by al(X) ....
, an(l)
are the roots
[ {aj(X) -xj} j=l
n n n (j) n (x1 X c n ( x - x k) = 0 , j=l xj) - y j=l 3 k=l n
where
[[ (J) (X-Xk) = (X-Xl) ... (x-xj_l)(X-Xj+ I) ... ( x - x n) 9 As the k=l n-I sum of the roots is equal to the coefficient of x with the opposite
sign we finally get n n i ~ ~ aj(X)= ~ xj + cj j=l j=l ~" j=l
n 1 n I {aj(k) -xj} = [3~i'= c.j j=l
or
Substituting the latter relation into (6.4) we get (6.2).
The operator
Theorem 6.2. Proof.
H*
D
is of weak type 1.
We shall prove that 128 m({x I H*f(x) >X}) ~ ~ IlfllI ,
(6.5) for all
X ~ R+
and all
f e LI(R)
with compact support (cf. a remark in
w 2 and definition 1.4). In general, (6.6)
f=f+-f-
, where
f+, f-~ 0
Ifl =f+ +f- 9 Furthermore,
m((x IH*f(x) >X}) ~ m({x I H*(f+)(x) >~}) + m({x ] H*(f-)(x) >~}) ,
so we may in the following assume that Let
and
X ~ R+
be given. For each
e 9 R+
E + = {x I supNyf(X) >X}
c
Y~
[Note that even if
f
'
is non-negative,
f>= 0 . we define the two sets E- = {x I sup (-Hy f(x)) > X}
s
Y~
H f need not be non-negative; Y E + and prove that
cf. (5.1).] We shall only consider the set
:
m(E ) ~ ~-~
f(t)dt , as the proof for
E~
is analogous.
35
For any finite interval of
I , and
Ic
Let us consider
I
of the real axis let
the complement
of
c(1)
denote the center
I .
the family of open intervals
I , for which
I f i ~ J I c f(t) 9 c(1)_------~dt > X .
(6.7)
By the definition
E + these intervals I cover E + , and as E + is c c E (here we use that f has compact support) already a finite number
bounded
of
of these intervals cover
E + . Using the same reasoning of Besicovitch type E as in the proof of theorem 2.1 we can find disjoint intervals I I, . . . , I n such that (6.8)
m(E~)
n !im(lj)
~ 4 J
and
(6.9)
1 I&l~ f(t) 9 )--------~dt c(lj1 > X ,
j = 1,2 . . . . . n
.
J The function tends to
0
gx(t) = ~ as
I
Itl + + ~
is uniformly . Thus,
to any
continuous
for
~s ]0,i[
Ix-tl ~ E , and it
we can find a decompo-
sition of the real axis into a finite number of small intervals
J
infinite
and
each
intervals
such that for each of the small intervals
j =1,2, ... , n
~i I c. f(t).
(6.10)
J
and two for
,
d t - ~ ji
n .=@ J f(t)dt, c(lj)-c(J)
J
< 6 9I .
J
We can even choose
the intervals J in such a way that for each I. either J j c I. or J A I. = @ . In the following we shall suppose that this has = j J been done. Due to the facts that f has compact support and gx(t) tends to
0
as
Itl ->+~
vals mentioned
we shall never need to consider
the two infinite inter-
above.
We define
g(x) = ~I
J f(t)dt 9 x-c(J) I
and
gj (x) = ~I J
. = j
The function
J f(t)dt 9 x-c(J)
36 g(x) - gj(x) ffi i
f(t)dt 9 x-c(J) Jn
J
is clearly decreasing in the interval get for
x=c(lj)
I. , and from (6.9) and (6.10) we J
,
g(c(lj)) - gj(c(lj)) > (I-6)X , for
x
in the left half of
I.. J
so
g(x) - gj(x) > (I-~)X
Thus we deduce that
n i .I ~m(lj) < m({x I g(x) > 89 ~=i =
+
n I m({x ] gj(x) < j=l
-
~(I-~)}) 2
.
An application of len~na 6.1 then gives !llm(lj)< __ 2 11 j =~(--(T~-6)J ~
2 ~i fJ f(t)dt+ j=l~~ X(I-~) J I. = J
f(t)dt<= X(I-6) 4 J
Combining this inequality with (6.8) we get, letting
~I -~~f(t)dt"
7
~ ~0 +
m(E +) _< ~32 I~f(t)dt _ as mentioned above. In a similar way we get
m(m~) =< ~ Letting
e-~O+
we get (for
f>O
)
m({x 'H*f(x) >X}) ffim({x , sup ,Hyf(x),>X}) y~R+ Finally, we get for
f=f+-f-
, using (6.6) ,
-< X~ j_cof+(t)dt + ~ which is (6.5), and we have proved that
H*
f (t)dt = is of weak type
We shall now establish that (6.11)
__<~--4~_~f(t)d~ .
H f(x) = lira H f(x) y-+O+ Y
IlfllI , i .
0
37
exists almost everywhere. If the function
f
is continuously differentiable
and of compact support, it follows from Hyf(t) = I [ +~ f(x-t)+f(x+t) dt ~ y t that the limit in (6.11) exists for all
x . For a general
f~LI(R)
we
set f(x) =
limsup Yl~O+, y2_~0+
IHylf(x) -Hy2f(x)
l
and obtain from theorem 6.2 that m({x Now,
~ f(x)
I~ f(x)>y})
does not change if we from
=< c11flll . f
subtract an arbitrary continu-
ously dlfferentiable function of compact support. Hence we deduce that m({x I ~ f(x) >y}) = 0 and so
~ f(x)= 0
for all
y >0 ,
almost everywhere. This means that the limit in (6.11)
exists almost everywhere, and so we have established the existence of the •ilbert transform, which clearly is a linear operator. Using lamma 5.3 we note that we also have (6.12)
H f(x) = lim Qyf(X) y~O+
Theorem 6.3.
If
f~L2(R)
, then
HfEL2(R)
a.e.
and
(6.13)
lira I l H f - Q . f l l y~O+
(6.14)
lim I I H f - H f l l 2 = 0 , y~O+
(e.15)
IIHfll 2 = Ilfll 2 ,
(6.16)
H(H f) = - f ,
Proof.
We start by proving (6.13). If we let
-Hf}2dx < -
2
=
Ifl "
= 0
J
a.e.
Yl ~0+
f-2P
in (5.13) we get
f~fldx
,
Y2
where we have used Fatou's lenmla, (6.12) and corollary 5.6. Another appli-
38 cation of corollary 5.6 gives (6.13).
We next prove (6.14). From lenmla 5.3 we have
lim [H f(x) -Qyf(X)] = 0 aly-~0+ Y most everywhere. Furthermore. it follows from (5.17). (5.15) and corollary 2.2 that llHyf-Qy fll~ ~ l[0fll~ ~ 321]fI]~ < +~ . and thus
llHyf -Qy f]I2 § 0
as
y ~0+
by dominated convergence. Together
with (6.13) this implies (6.14).
From (5.14) follows that (6.17)
,,Qyf,,2p = I ,Qy f,2dx = If .P2yf dx .
so an application of corollary 5.6 and (6.13) gives (6.15).
At last we prove (6.16). Using Minkowski's inequality we get (6.18)
llH(Hf)+fll2~ NH(Hf)-Qy(Hf)II 2+ llQy(Hf)-Qy(qyf)ll +lIQy(Qyf)+fll 2 9
From theorem 5.4 and (6.17) we deduce
Using the inequality and (5.10) in (6.18) we get ]IH(Hf) + Eli2 ~ IIH(Hf)-Oy(Hf)]I 2 + ]]Hf-Qyfl] 2 + I]-P2y f+ fl]2 9 Using above]
]IP2yf-fll ~ 0 for
Theorem 6.4.
(e.29)
y§
[corollary 5.6] and
we infer that
l l ~ f - H f]I ~0
I]H(Hf) + f]l2 = 0
proving (6.16).
If f ,gcL2(R) , then
9 Hg~
=
-
.Hfdx
[(6.13) proved
.
39
Proof.
We start from the identity (5.12), let
It follows from theorem 6.2 that theorem 6.3 we have that
H
H
y+0+
and use (6.13).
is of weak type
is of type
1 . From (6.15) in
2 , and hence also of weak type
2 . The Marcinkiewiez theorem (theorem 1.9) then gives that p
for all
stant
pE ]1,2[ , hence for all
H
is of type
p c ]1,2] , i.e. there exists a con-
Cp E R+ , such that
(6.20)
If
D
llHfllp =< cpllfllp
feLP(R)
so letting
, l < p _ < 2 , and y-~0+
1I+~ f ~
for all
geLq(R)
and using that
that (6.19) also holds for
(6.21)
for all
ISg .Hf dx[ = < cp llfllp llgllq we conclude
fs
and all
and
g~Lq(R)
H
. Especially we get
__
l
f cLP(R) , so
l[HglIq < sup [Iif.Hgdx[ = llfIlp<__l proving that
(I
, the identity (5.12) is still valid,
= II i g . g f d x l
geLq(R)
f ELP(R)
is of type
qE [2,+~[
< CpIlgllq =
, where
c = c ,i§ q P P q
1
.
~hus
we have proved
Theorem 6.5.
The operator
H
is of type
p
for all
p E ]I,+ |
We shall at last in this section prove that the maximal Hilbert transform H*
also is of type
Lemma 6.6.
For
p
for
f (LP(~
pE]l,+~[
, p E ]l,+~[
H~fCx)
where
e
. First we prove
, we have
< e f(x) + e (~f) Cx) ,
is the Hardy-Littlewood maximal operator.
Proof. From (5.11) in lermna 5.1 we get (6.22)
Py(Qy2 f)(x) = Qy+y2f(x) .
40
When
f ~Li(R)
we deduce from (6.13) in theorem 6.3, letting
Y2 + 0
in
(6.22) that Qyf(X) = PyHf(x)
.
Using (5.17) we have
IHyf(x) I ~
IHy f(x) - Qy f(x) I + ]Qy f(x)[
= IHy f(x) - Qy f(x) l + IPyHf(x) I ey(Ifl)(x) + ey(IHfl)(x) 9 By an application of (5.15) we then get the result in lemma 6.6, first for all
f cL2(R) n LP(R) , and then for all
As both
H
and
0
are of type
p
f ELP(R) .
for all
D
p E ]i,+=[
it easily follows
from lemma 6.6 that we have
Theorem 6.7. p ~ ]1,+-[
The maximal Hilbert transform
H*
is of type
p
for all
.
~ 7.
Exponential estimates for ~he Hilbert ~ransform and th9 maximal Hilbert transform.
We shall now prove some exponential estimates similar to the result given in theorem 2.6 for the Hardy-Littlewood maximal operator. We shall only consider functions, which are essentially bounded and which are equal to zero outside a finite interval of length
Theorem 7.1.
A .
There exist positive constants
cI
and
c2
such that, if
f
is any essentially bounded function, the support of which is contained in an interval of length (Z.1)
Proof.
A , then
m({x I [H f(x) I > ~}J <= c I " A" - - - ~
We may assume that
theorem 6.2 we have e.g.
IIfll==I . Since
9 exp -c 2 9
9
H
I
is of weak type
[using
m({x I !H f(x) l > ~}) < I ~ [[fIll] it is enough to
41
prove (7.1) for large values of
X . Let
E~ = ~xlnf(x)<-X}
E += l {x ]H f(x) > k} , We shall give an estimate for
m(E~) , the estimate for
. being si-
m(E[)
milar. We have, applying theorem 6.4,
(7.2) As
have
Xm(E~) =< I _+| XE~ (x) .H f(x)dx = - I+~f(~) "HXE ~ (x)dx .
H
is of weak type
i
and of type
2
(according to theorem 6.3 we
[IHfll 2 = llf[l2 ) we conclude from the Marcinkiewicz theorem (theorem
1.9) that for
p e ]1,2[
<
g --~-[q +
which again is smaller than
iO24,qm(E~)
provided that
,
q>3
.
Using H~ider's inequality on (7.2) followed by the estimate above we get for
q >3
that X.m(E~) =< A I/q IIH•
< Al/q [ 1024 hl/Pr +.]I/p = 9 ~ q J Lm(E k)
which again gives A~ / 1024 hq i024q " ~--~--~qJ
m(E A ).+ . __
1024 ~ >3e 9 - - ~
= -
we may put
~ q = i024e
, so we get
m(E~) < ~exp(-10--6-~e)
.
Similarly we get
m(El) < ~
exp (- 10--6-~e)
and taking these inequalities together we have for large values of
(
m({x I [(Hf) (x) I >~}) < 2 c A 9 exp - I0--6~ = II proving theorem 7.1.
D
)
42 Theorem 7.2. f
There exist positive constants
cI
and
c2
such that, if
is any essentially bounded function, the support of which i8 contained
in an interval of length
A , then
(?.$)
m({x I H~f(x) >X}) ~ c I A---~--exp~- C2]T~T~~ ] .
Proof.
We remark that it suffices to prove (7.3) for large values of
We may assume that
f ~0
(write
f= f+-f- )
E + = {x I sup Hy f(x) > X}
and that
E~ = {x I sup (-Hy f(x)) > X}
y~ where
mate
X >0 for
and
m(E )
X .
IlfIlm = I . Let
y~
g >0 . We shall give an estimate for
m(E~) , the e s t i -
being similar.
~s in the proof of theorem 6.2 we can find disjoint intervals
II, 12 , ..., In ,
such that n m(m~) <= 4 j[im(lj)
(7.4) and (7.5)
~I II c.f ( t ) ~ d t
> X ,
j=l, 2 .... ,n .
J Let
fj(x) = f(x) Xl.(X) , j= i .... ,n , and consider the function
I[
f(t) dt x-t
gj(x) = H f(x) - H fj(x) = ~'I~ J The function
gj
by (7.5), hence
is decreasing in gj(x) >X
lj
(because
in the left half of
f ~0 ) and
gj(c(lj)) >
lj . Thus we infer that
n
I
X ~m(lj) < m({x I Hf(x) > m({x I H fj(x) < - { } ) j=l = ~ }) + j=l From theorem 7.1 we deduce that n
2
12 J~I. = m(lj) <= ClA. ~exp
2c'1
9
-c 2.
(c2)
__<--.-~- Ae,p --F-),
+ j=l [ m(lj) 9 ~--~--exp (- c 2
In
+ z [ m(I.) " j=l
~ '
)
43
provided
%
is sufficiently large. Rearranging we then get 8c I / c2 \ m(lj) < --~--Aexp~--i-X / 9 j=l
Combining this estimate with the inequality (7.4) we get m(E +) __<---~--Aexp We have the same estimate for
m(E~)
-
%
and thus
f(x) I > %}) < --~--A exp m({x I sup y~e IHy The result then follows by letting
e+0+
.
D
CHAPTER III.
We shall now consider functions from
LI(]-~,~])
. For technical reasons,
however, we shall extend them by periodicity to the larger interval ]- 4~, 4~] . Now a reasonable approach would be to consider smaller and smaller intervals, so we introduce the so-called dyadic intervals in w 8 depending on a rather stiff partition of the interval. Since we also need to take care of what is going on in the neighbouring intervals we introduce what we have called smoothing intervals. This partition of
]-4~,4~]
into
dyadic intervals is in some sense very convenient, but it means that we have to modify our different kinds of Hilbert transforms. This is also done in w 8 . In w 9 we generalize the concept of Fourier coefficients. The application of these also has a smoothing effect on the later results, especially in chapter IV. We prove some estimates for the generalized Fourier coefficients and for the constants derived from these. Finally, in w i0, we shall review the operator
M
defined in w 4 . In order to give an estimate of
we define a new operator S~(x; f ; ~ )
M
M ~ , which is derived from some functions
. [In the definition of
S~(x; f ; ~ )
we use both the smoothing
intervals introduced in w 8 and the Hilbert transform.] It follows that it is enough to consider the operator preliminary estimates of estimates needed for
M ~ , and we shall therefore give some
S~(x; f; ~ )
S~(x; f; ~ )
. We cannot, however, prove all the
. These are postponed to chapter IV.
46
w 8. It follows
The dyadic
intervals
from the f o r m u l a t i o n
that we are only interested ciple it would be enough the H i l b e r t means
transform,
and the m o d i f i e d H i l b e r t
of theorem 4.2 w h i c h we still have to prove
in periodic
to consider
however,
. For some other technical
interval
]-47,47]
For each
ve No
functions
of period
the spaces
we implicitly
that we also have to consider
]-27,27]
transforms.
LP(]-7,7]
27 , so in prin. W h e n we use
p e r f o r m a convolution,
the periodic
extension
of
reason we shall extend
f
f
which
to
to the larger
.
we divide
]-27,27]
into
2 9 2~ =
2 w+l
half open inter-
vals of equal length, (8.1)
wjv = ]-27 + 2 ~ ( j - I ) 2 -v , -27 + 2 7 .j 9 2 -~]
where
m(wj~)
= 27 9 2 -~ . These w-intervals
dyadic intervals from level It is natural level
to define
It is obvious
shall
~ + i
(8.2) where
need not,
o
however,
introduce
For
of
w*-type)
and
and
as the dyadic
~ c N
interval
from
IV, so w h e n
o
w*
for
= ]- 2 7 + 27(j-i)2 -~
-I
from level
~ . We
- 27 + 27(j+i)2 -~]
,
. Thus
= 47 9 2 -~
W*l = ] - 4 7 , 4 7 ] ~=
interval
smoothing intervals from level
by
j = I , 2 .... , 2 v + l - I
we define
(8.3)
In the following interval
that
be a dyadic
m(w~) v = -i
]-27,27]
the so-called
(8.3)
ment with
also called
w.J~ is the u n i o n of two n e i g h b o u r i n g dyadic inter~ + i . The union of two n e i g h b o u r i n g dyadic intervals from
w~ = w. u J~ J~ Wj+l,~ ~ EN
,
that any
therefore
(or intervals
are in the following
is only used once in chapter
else is said we shall suppose
vals from level level
W_l
j = i , 2 . . . . . 2 ~+I
o
Wl,_l = W _ l =
-i . This interval
nothing
~ EN
,
. Note that
m(w*l)
= 87
in agree-
.
we shall often use the short n o t a t i o n for any smoothing
interval.
w
for any dyadic
47
By the construction v c {-i} u N
o
(8.4)
~
m
that we to any
from level
m c ~*
Furthermore, of
it is obvious
can find an
v+l
and
that
x
from level
4m(~) = m(~*)
can be chosen as one of the two intervals
m* . These simple facts are essential
By the phrase
~*
, such that
" x belongs
belongs
in the middle
for the proof of theorem 4.2.
to the middle half of m* "
to one of the two middle
we shall understand
intervals of
u-type satisfying
(8.4).
We shall now introduce for two reasons.
the modified Hilbert
transforms.
The first one is not very difficult
These are necessary
to handle.
In fact,
we shall only consider a finite interval
instead of the whole real axis,
and it will only cause minor corrections
in the results previously
ed. The second reason, however, also more difficult choice of method,
is more essential
to obtain the crucial estimates.
intervals
introduced
these rather stiff partitions
f c LI(] - ~, ~]
Let
w*
above,
so the modified Hilbert
and let
be any smoothing
transform H , f
of f
fo
from level
H , f = H(f ~
(8.6)
and
H~.
.
extension
v ~N
o
to
]-4~, 4~]
. We define
.
the Hilbert
by
X~o,) ,
~, f
of f
H~, f = H*(f ~ immediately
transforms will be sub-
]-2~, 2~]
with respect to w*
and the maximal Hilbert transform
H ,
of
be its periodic
interval
(8.5)
It follows
It is linked to our
because we shall only use the dyadic and the smoothing
ordinated
Let
obtain-
for the proof, and it is
with respect to ~* by
X~*)
from theorem 6.5 and theorem 6.7 that the operators
are of type
p
stants involved are independent
for all of
p ~ ] I,+~[
~* .
, and even that the con-
48
A closely related operator, which we shall denote by Ll(] -7, 7])
, is defined on
by
(8.7)
H f(x) = sup [ i (pv) Ix+8 fo(t ) dtl ~R+ Jx-8 x-t
Lemma 8.1.
Proof.
The operator
H
is of type
p
for all
p ~ ]I,+ ~[
It follows from the identity 1
(pv) ~x-6
f~ x-t
dt = 7(pv)
x-t
f~ x-t
dt -
dt
{Ix-tl_>6} tha t
(8.8) As
IIH
llf~
H f(x) ~ [Nf~
f~
~ Cp Hf~
and
IIH* f~
l + H * f~ ~ c*I1f~ P
= 41/Pllfllp , we get from (8.8) for IIH flip
lIH f~
+ llH* f~
, say, and
p e ]I,+~[
,
~ 4 I/p (Cp + C;) llfllp
Finally, we shall introduce the modified maximal Hilbert transform
A H
sub-
ordinated the dyadic intervals introduced above. Consider a given
m~j v
and let
x
be an interior point of
~ jv , x E intw~j v . r (Ox)r=~,~+l,.. " of inter-
Then there exists a uniquely determined sequence vals, such that each
o r is a smoothing interval from level r , o~ c~* x belongs to the middle half of each o r . Note that x depends on x and that ~ ~v . A The modified maximal Hilbert transform H. with respect to ~J~ is dejv and such that
x
fined by (8.9)
where
Hj~ f(x) = sup [ I (pv) f~ r_>~ or x-t x f ~LI(] -7, 7])
and where
fo
dt I
'
x ~ int m
,
as usual denotes the periodic exten-
49 sion of
f
to
]-4~, 4~] . We shall sometimes for short write (8.9) in the
form f(~)
=
I I sup l;(pv) o x
A H
We shall prove that
f~x-t dtl q
also i s of t y p e
x
p
for every
normally is a skew interval with respect to
p E ] 1 , + ~ [ . As r x x , we shall first prove the
f o l l o w i n g lemma.
Lemma 8.2. f
to
Let
f eL1(]-~, ~])
1- 4~, 4~] . By
A=
]-a, b[ , - ~ < - a
by
~*
~
and let
~
be the periodic extension of
we shall denote the family of intervals I a , satisfying the condition ~ < ~ <= 3 , and
we shall denote the family of sym~netric intervals
contained in
(s.lo) sup I~ (pv) A~ ~
t
<
A
Proof.
Let
(8.11)
I ~I (pv) I
IT (pv)
A = ]-a,b[ r ~ , where f~ t
]- a, a[ c ~ *
dt i +
= Ir
t
0
, so
I < ~a <= 3 . Then
dt I =< II (PV) [a -fO(x+t) -t dt i + i i i ~ fO(x+t) dt i ' t --a
A where
I -- ]- 6, S[
]- ~, ~[ . Then
. Let
Fx(t ) = St f~
. By a partqlal integration
we get
l/sing that
11b
f~
IFx(t)]
~ 21tl ef~
[of.
1emma 2 . 4 ] we deduce t h e e s t i m a t e
dt I -< 7 g IFx(b) l a
=< W
"a--
~
~
t2
<
dt
<
When this inequality is substituted into (8.11) we easily get (8.10). The case
I =< ~a ~ 3
is treated similarly.
50
~*
is of type
Proof.
Let
x
~*
and
A
The modified maximal Hilbert transform
Theorem 8.3.
p
for all
xEint~*
H
with respect to
pc ]I,+~[
and let
in (8.9). Then
a be any of the intervals associated with x a x -x = A is an interval from the family
introduced in lemma 8.2, so using lemma 8.2 we deduce that (8.12) As both
AH f(x) ~ Hf~
+ 3 0 f~
.
H
(cf. lemma 8.1) and @ are of type p for all p ~ ]i,+~[ we A easily deduce that H also is of type p , following the same line as the proof of lemma 8.1.
D
It follows immediately from (8.8) and (8.12) that ^H , f(x) =< IHf~
for any
{xE~*
~* #~[i
I IHf~
' so
{x~w*
it
I~ f(x) >X}
l >-~} U {x~
With trivial modifications
l + H* f~
+ 3@ f~
is contained in
[H*f~
> ~}
this is also true for
U {x~*
I ef~
> ~}.
w* -W[l
9 Thus, if
f ~ L~
f o l l o w s i m m e d i a t e l y from theorem 2 . 6 ,
we also have an exponential
Theorem 8.4.
t h e o r e m 7 . 1 and t h e o r e m 7 . 2 t h a t ^ estimate for Hw, . In fact,
There exist positive constants
is any essentially bounded function and transform with respect to (s.13)
~
cI
and
c2
such that, if f
is the modified maximal Hilbert
~* , then
m ( { x e ~ * [~f(x) > x}) -! ci" m(~*) 9 T
" exp -c a
A We note that m ( { x ~ * I H f(x) > X}) < m(m*) , and as the function i ~(t) = ~ e x p ( - c 2 t ) , t E R + , is decreasing and ~(t) ~exp(-e2t) for we can omit the factor
1
llfll.
t
1
in front of the exponential
the right hand side of (8.13), provided have the following corollary.
that
c I ~ exp c 2
t~l
function on
. Thus we also
,
51
There exist positive constants
Corollary 8.5.
f
is any essentially bounded function and
bert transform with respect to
The constonts
cI
and
c2
and
c2
such that, if
is the modified maximal Hil-
~* , then
I~f(x)> X} ~
m({xe~*
cI
H
clm(~*)exp
-c 2 9 ~
do not depend on the choice of
~* .
w 9 . Gene_ra_alized Fourier coefficients. In the following
~
will always denote a dyadic interval as introduced in
w 8, and
m.jv will denote one of the dyadic intervals of length 2~. 2 -v contained in ]-2~, 2~] . Similarly, ~* will denote a smoothing interval (including
m~-i = ]- 4~, 4~])
intervals of length For each
n eN
o
m~. jv
4~. 2 -v .
[n~ 2 "~r
equal to ~jv
denotes the greatest non-negative
is any smoothing interval from level
(9.2)
~*(n; ~ m* =m~l
Remark 9.1. ~(n; ~jv)
jv
integer less than or
v c No ' let
mk,v+l
n*[~v]
for
~*(n; ~;v)
n[~jv]
for
. This has been done in order to
For the same reason we have avoided the use of
the Lebesgue measure,
.
~*(n; m*l ) = n , which is consistent with (9.2).
Here we have changed the usual notation, which is and
be any
(8.4) . We define
n . ) = ~(n; ~k,v+l ) = [n * 2 -v-l] = [~-~m(mjv)]
we define
avoid confusion.
nlml
'
n 9 2 -v
of the four dyadic intervals satisfying
and
m. we define j.v
n m(~jv)] = in " 2-v] ~(n; ~-'~)3 = [ ,2~_
where
For
will denote one of the smoothing
and each dyadic interval
(9.1)
If
and
since it may be difficult
. In our notation they are written
I'I
for
to distinguish between
~(n; m)
and
n .m(m)
n[~]
. In
chapter IV they will both occur in formu]m, which are very much alike.
52
For an arbitrary function
f ELI(] -~, ~])
the periodic extension of
f
We define, for
e E R , e = e . 3~
Fourier coefficient
e =n E Z
we let (as before)
]-4~, 4~]
and
ca(u;f) f~
We note that if
to
with period
f E LI(] - ~, ~]) , the
~'th generalized
exp (- i 2V~x)dx = ~
f~
exp
- im--~/ax 9
we have the ordinary Fourier coefficients
of
f
~ ) , but in the following it will be neces-
sary also to deal with the generalized coefficient, ~ = ~ , p EZ
denote
by
(with respect to the interval
case where
fo
2~ .
especially with the
e
We first remark that we of course have the estimate
(9.4)
lca(co;f) l <__~
If~
ldx < ~ - Ilfl] l 9
Next we notice that
[
(9.5) which ensures that for
(9.6)
nEZ
Cn(~;f) = ~
I
I~
0 ~ Cn(m;f) ~ sup ICn+
following way. First we consider
(~;f) l " ~
i
,
above that
(m;f) l < m(--~f [f~
We also have to introduce the constants
(9.8)
,
we can define
and we have from the inequalities
(9.7)
1
i+-'/'~'~ < I 0
C~(~;f)
Idx
9
. This is done in the
W~l . We put
C ~ ( ~ I ; E ) = Cn(~l,0;f ) = Cn(~2,0;f)
(The latter equation follows from the fact that
fo
.
is a periodic exten-
53
slon of
f
and that
~i,0 = ]-2~' 0]
For all other intervals of (9.9)
C~(~;f)
i.e.
C~(~;f)
~-type
(9.1) and (9.2)
4m(~) = m ( ~ ) }
u-type satisfying
$(n;~)
and
numbers, as there are
(8.4).
~ ( n ; w ~) , n c N O , introduced
C~.(n;~.)(~* *;f) = max{C~(n;~)(~;f ) I~ ~ * ,
f
all coefficients versely,
is equal to c (~;f)
c
n§
, -3p E Z
(~;f) = 0
0
in
n EZ for all
and every fixed
4m(~) = m(~*)}
almost everywhere
and all numbers
if there exists an
(9.6) that
,
, are defined in such a way that
If the function
all
Imce*,
is the largest of four well-defined
We note that the functions
n~Z
~2,0 = ]0, 2~].)
we put
= max{Cn(~;f)
exactly four intervals of
(9.10)
and
Cn(~;f)
such that ~ ~Z
in
Cn(~;f) = 0
, i.e.
Cn(~;f)
0 . Con-
we infer from
Cn+p+(~_3p)/3(~;f) = 0
p r Z . This shows that
provided that just one of the
~ , then of course
are equal to
equals
Cn(~;f) = 0
for
for all
0 .
Furthermore we conclude from the condition
Cn+~(~;f)
= ~
f~
exp
-im--~
(o for all
~ (Z
everywhere
in
(actually it suffices with
~ ~ 3Z)
that
f~
=0
almost
~ .
We have thus proved the equivalence of the following three conditions:
(8.11)
~(x)
= 0
almost everywhere in
~8.12)
there exists an
(9.15)
Cn(~; f) = 0
n ~Z
for all
such that neZ
.
~ ; Cn(~;f) = 0 ;
54
We shall now prove some results concerning the coefficients
Cn(~;f ) ,
which will be needed later on. In order to shorten the notation we shall sometimes in the proofs write or
Cn(~)
instead of
c a or
Cn(~;f)
c (~)
instead of
c (~;f) , and
where no misunderstanding
Cn
is possible.
First we shall prove a general lemma. In what follows we shall need some positive constants. These will be denoted by be confused with the shortened notation
c
Cl, c2, ... or
c
and should not
for the generalized
n
Fourier coefficients.
Lemma 9.2.
To any
~ E C2([0,27])
P2(t) , such that the function
~(t) =
there exist polynomials
P1(t)
and
@(t) , defined by p1(t)
for
x E [-27,0]
~(t)
for
x E [0,2~]
P2(t)
for
x E [2~,4~]
satisfies the following conditions @eC2([-27,47])
,
and for some constant (9.14)
max
r
I1~11|
= 0
for
k=0, I,2,
01 > I
l~(t) I +
[-~7, 4~] where
= r
max
c1{II~P{I~+ II~Pf t II~} ,
l@"(t) l <
[-2~, 47 ] l~(t) l for any function
max
~E~([0,27])
.
[0,~7] Proof.
We shall prove (9.14) by proving it in each of the intervals
[-27,0] , [0,27]
and
constructr a polynomial p~k)(-27) = 0 of class ~"(0)
C2
, ~'(0)
for
Pl(t)
k=0,1,2,
also for and
[27,47] . It is trivial in of degreer and
5
in
[0,27] . It is easy to [-2~,0] , such that
p~k)(o) =~(k)(o)
t = 0 . This polynomial
, k=0,1,2
Pl(t)
, so
is
only depends on
~(0) , which may be arbitrary numbers, so there exist
constants, such that
(9.15)
IPl(t)l ~ c~l~(O)l § c~l~'(o)l + c~l~"(o)l ,
(9.16)
Ip~'(t)l _< col~o(o)l + c~l~p'(o)l + c~l~p"(o)l
55
In order to estimate
I~'(0) I we use the following Poincar~-like method.
From the equation 12~ (t-2z) ~"(t)dt = o
[(t-2~) ~'(t)] ~2~ - I~~ ~'(t)dt = 2 ~ ' ( 0 )
- {~(2~) -~(0)}
we get after a rearrangement
Im'(O) l ~ l$z~L211mll~ + llm"ll,
lt-2~ldt
=
llmll.+ ~211m"ll.
o
When we substitute this inequality and
l~(0) l ~ It'll= and
Im"(0) J ~
into (9.15) and (9.16) we easily derive (9.14) in the interval for a suitable As
P2(t)
Cl> I . [2~,4~]
in
is defined in a similar manner, the lemma is
D
proved.
Remark 9.3.
If we choose
Pl(t)
= 1--~3 ( t § + ~ 1
in
[-2w,0]
and let
max
P2(t)
Let
(t§
a t.
( 2 ~ - 3 t ) tp'(O)
be similarly defined in
[2v,4w] , it is pos-
satisfies the conditions of the theorem, and that Ir
[-2~, 4~ ]
Lemma 9.4.
§
(t+ 2~) 3 (3t 2 -3~ t+ 2~ 2) ~0(0)
sible to prove that
~C2(~)
l+
max
Jr
1 < 21I~011 + 81J~0"II=
[-2~, 4~ ]
, ~=~.
. Then 3~
~EZ ~ exp
where
II~"lI=
[-2~,0]
for
some
constant
9
,
,
c2 ,
(1 +p.2) l~(~J < ~2 ~ {maxlkoJ + 2 - 2 " r o
. oJ
58
Proof.
We may assume that
variable
~ = [0,2w] , using if necessary the change of
t = 2 -v T + ~ . Let
If we expand
~
~
be the function introduced
in a Fourier series over
[-2~, 4hi
in lemma 9.2.
we get
~eZ and especially, ~O(t) =
Furthermore,
~ ypexp(i~t) ~eZ
,
t~ [0,2~]
9
we have exp(i ~t),
tE [-2~, 4~]
~cZ This gives that
IY~l
~
max [-2~,4~]
E Z , and so according (I
+~2)IY~l < 9{ =
so we may choose
I+I
and
p2 -~- IYgl _<
max [-2~,4~]
= ~. av
for
to lemma 9.2,
max [-2~,4~]
I~I +
max [-2~,4~]
I~"I} < 9ci{ max =
e 2 = 9c I . The factor
2 -2v
There exists a constant
c3 > 0
Iml + max
[0,2~]
Im"l} ,
[0,2w]
in the len~na is due to the
change of variable and the fact that we differentiate
Lemma 9.5.
I~"I
twice.
such that we ~or
D
n ~ NO and
have IOn.2_v(~;f) I < c 3 9 C~(n;~) (~;f) 9
Proof.
Let
8 =n-~(n;m)
9 2 v = n_2VLn.2-Vjrl
Using lemma 9.4 on the function t E ~ . jv
k0(t) = e -iBt
. Then we have and
~=m.
jv
0 &~8 < 2
~
we get for
'
~ Z Y~ exp
~
~!Z ~p exp
with (l+~2) Iyp[ ~ c2~ {maxl~] + 2 -2v maxl~"]} ~ c 2. {i+2 -2v 82 } ~ c2{i+2-2v'2 2v} = 2e 2 9
57 Then we have _~(m;f) = ~ I
e
Ir176
I I e-iBt exp[-i~(n;m)2vt] 9 f~ (t)dt = m(m)
~EZ U ~(n;~)+~ and hence ICn.2_~(m;f)[ < ~ Iy~['Ic
:~EZ
~(n;~)+~
(w;f) l < 20c2.~0
=
~ Ic
p~Z ~(n;~)+~
(m;f) l. ~ I
I+p2
= 20c 2. C~(n;w) (~;f) proving the lemma.
D
Lemma 9.6. Let n E Z and feL2(~) and M ~ 2 be constants such that
be given where
~=~j~ . Let A, B c R +
I If(t)[2dt ~ A 2m(w) and [Cm(~;f) I < B
for
In-ml < M .
Then we have
Cn(oJ;f)<-- 9.{ A-A-+ BlogM} . {g Proof.
We have
cn(~;f) = ~o
X I=
~s
n+~
(~;f) 1 9
I
I+~2
and (~;f) = ~ n+ ~where
IL
= = 2~(n+~)
(
)
f(x) .exp - i 2~(n+~)x dx = ~
if
~0f(x) 9 exp(- i~ x)dx ,
.
By a Fourier expansion we get exp(i~ x) =
[ ~ , k " exp(i 2Vkx) , kel
58 where i Io~exp ( i(cL~-2~k)x) dx = m--~I~ Iw e x p(- i 2 ~{k- (n+~) }x) dx ~ ,k = m(m)
By an integration we get that ~ ,k=0 if k - ( n + ~ ) = 0 and that
if k - ( n + ~ ) c Z\{0}, that ~ ,k =
i~, k] _-< 2 9 Ik-(n+~)[ -i
if k - ( n * ~ ) ~Z .
Hence we have ,k [ = < ~ 4
I
,
I
~ ,k e Z
l+[k-(n +~) As
f(x) =
~ Ck(m;f) exp[i 2vkx] keZ
Cn+~(o];f) = ~ i
we get
10j x ! Z Ck(W;f) ) dexp(i2~kx) X k 9 exp(ia
Ck(W ;f) exp (i 2~ kx) I ~- , s exp(-i" 2v s s
kEZ =
(in L2-sense)
~ Ck(~;f) k~Z ~,k
'
and we have the estimate
[c n+ ~ 3
(~;f)l < ~ Ick (~;f)1"[~,k[ < ! ~ [Ck(w;f) I 9 I = kEl = ~kEZ l+Ik-(n+~)I
We now consider the two cases: first case we get as M > 2
I I ~I _<_~ M
I)
and
2)
~ i I 71 > ~ M
. In the
=
,
Ik-n[<M
i
l+Ik-n-~ I =
i
Ik-nI<M
< 2B-(l+log2M) and
l+Ik-n-~ I = l
_
59 1 Ik-nI>M
ICk(~;f) I 9 , i l+]k-n-~I
1
=
Ik_nl >M
(1+ [ k - n - - ~ I ) 2
1
2 Hence we get for
n+~-
M I "~I < ~" 9
le
(~;f) I _< A
.
Then we g e t Cn(~;f ) :< I
ul "
~ ~Mic
I =< q ' ~ - 6
~ lJ Z
:
I
M ICn+
I I i-~i-~2+ A ' - " i0
Substituting the value of
q
!
and the len~a is proved.
D
Le~ma 9.7.
ond
/f
~=~.
I~- ~ - n ]
Proof.
As
[f(x) I = i
Cn(m;f) ~ I where
for all
we have ncZ
I
A
J~
BlogM}
then
9 CnO~;f) <= I .
Ic
~;f)[
J i
and hence also
. It is therefore enough to consider the case
12-V I - nl >= i .
We again consider two cases: i)
I
i
I _< yI 12-~ ~ -nl
and
2)
I
=< q + ~ " ~ <= q+ ~ A " - -
we get the estimate
=
f(x) = e i~x
I
II~-~ M l+IJ2
Cn(~;f) < 9 { A +
If
I" ~ i
~ > ~1 I2-~ ~ -nl I ~I
"
60 In the first case we have
IOn+ ]/ ( { ~
= ' m ( ~ ; ~ exp(ilx) e x p ( - i 2 V ( n + ~ ) x ) d x ]
3 = [ m--~)[ e x p ( i 2 v { 2 - V l - n - ~ } x )
which is smaller than
12-~ l - n i
2 i 21; 9 12 -~) I- n - ~ l
~ SO we get
- -
2 " ~-~ 9
9 Ic I ~ 12-vl-nl n+~
I
2
<
--
=
12 -v X - n - ~ [
1 I ~~I > ~ 12-v x -nl
In the second case, where
,
dxl
, we use that Ic
and we get
~
I;
(~;f)
n+ ~3
1 9 ~-~ 9
12-~ X - n l Cn(~O;f) < 12-v X - n [
~
ICn+
-
1
[ "
I~1__< 12-Vt-nl 9
+ 12"~X-nl < 2 = -1;
"
Let
and let ~
[0,+~]
1~l>89
iCn+21 3
[
9
1 ~
I
~
I~[> y 12-~;~-nl
I+~2
. In w 4
S~(x;f;~*)
2 <=
and the operator
1 +
~
< I
M~ .
we defined an operator
M :LP([ -~, 1;] + ~
sup ISn(x;f) l , neN o
Sn(x;f )
~
denote the class of measurable functions with
Mf(x) =
where
1
1+~2
D
The functions
p s ]I,+~[
values in
9
1 ! i I i-~ ~ Z I+-~ + 12"~ l - n l ~ I 0
proving the lemma.
w i0.
I
i'~
is the partial sum
Sn(x;f) =
n ikx [ cke , k= - n
of the Fourier series for
xe[-~,u]
f . We note that
,
neN o ,
by
'
81
(i0.I)
Sn(x;f) -Sn_l(x;f)
= c n e inx + c - ~ l e -inx
We still have to prove theorem 4.2, i.e. that p e ]I,+ ~[
is of type
"
p
for all
. Instead of giving a direct proof we shall in this section in-
troduce another operator (10.21 where
M
n 9N
J
If* :LP([ - ~,~ ]) § ~
and prove that
llM-fIlp ~ 511fllp+ lll~*fllp , ll'llp always denotes the p-norm of
LP([ - ~, ~]) . The operator
is dealt with in chapter IV. We may note that we do not prove that self is of type
If
p , but only of restricted
f ~ LI(] - w, ~])
type
is any real valued function from denote the periodic extension of
The definition of
M*
L1 f
we always let to
m[l = ]-4~' 4~].
depends heavily on the Hilbert transform. Let
be any interval of smoothing type introduced in w 8, and let
(10.3)
i.e. Since
S~(x;f;~0 ~)
Sn~(X;f;~)
S~n(X;f;m* ) f
on
I f = ~ (pv) j ~
]-~, =] e -int f~ x-t
is the Hilbert transform of
where the bar denotes complex conjugation. IS~n(X;f;~)l
The operator (10.4)
by
at ,
x c
e -inx f~
M* :LP(] -~, ~]) + ~
hE7
]-~, ~]
,
Xm~(x)
9
,
Especially,
= ISn*(X;f;m*)I
,
ncZ
~
n ~ Z . We
is supposed to be a real valued function we immediately get S ~ ( x ; f ; ~ ~) = S~(x;f;~ ~) ,
it-
p , p E ]i,+~[
fOcLl(] -4~,4~])
define the functions
I~
M~
.
is now defined by
M~f(x) = suplS~(x;f;m!l) l = sup IS~(x;f;~l) l 9 nel neN o
62
We note that
M*
estimation of ~
is only defined by means of
~ i = ]-4~, 4~] , but in the
M ~ , however, we shall also need
S~(x;f;m*)
for general
.
Dn(Y) , n E Z , denote the Dirichlet kernel defined by
Let
1
sin(n+ ~ )y for
y E [-~,~]\{0}
for
y =0 .
2 sin2Z
Dn(Y) =
I
Then it is a classical result (which is easy to verify) that for n ikx I [ [ ck e = ~ j f~ k= - n Ix-tl<~
Sn(x;f) =
In order to prove (10.2) we shall try to estimate 1-! IIfllI [< (2~) P Ilfllp]
kernel
Fn(Y) , n ~ Z
and
Then for almost every I ~"
[ J
fo(t )
I
(pv)
ISn(x;f) l by means of
sin n~ Y
for
y ( [- 7, x]\{O}
n
for
y=O.
x E ]-~,~]
I Fn ( x - t ) d t = ~(pv)
I
mEZ
,
J[
e-imt f~ x-t
,
,
e -int f~ x-t
F
ein(x-t)_ e-in(x-t)
J Ix-tl<=
dt -
f~
2 i (x-t)
e -inx (pv) 2~---l-
Ix-tI<~
Now for all
Dn(X-t)dt
IS~(x;f;m'_l) l . To do so we introduce another
tx-tl<~ e inx (pv) 2~ i
O
, by
Fn(Y) =
(10.5)
n EN
I
e
int
f~ x-t
"dr .
Ix-tl<~
dt = S~(x;f;~* I) I -
Ix-tl<~
I
Ix-tl_>~
It I<4~ so from (10.5) we get the estimate
e-imt f~ x-t
dt ,
63 (10.6)
i i ,iT
[S~(x;f;~,_l) i+
fo(t ) Fn(x_t)dtl <=
__filflil
9
ix-tl<~
For all
Lemma I0.I.
yc [-7, 7]
and all
ne Z we have
IDn(y)-;n(y)[
Proof.
We introduce the function
g(y)
by
1 cot !2 - !y
g(y) =
< I .
0
for
yE [-~, ~]\{0}
for
y=0 .
It is easy tO check that g is continuous and decreasing. Especially, i ig(y) l ~ ~ for y ~ [-~,~] . For y = O the lemma is trivial. For y ~ O we get sin(n+ g)y i
1
= ~ s l n n y .cot
Dn(Y)
y
1
+ [cos ny
2 sin2~
= sin ny 9
{1
[cot
~
-
i}
+
sinn[ Y
I + ~- cos ny
I = g(y) * sinny + Fn(Y) + [ c o s n y so I I I IDn(Y) -Fn(Y) I -< Ig(Y) l + ~ -< ~ + ~ < I .
If
Theorem 10.2.
feLP(]-~,~])
, pc]l,+~]
D
, then
llMfi[p < 511flip+ RIM*flip 9
Proof.
For every
Sn(x;f ) =
I
neN
[
o
and every
xc ]-~,7]
we have
fo(t ) Dn(X-t)dt
;
Ix-tl<7 fo(t){Dn(x_t ) _Fn(X_t)}d t + I
7 Ix-( <7
I
fx-tl<7
fo(t ) Fn(x-t)dt
64
Using lenmm i0.i and (10.6) we thus get the following inequality for almost every
x ( ]-~, ~] ,
]Sn(x;f) l _<_~ 9 i" llfNl+ IS:(x;f;~_l) l +
I]fllI
1-! <~
1(1+4
T)
9 (2~)
PNfIlp+IS:(x;f;~_l)
l ,
8o Mf(x) = sup ISn(x;f) l ~ ~i (I + ~ nEN o
i P II
flip+ M~f(x)
) 9 (2~)
,
a.e. ,
from which we get, using Minkowski's inequality, 1
I
1----
llMfIlp _< TI (I + 4 )
9 (2~)
= 2(l+~)Ilfllp+
We shall now give some estimates for
m
11fllp+ lIM*fllp
P " (2~) p
IIM*flIp__< 511flip +IIM*fHp
S:(x;f;~*)
, where
Q
9
~* = ~ju U ~j+l,v
is any smoothing interval composed of two neighbouring dyadic intervals, ~ju
and
~j+l,u' each of length
2~ 9 2 -~ . Let
f ELI(] -~,~])
. We shall
in this section consider I
[
s~(x;f;~') = u (pv) ]
for different values of in
the
Lena
following
10.3.
such that
~.
n . Recall that
l e m m a we s h a l l
only
e -int f~ x- t
dt
S* (x;f;~*) = S:(x,f,~*) -n
consider
n, n
o
E N
o
Let n, n E N and suppose that there exists an o n o =2 ~+I m o . If In-nol <- 2~+1 then
leinX~(x;f;~ *) -e
inoX
m o 9 NO ,
S*n (x;f;~*)l o
< c4"max{O,(no;~j~)(~j~;f);
C
, so
.
. ;f) ,(no;~j+1, v) ( ~j+l,~ } "
65
c 4 E R+ is a constant independent of
where
and
f
n, no, m ~
and
v
and
~*
.
Proof.
We note that
[of. (9.1)]
First we consider the function
we have
~(u)
$(no;mjv)
=
~(no;mj+l,v)
=
2m ~
defined by
e iu- 1 u
for
u ~ R\{O} ,
for
u = 0
~(u) = i
Then so
r E C|
II~ll=
, and
~(t) = ~
Now, let
modifications for
Ilmll=
=
r
+0
' c1
, I1~"11=~
r
+0
xp (i{n-no}t) -I
for
u§
= or
u§
= (n-no) ~({n-no}t)
9
2~
c I'
IIm"ll| = In-nol3" 11~"11|
,
-
m.jv . Using lem~na 9.4
we have
where
(1 p2) lyp I +
< c 2 " Imaxl~] +2-2VmaxI~p"l}
jv
-~jv
c2
.{ 2 9 2~ c ~ + 8 .
2~
.
Then we have the following computations
Aj = e inx
e-intx-tf~
dt - e in~
-0J. jv
= Im. {e in(x-t)
I
e
(d. jv
-
e in~
f~
-in t o
f~ x-t
dt
JV = fm. e in~ 9 ~(x-t)f~ 3v
,
with trivial
p~Z
(10.7)
|
c{ ~ R+ .
t = 0 . From the above follows that
I.n-nol 9 I1~11| ~ 2
Let us consider
and
for some constant
.
=
p~Z
.iv
dt
c
,.,"
= c2
2 3v
*
c I'
66
[ ypexp{ i(n~
I~
exp{-i29(2m~
" f~
pEl
= ~EZ[ Y~'exp{i(n~
P"~)x} 9 c
+~ (m.j~;f) 9 m(~0jg) , 2mO
where we have used (9.3). Hence, according to (I0.7), IAjJ ~ m(~j~) ~!Z IYpl" Ic2m P (wj9 ;f) J o+~
(lO.8)
c~.2 V < m(mj~) 9 [ - " Ic (mj~;f) l pEZ l+p 2 2m~ +P-~ = 2z 9 10c 2' .C2m (mj~ ;f) o
=
20~
9
c2'
9
C2m ( m j v ; f ) o
.
Similarly, if A.
J +1
=
inx I e-int f~ e ~j+l,~ x- t
dt -e in~ I e ~j+l,~
-in t x~ tf~
dt
then also JAj+II ~ 20~ 9 c~ 9 C2m (Wj+l,~;f) o and therefore leinx S~(x; f;~*) -e
in x o S*n (x;f;w*)l = ~I IAj +Aj+ll o
40 c 2' 9 max~C. I zmo (~.3~ ;f) ; C2mo (~j+l,~;f)} , and the lemma is proved with
c4 = 40 c2' .
D
Remark 10.4. A careful examination of the function r above gives that we may choose c4 = 13,500 .
67 Remark 10.5.
For later use we note that (10.8) is still valid, if
is replaced by one of its dyadic subintervals from level ~k,~+l
is one of these subintervals,
co. J~
~ + i . If
then -in t
(10.9)
[einx
f ~k,~+l
e-int f~ x-t
dt- e inoX
; (Ok,~+i
e
f~
~
dt I
x-t
20~T. C~" C2m (C0k,,~+l;f) . O
Corollary 10.6.
m oE NO
9
Let
Let
F c= ] - %
~* = mjv u mj+l,v ~]
Proof.
n o = m ~ 2v+1
be a measurable set. If
l[Sn(X;XF;m*)[-'~o (xTXF;~ where
and
for a suitable
[n-nol =< 2 ~ + 1 , then
<= c4"max{C2mo(~jw;XF);
C2mo(~J+I'~)IXF)} "
c 4 E R+ is the constant introduced in lemma 20.3. The corollary follows immediately from lemma 10.3, as
I[s~(x;xF;~*)l- Is~ (x;XF;~*)I[
lle inx S~(X;XF;(o*)l - [ein~ S* (x;XF;(O*) 1 no
O
in
leinXs~(x;XF;(o *)
-
e
x
o S*n (X;•
I
.
O
Lemma 10.7.
In -nol <_ 2 ~ + I
Suppose that , then
n o =2 ~+Im o
for a suitable
m o EN " If
~n x
leinX~nn(X;f;m*)-e
o S,no(X;f;m,)l =
Proof. Let ~ = (olU m2u (o3u m4 , where 4m((o6) = m((o*) = 4 .2~ ,2 -(v+l) J +I , j =i, ..., 4 . Then by j =I, ..., 4 , i.e. m. belongs to level J (9. lO),
( C*mo((o*;f) = max~C2mo((oj;f) I j =i, ... ,4}
68 so we get by using (10.9) in x
leinxS~(x;f;~*)-e
o S*n (x;f;~*) I <= 4 9 20 C ~' 9 C*m (m*;f) .
o
Remark 10.8.
L e n a 10.9.
But using the exact value of
Let
Let ~
c~
we get
be a smoothing interval, and let
,,,*
belongs to the middle half of
Proof9
o
c 5 = 80 c 2' =<27,000
m, n e
Z
. If
x
w* , then
be the biggest subinterval of ~*
of the form
~-~-= ]x-a, x+a] . Then, (cf. the proof of lennna 8.2),
I s*(x;J'~;,~*) I
e
-int e imt x-t
l(pv) I~ , -ei(m-n)t --~_tdt
I + 3NO(eX(m-n)x)II|
1 (pv) I~, ei (m-n) t
I
x
=
dt
1 i(m-n)x.
~e
--------{-dt ~I
~pv) ~*
+
ei(m-n)(t-x) I x--t dt + 3
-a
=
I ~2 lim /a(m-n) sin 2 r uU, du +3 =<--~. ~ + 3
-a
= 5
B
CHAPTER IV.
In this chapter we shall prove theorem 4.2, which remains of the proof of the Carleson-Hunt theorem. (This final proof is given in w 18.) We need, however, some technical results, before this can be done. As mentioned in the introduction to w 10 we shall prove that the operator (10.4) is of restricted type
p
for all
p ~ ]I,+=[
M*
defined by
, since we then can
use interpolation theorems from chapter I. Our goal is therefore to prove the existence of a constant (IV.l)
Bp 9 R+
for
p r ]I,+=[
, such that
m({x ~ ]-7, 7] IM*XF(X ) >y}) <= B py-pm(F) p
for every
y c R+
and every measurable set
constructing an exceptional set (IV.2)
IS*n(X;XF;m*I) I_
=< y
F =c]- 7, ~] . This is done by
EN __c]- 47, 4~] for
for each
x ~ ]-~,w] \E N
N ~ N , such that
and
Inl <= N
and (IV.3)
m(EN) < = pB P y ~ m ( F )
,
since (~.I) follows from (IV.2) and (IV.3). This exceptional set the union of some sets
S*
(introduced in w 12) and
and T*
V~
and
(introduced in U*
w Ii) ,
EN
is
and
(introduced in w 15) 9 In each case
we prove an estimate similar to (IV.3). In order to define introduce some auxiliary functions
Y*
~
we have to
Pk(X;W) in w 12 and prove some esti-
mates for these in w 13. We shall use these functions to define some index sets
G~
and
G , which will be used for defining a splitting
of the interval
m*
into disjoint subintervals, on which
n(p*;k) ,o C~,(n;~,) 9 (~ ;XF)
behaves fairly well. In w 17 we give the final estimate for which is needed in the proof of
M*
being of restricted t ~ e
S~(x;XF;m~I) , p , but be-
fore that we have also given (in w 16) a fairly technical proof of some estimates for elements not contained in As
S~(x;XF;m~I) = 0
throughout and
w167
F~]-w,7]
and thus
M~XF = 0
GEL . if
assume that we are given measurable, where
m(F) >0 .
F
is a nUll-set, we shall NeN
, pc]l,+|
, yeR+
70
w II.
Let
m
Construction
denote any dyadic interval and
we denote by
~
~ x~
m~
and
V~------"
any smoothing interval. For
the set
(ii.i) where
S*
of the sets
= {x c ]- 2~, 2~] I dist(x,~)
is the left end point of
~ 3 " m(~)} \ {x~}
~ . Thus, normally
~
is the union
of seven neighbouring dyadic intervals from the same level with middle.
If
~
~
in the
lies too close to one of the end points of the interval
]- 2~, 2~] , then
~
is still the union of some neighbouring dyadic inter-
vals, but the number of these may be less than seven. We define (11.2)
S = U{~ ] y - p l
We note that Y-Pl
S=@
o XF(X)dx ~ m(~)} }
and hence also
o XF(X)dx < 1.m(m)
for
y >l
S* = @
and
if' y > l
. In general
S* = U ~
~cS
.
. In fact,
we h a v e t h e f o l l o w i n g
esti-
~te.
Theorem 11.1.
Proof.
r n f ~"~) < 14 . y - P m f F )
A direct computation
gives
f2~ o re(S*) <7re(S) __<7y-p J-2zXF(X)dx
= 14Y-P
In the sequel we shall only consider in
S . Let
.
I~
XF(X)dx = 14 9 y-Pro(F)
the dyadic intervals
II" Ilp,m denote the p-norm over
m .
~
not contained
71
zf ~#s
Leu~na ii. 2.
then
IIx~llp,~ < {m(~)}l/PY
(n.3) and (12.4)
Proof.
Cn(~]X~ ) < y
It follows from (11.2) that if
I
Y-P
.~d as
for all
o
]x~(x)lp = XF(X)
Using (9.7), HSlder's
0 =<
Lemma 11.3.
~S
.
nEZ
then
o XF(x)dx < re(w) ,
we in~nediately get (11.3)
.
inequality and (11.3) we finally get
Cn(~,X~)=< ~ 1
I~ I•o
Suppose that
~* ~ S ~
one can find a constant
k ~N
=< ~ 1
and that
~m(~)}llq
IIx~llp,m
X F " X~, ~ 0 . For every
nEZ
such that
(11.5)
Proof.
It follows from the equivalence of (9.11) and (9.12), and the de-
finition
(9.9) of
C~(~*;XF)
that
C~(~*;XF) >0
obvious that there exists an integer fied. We have to prove that From the condition intervals
~ a~*
~* ~ S* for which
kr
for all
ncZ
. It is
k E Z , such that (11.5) is satis-
.
follows that 4m(~) =m(w*)
w #S
for all of the four sub-
. Hence
Cn(W;X~) < y
for each
of these subintervals according to lemma 11.2 and then also
0 < Cn(W * ..,XF) o = max {CN(W;• and (11.5) becomes trivial with
k EN .
wow*
, 4m(~) = m(w*)} < y , 0
72
If
Lemma 11.4.
tire integer
~* ~ S ~
then for every
p E ]I,+ ~[
there exists a posi-
L=L(p) ~N , such that
~-ky ~_ c;d~,;xOd
~, d~/2~= d~/~y .
We may choose
for (11.6)
Proof.
L = L(p) =
If
for which
m* ~ S*
then
4m(m) = m(m*)
100
for
I
[2p]
for
50
m ~S
for each of the four subintervals
Cn(m;XF) <= ~ i
< yP I ~ XF(X)'dx ~
where we have used (9.7) and the definition
(11.7)
If
(11.2) of
i
y
y > 2 -k/(p-l)
~2 - i=< 0
~2 -1
=<2
and so
-
=
22(p_i ) k
L(p) => 2. ~
so the lemma is proved in this case, if
] i00} , (b)
If
2
I
i
O
X F(x) dx =< I ,
2 - k y < . . o < i , from which follows that = Cn(~,X F) =
we get
, say
we use that
o 1 Cn(0J;X F) =< m - ~
SO
S , so
. ..,XF) o = max {C n(~;xF) ,coc~j* , 4m(~) = m(~*)} < yP 9 2 - k y __< Cn(m
From (11.7) we conclude that
(a)
m cm*
. Thus
y __<2k 9 Since ~2 -I=> 0
73
s y
so in this case we may choose
~k < 2
,
=
L(p) s
, say
L(p) =max{[2p] , I00}
0
It is for technical reasons in the sequel that a positive integer
L(p)
has been chosen as
~ I00 . A closer examination of the places where
is applied would reveal that it suffices with
L(p)
L(p) ~ 36 .
We define O
(11.8) where
%~L = {xe~!l I~ XF(X) > L y } L =L(p)
is the constant defined in (11.6), and where
modified maximal Hilbert transform of
X~
with respect to
O
~ XF ~I
"
Theorem Ii. 5.
Proof.
From theorem 8.3 follows that
pc]l,+~[
A
H
is of type
p
for all
, i.e. A IIHfllp =< A*P IlfllP ,
and hence also of weak type m(V~) =,kA o(LY) HE F
p
[cf. (1.5)] , and as
[cf. (1.2)]
we get o p
f_i
p
m(V~) <_ (A~)p. (Ly)-P. IIXFIIp,~I = 4 ~ L ( p ) J " y - p " re(F) .
is the
74
w 12. Construction of the Pk(X;~)-functions and the sets
Gk
and
Y*
and
~
and
X ~.
In this section we shall define the set where the Fourier coefficients of XF - roughly speaking - are fairly large and prove that this set satisfies an estimate of the same type as
First we consider of. (8.1). If
mrO ' r = 1,2,
f e L2(] - ~, z]
is convergent in
S*
in le~na ii.i.
where
9 1 0 = ]- 2~, 0]
and
e 2 0 = ]0,2~] ,
we have a Fourier expansion for
L2(] - ~ , ~ ] )
f , which
, hence ICn(Wr0 ;f) 12 < + ~
nEZ Thus, given a n y ? o s i t i v e
constant
=ER+
, then
ICn(mr0;f) I ~ e
for only
a finite number of the Fourier coefficients, if any.
Let
k ~N
(12.1) Then
be given. We define Gk(~r0) = {(n;Wr0)
Gk(~r0 )
n~Z
, [en(~r0;X~) I t 2-kyp/2}
has a finite number of elements, if any. Furthermore, as
o ICn(~r0;X~) I = IC_n(~r0;XF) I , it follows that only if
(n,~r0) ~Gk(~r0)
(-n,~r0) ~ Gk(Wr0 ) .
Having thus defined Pk(X;Wr0)
Gk(~r0)
[polynomials in
Rk(X;Wr0) =
we introduce the functions e ix
and
-ix] e
Rk(X;~r0)
o inx ~ Cn(~r0;XF) e (n,~r0) r Gk(~r0)
Pk(X;~r0) = R~(X;Wr0) , Gk(er0) = @
then of course
and
by , x E ]-2~, 2~]
and
If
if and
x ~ ]-2~, 2~] .
Rk(X;~r0) E 0 .
,
75 We see that we in the definition of terms in the Fourier expansion of
Rk(X;mr0) X$
just have picked up the
which are large in some sense.
Next we consider the intervals Then we expand the function and define
~sl from level ~ = i . Let msl c mr0 " o XF-Pk(.;~r0) as a Fourier series over msl
Gk(~0sl ) = { ( n , m s l ) J n e Z , I t is obvious that
Gk(~sl)
,c n(tosl ; XF-P k(.;mr0))[ > 2-kyp/2} .
only contains a f i n i t e number of elements,
if any, so we may define
~(X;%l)
:
en(msl; x ~ - P k ('",mr0))9 .. i2nx , x 9 ]-2~,2~]
X (n,~sl) ~ Gk(msl)
and Pk(X;msl) = ek(X;mr0) +Rk(X;msl) = Rk(X;mrO) +Rk(X;msl)
In general, suppose that Pk(X;ms
(12.2)
~jv cms
9 x 9 ]-2g,2=].
I , and suppose that the function
has been constructed. We define
Gk(~jv) =
(n,~j~)
n9
o , [Cn(~j ~ ; XF-Pk(.;ms
~ 2-ky p/2 } ,
which only contains a finite number of elements, if any, and .. i2~nx , xc ]-2~,2~] , (12.3) Rk(X;~j~) = (n'~J v) !Gk(~j ~) Cn(OJj~ ; XF-Pk(.;m~,~_l~)e and (12.4) If
Pk(X;mj~) = Pk(X;~0&,~_ I) +Rk(x;c0j~)
mjv cms
(12.5)
, x 9 ]-2~,2~]
c... Cmsl C~ro , it is easy to see that
Fk(X;mj~ ) = Rk(X;mr0 ) +Rk(X;msl ) +... +Rk(X;ms
+Rk(X;mj~)
.
76
In this way we continue, interval
m
functions
and we see that we for any
have a set Pk(X;m)
Remark 12.1.
Gk(m)
and
Rk(X;m)
k EN
and any dyadic
with a finite number of elements and two ix -ix , which are polynomials in e and e
Note that by the construction of
Rk(X;~jv)
in (12.3) we
have removed all the large terms of the Fourier expansion of over
X~- Pk(';ms
mjv ' so using (12.4) and (12.2) it follows that ICn(~jv; X~-Pk(';mjv))l
Remark 12.2.
In general,
for all
n~Z
.
the notation above is a little awkward, so we
an(U )
shall often write
< 2-ky p/2
for short instead of
Cn(mjv; x ~ - P k ( ' ; m & , v _ l ) )
,
thus Rk(X;m) =
~
(n,~) where of course
an(U) 9
.
.
even repzace
.
an~m)e
also depends upon k
12 nx
help in the following. consider a particular
an(~)ei2~m(~)-inx
.
Dy
a e
i~x
and
F . Sometimes we shall
It should always be remembered, term
a e i%x
never collect terms from different ~
k
_.. . ~nls simplification
from
that it originates from one and only one
same
x E ]-2~, 2~]
~(~)
is of great
however,
that if we
Pk(X;m)
we shall always assume
Rk(X;m')
, where
m' ~ m , so we
Rk-functions , although terms with the
may occur in more than one of the
Rk-functions.
We now collect all the sets defined in (12.2) in the definition (12.6)
Gk =
U Gk(m) m : ]-2~,2~]
,
ken
which of course is a disjoint union. In particular, uniquely determined
in Rk(X;~)
.
term
an(U)
,
(n,m) e G k
, namely the coefficient of
defines a
e i2~ m(m)-Inx
77
X
Len~na 12.3.
[an(~)]2 .m(~) ~ 2m(F) .
( n , ~ ) ~ Gk
Proof.
Consider any
=0 put we get
v r NO
and assume that
~g,-I =~-I = ]-2w'2~]
r
and also
~jv c~g,v_l
' where we for
Pk(X;mg_l) =0 . Using (12.4)
~.
j~
jx~
Using the Fourier expansion of
•
0
-Pk(X;Wg,v_l)
and
Rk(X;mjv)
over
~jv , where the latter expansion is given by (12.3), we conclude that O
Rk(X;mjv) and XF(x) -Pk(X;~s other in L2(~jv) , so
-Rk(x;~jv)
jx~
are orthogonal to each
j~
IX~(X)
=J
-Pk(X;~,~_l)[2dx
mj~ After a rearrangement of this equation we get by using (12.7) that f
(12.8)
~
= For
v ~0
lan(~j~)12mC~j~) = ]
I~jv
-Pk(X;~z,v_l) l2dx- I
~jv[x~(x)-Pk(X;~jv)12dx
the first term on the right hand side of (12.8) is equal to
two [• -0[2dx=mCF) , so if we use that each ~Z,~-I c o n t a i n s ~rO intervals from level v , we get by a stm~natlon of (12.8) over all m
J
from the same level
u
that
(~,~)~Gk(~) " m(~) =2~. 2-~ -
~ _~ It~[XF(X) -Pk(X;~)12dx m(~) =2~* 2
,
9
78
or by iteration
I
ian(~)i2m(m) = 12~
-2~
(n,m)CGk(m)
IXF(X)[2dx
m(m)>2~.2 -~
-
I
2m(P)
.
m(e)=2~.2 As this is true for all
~ eN
we finally get for
~§
that
O
I Ian(m) I2m(m) ~ 2m(F) . (n,m) E Gk
~ m(~) ~ 22k+ly-pm(F) (n,o) ~ Gk
Corollary 12.4.
Proof.
If
(n,~) ~G k , then
Jan(m) I ~ 2-kyp/2
D
.
and so i ~ 22ky-Plan(~)I2 .
Prom lenma 12.3 follows that m(m) ~ 22k 9 y-P (n,~) E G k
~ lan(e) 12m(~) ~ 22k+l y-Pm(F) . (n,m) e Gk
We shall now introduce the exceptional sets Gk
we define for each
(12.9)
ken
a function
X*
Ak(X)
and on
lan(~) 12XJX)
Ak(X ) =
Y* . Using the sets ]- 2~, 2~]
by
,
(n,m) E Gk and we define the set (12.10) If
Xk
by
X k = {xe ]-2~, 2~] IAk(X) > 2kyp}
x e X k , one can find a finite subset lan(m) 12X~(X) > 2k y p
(n,~)
I of the index set Gk, such that
, and as the left hand side is a step func-
~ I
tion, there exists a dyadic interval for all
~' such that x~ m' and Ak(Z)> 2ky p
z ~ m' . Hence we have proved the existence of a dyadic interval
79
m' , such that
xs
=c Xk,
so
is a union of dyadic intervals. Using
Xk
the notation from (ii.i) we define
(12.11)
x~ =
u g
and
X~ =
~X k
m(~)
Theorem 12.5.
Proof.
U ~. ~L k=l
o
<_ 24 9 y-P re(F)
The theorem follows from a small computation: §162
+co
m(X*) __< ~ m(~) < 7 I m(Xk) k=l
k=l
7 ~+~ 12~ 2 - k y - p A k ( x ) d x = 7 .y-P +~ ~ 2 -k ~ lan(~)12m(~) k=l -2~ k=l (n,m) ~ G k =< 7y -p [ 2 -k. 2m(F) = 14y-Pm(F) k=l where we have used lemma 12.3.
Let
m
D
be any dyadic interval, say
Fk , k e n
,
m = ]27 9 j .2 -~ , 2~(j+l) .2 -~] . By
, we shall understand the set
~0
F k~ = ({2~ ~ j "2 -~} + ] - 2 ~ .
i.e.
Fk m
is
t h e u n i o n o f t h e two d y a d i c
the left end point of from level
v + 3k
Then e s p e c i a l l y
(12.12)
2-u-3k, 2~~ 2-9-3k])
m
intervals
from level
,~ + 3k
having
as common end point, and the two dyadic intervals
having the right
end p o i n t
m(F k) = 4 9 2 - 3 k m(~)
Y* =
U k=l
of
m
. We d e f i n e
U Fk . s (n,~) ~ G k
a s common end p o i n t .
80
Theorem 12.6.
m(Y*) <_ 8 .y-Pro(F) .
Proof.
Using corollary 12.4 we get
m(Y*) ~
~ ~ m(F~) = 4 ~ 2 -3k I m(~) ~ 4 ~ 2 -3ko 2 2k+l y-Pm(F) k=l (n,~)EG k k=l (n,m)~G k k=l = 8 .y-Pm(F)
w 13.
Estimates of
Pk(X;~)
and introduction of the index set
From the given measurable
set
and the sets
Pk(X;~)
Pk(X;~)
Xk , where
D
F
we have constructed
Xp [cf. the definition
(10.4) of
the functions
Pk(X;~)
is too large. These auxiliary functions ~.O.~
do not, however, define the terms
O
G~ ~_.
Sn(X,XF,~_I)
associated with
M*] , so we shall need some method to
estimate those terms, which in some sense are not too far away from the terms of
Pk(X;~)
. This sense is defined by means of the size of the set i~x factor ~ in the exponent of each term a e
and the corresponding from
Pk(X;~)
, so in general we shall consider pairs
n EZ
and
is a dyadic interval,
by
~
(n,~) E ~
being close to some element from
the functions As
X*
and
Xk
G k , i.e. the index set for
already have been defined as exceptional ~
If
~ ~ Xk
Pk(X;~)
sets we shall on-
not included in the set
fined by (12.10). First we need an estimate of of the number of terms in
(13.1)
Pk(X;~)
then
Pk (X;~)
contains at most
2 3k
terms;
J =
~
for
=
x ~ ]- 2~, 2~] ,
where
J
(13.3)
[ adexp(i~jx) Pk (x; ~) = j=1
.
Xk
de-
itself and also
.
and (13.2)
where
Pk "
ly be concerned with dyadic intervals
Len~na 13. I.
(n,~) c ~
and we shall try to define what is meant
81
Proof.
From (12.2) -(12.4) follows that
l__
so
for
i <j <J =
-
,
Hence,
J <
J J I Ia~12" 22ky-p = 22k. y-P =[llajl2 .
= j=l Choose any
[aj[ _>2-kyp/2
J
j
x O ~ ~\X k . By (12.9) we have J I lajl 2 < Ak(X o) , j=l =
(13.4) and by (12.10) we get (13.5)
Ak(Xo) < 2k yP
SO
J<
2 2k y - P ~ ( x o ) =
,
< =
2 3k
proving (13.1). Now,
I ~ [aj[ .2 k y - p / 2
for
- (13.5) follows, using any
I ~j ~ J x ~ ~ ~\X k
by construction, so from (13.3) that
J y-p/2 J 12 < 2ky-p/2 22kyp/2 . [Pk(X;~)[ _-< I I 9 lajl < 2k. I [aj = Ak(X o) _< j=l
Let If -
a e
~ =~j9 a e ilx
-i~x
=
j=l
he any dyadic interval from level is any term from
is also a term from
v
not contained in
Pk(X;~jv) , then [cf. w 12] Pk(X;~)
Xk .
by construction
. This means that in the sequel we
J
need only consider exponents
I~N
. Also by construction, there exists o lax another dyadic interval ~s == ~ j~ , ~ =< v , such that a e is a term from Rk(X;~s . Hence there exists an integer n ~ N o , such that 1 = 2~n , i.e. [cf. (9.1)] I and (#(l;ms
If on the other hand (n;~s i~x ae where I = n 9 2~
ms
~ Gk
= (n,~s
then
~ Gk
Pk(X;~s
.
contains a term
82
We shall now define the set are close to the set
By
Definition 13.2. hEN
o
of indices from
G~
which in some sense
~
Pk(X;m)-functions.__
we shall understand the set of
(n,~) c ~
~X k , for which there exist a dyadic interval
and
and a term
Gk
KG" associated with the
a'e i~'x
from
Rk(X;W') , i.e. In-~(~';~)l
(~S.S)
,
~' =~
(~(~';~'),~') EG k , such that
< 21Ok
and re(w) > 2-10km(w')
(~s.?)
m(~)
It should be noted that we write Furthermore, if
~=~3~"
and
~/(~';c0) (= ['~(~'; m ' ) " m ~
~' =~s
at most
10k possible choices of the
m' . On the other hand, for given
i.e. for given
~'
and
2 9 210k- i
w'
]) in (13.6)
it follows from (13.7) that
<~ <~ +10k , so we have in this case only = dyadic interval
.
non-negative integers
(~(l';w'), ~')e Gk
~'
we can for each
n EN
,
satisfying (13.7) find fulfilling (13.6), so if
O
Gk(A' ,~') denotes the particular pair
set
of all elements
(n,~) E G~~ associated with one
(~',~') , then ~+10k-i
l
m(,,,) =
~+10k-I
I
m(~) ~
l
X
(2.2 l~
m(~')
m(~)ffi2.~.2 '-v < 10.k.2
.210km(~')
where we have used that all different joint and included in
(13.8)
l (n,m)(G~
< 2 0 . 211km(to ' ) , m
from the same level
~
~' . Hence,
-
l
I
(~(A';~'),m')EGk (n,~)(G~(%',~') < 20 9 2 llk
~
m(~') E 40 .213k.y-pm(F)
(r,~')EGk where we have used corollary 12.4.
are dis-
83
When comparing elements from In-~(%';~)[ < 2 1 0 k m'
P
is too big compared with
term
with elements from
could be satisfied for some
Gk
the condition
(~(%';~'),w') c G k , while
m . In this case we shall search for another
(@(h";w"),w") c G k , such that - roughly speaking - we have
(13.9)
210k < [n-~(~";~)l
< 220k ,
=
i.e. we extend the range of the exponent, when the set
m
is too small.
This is the essential fact in the following definition, where we so to speak "approximate" with an exponent in the range [0,210k[ other one in the range
[210k, 220k[
and with an-
. For technical reasons, however, we
shall use another description.
By
Definition 13.3.
and
~
we shall understand the set of
~ # X k , for which there exist a dyadic interval
e~d two terms
a' e i% 'x
and
a" e i~"x
from
In-~(x';~)I
(n,m) ~ ~
, n E No
~' , such that
Pk(X;W')
~ =c~, ,
, such that
< 21Ok
and (13.10)
2-10k < 1~, _~"1
Using the triangle inequality and that
9 m(~) < 220k
~(%";w)
is an integer it is easy
to derive (13.9) from (13.10). We shall, however, not use this fact.
If
m~X k
it follows from lemma 13.1 that
terms and hence at most
26k
Pk(X;W')
pairs of exponents
contains at most
(%',%")
23k
as defined in
definition 13.3. From (13.10) we get a substitute for condition (13.7) in definition 13.2, because it follows from (13.10) that given any pair (%',%")
of exponents the dyadic interval
30k levels, say
~ +I
, ..., ~ + 3 0 k
, where
w
can only belong to at most ~
also depends on
(%',%") .
Using the same technique as in the derivation of (13.8), i.e. first considering all
(n,w) e G ~
furthermore all
~
originating from the same pair
(%',%") , where
belong to the same level, then collecting all the
30k
84
levels and finally using corollary 12.4 once more, we get (13.11)
I
m(~) ~ 120 9 219k.y-pm(F)
(n,~) ~
The extra factor
26k
~
comes of course from the number of pairs
(A' ,A")
.
We write for short
13.12) ~
Lemma 13.4.
m(~) ~ 125 9 219k 9 E-Pm(F)
(n,~) c~ k Proof.
As
k 9 N , the lemma follows immediately from (13.8) and (13.11).
0
For each
k c N
(13.13) where
we define
C.~ =' {(n,a~*) 13m : (n,aO ~-"k ^ ~ cm* ^4m(a~) ,,, re(w*)} , ~
as usual denotes a smoothing interval.
re(w*) <= ,500 9
Lemma 13.5.
Proof.
219k y-Pro(F)
.
This is a trivial consequence of lemma 13.4 and (13.13).
Remark 13.6.
If
~
does not lle too close to either
are exactly two smoothing intervals with
~
-2~
or
D
2~
there
in the middle half, such that
m(~ ~) =4m(~) . Furthermore, if (n,~) ~ Gk
~'~
[cf. | 12]
and
for each of the four intervals
(n,~') ~ G~ , then ~c~
~X
k
and
, for which 4m(m)=m(~*).
85 We ~ote that
(n,mll) ~ G~
if and only if
where we have used the periodicity of
Suppose that
(n,~) ~ Gk
(n'ml,0) ~ ~k
and
(n,m2,0)
~Gk'
X~ 9
Then
"
k +Q~(x;m) Pk(X;~) = Q0(x;~) k Q0(x;m)
where
a ,eil'x
contains those terms
from
Pk(X;m)
for which
In-~(l';m) l < 210k , i.e. for which (13.6) is satisfied. As not be fulfilled, so either Jl'- ~"I" -m(m) - ~ > 220k
(n,m) ~ G~~ , condition (13.10) can-
I~'-~"I" m(m) --i-~- < 2-10k
for any two exponents
~'
or
and
~"
occurring in
Q~(x;~) . But I~'-~''I" re(m) 2~
I~ , m(m) _ ~,,. m(m) I
=
"
2~
-f#-
~ I~(~';~) -~(~";~)I
In-~(%';m) l + In-~(%'';m) I +I < 2 .210k+l
+ 1
< 220k ,
so the latter possibility is ruled out, and we get I~'- X"I" m(m) --i~--< 2-10k
(13.14)
We may of course write I j=l where
il.x J
'
j=l
I <= J , so the terms with index
Le~ir~ 13.7.
Let
exists a constant
~
~
j =I+I,
... , J
be any exponent occurring in
il.x J
'
belong to
is defined by (11.1).
. 2
k
Ql(X;~) 9
Q~(x;~) . Then there
p , such that
eiXX l Here
J
for
X E~
~
86 N
Proof. Let x ~ be the midpoint of m and suppose that % = ~i " For and j = I, ..., I we have the series expansion exp[i(%j -ll)(X-Xo)] = I + Using (13.14) we get for
xE
[ n--[l, in[(lj -%l)(X-Xo)]n . n=l
x~m
7 7 9 2-10k 2-10k l~j-~l I " ix-xol ~ ~ ' l ~ j - ~ l I "m(~) < ~ .2~ < 22. SO
(13.15)
Iei(lj-Al)(x-x~ n= 0 (n+l) '. < 23 9 2 -lOk
kEN
Now,
j-1
J
+ e i%Ix
~I a.e i(%j-%l)x~ {e i(%j-~l)(x-x~ -I} , j=l J
SO
Qk(x;m)-{ ~j=l aj ei(Aj-%l)X~ eillx
=< j=~lllaJ I 9 e i(%j-%l)(x-x~ - 1
I _< ~ lajl 9 23- 2 -10k .= j i _< 23.2 -10k
J
lajl ~ 23. 2-10k. 22k- yp/2 = 23- 2 -8k. yp/2
j=l I i(lj-ll)X ~ where we have used (13.15) and (13.2). Putting p = ~ a.e lemma is proved. D j=l J
Lemma 13.8. If
(n,~*) ~G~
and
~ * & X ~ u Y * , then the functions
associated with each of the four subintervals 4m(m) =m(~*) , are identical.
~
of
the
k Qo(X;~)
~* , for which
B7
Proof. iXx a e
Let
~ and ~ be any two of these subintervals. Suppose that o k is a term from Q0(X;~o) , i.e. there exists an ~', such that
~0 =c~'
and
(~(X;~'),~') ~ G k
finition 13.2 of
G~
[cf. remark 13.6]
In-~(X;~o) I <210k
is satisfied for
condition
(13.16) If
and
. But since
(13.7) cannot be fulfilled, m(~') > 210km(m)
~ %m'
then
~ cy*
In fact,
Fk
2 9 2-3k m(~') > 2 9 27km(~ o)
length
(n,~o)
moU~Cm*
, where
so (13.6) in de(n'~o) ~ ~k
so
.
consists of two intervals each of ~o c~'
and since
and
~'
and
m(~*) = 4m(~ o) <2 9 27km(wo) , the smoothing interval
must contain an end point of
~' , and we conclude that even
w*
~* c=F~, c=y* ,
which is contradicting our assumption. Hence ~' =~* and especially m' ~ ~, iXx and the term a e must also belong to Pk(X;~) . As Wo and ~ were arbitrarily chosen,
the lemma is proved.
0
It follows from the two last lemmata that the approximating in lemma 13.7 may be chosen with the same
p
and
X
term
p e
iXx
for all four subin-
tervals.
w 14. If
n ~N
Construction of the splitting
we shall write
p* = (~*(n;~*),~*)
g(p*~r)
of
m* .
for short in the following.
O
Let
r EN
and let
be fixed. Let G~
L = L ( p ) ~N
be the constant defined in (11.6),
be the index set defined by (13.13). Since
L 9r c N
we may
define (14.1)
G(r) =
{p* =(~*(n;~*),~*)
~G* rL
Using this subset
G(r)
ting
m* , i.e. we shall define a disjoint covering of
If
~(p*,r)
of
of the index set
,o
C* (~ ;XF) <2 -2-ry ~*(n;~*)
p* = (~*(n;~*),~*) ~ ( r )
(14.2)
~
we shall define our splitw* .
it follows innnediately that
C~(n;~)(w;X F) < 2 9 2-ry
for each of the four dyadic subintervals Let
G;L
}9
be any of these subintervals.
~ c~*
, for which
4m(~) =m(~*)
We consider the two dyadic subinter-
.
88
vals ~
~i'
and
~2'
of
~ , for which
2 m ( ~ ) =m(~) , i = 1 , 2
~I'
. If
and
both satisfy condition (14.2), we shall iterate this process on each
of the intervals
and
~i'
~
.
If at least one of the intervals shall say that
~
~I'
and
~2'
does not satisfy (14.2) we
is an interval of the splitting
~(p*,r)
In this way we continue until either we have got an element
. ~
satisfying
(14.2), such that (14.2) is not fulfilled for at least one of the two sub~i'
intervals
and
~2' ' or until we have reached level
m(~) = 2z 9 2 -N , where
N
is the constant given in the introduction to
chapter IV. For these intervals from level as elements of
~(p*,r)
Len~a 14. I.
~ 9 R(p*,r)
Let
N , which we also shall define
, we of course have that condition (14.2) is satis-
fied. Then it is obvious that
The intervals
N , i.e.
~(p*,r)
defines a disjoint covering of
~* .
are characterized by the following obvious le~na.
~ 9 n(p*,r) . Then the following three conditions are ful-
filled: (14.s)
m(~) 9 2~ . 2 -N . =
(14.4)
If
~I m(~*) => m(~) => 2 .2~ 9 2-N
dyadic interval
then one co~ find at least one
~, c~ 9 such that
2m(~') =m(~) , and such that
(14.2) is not fulfilled, i.e. ,
o
C@(n;~,)(~ ;XF) >= 2.
(14.5)
If
~'
x
~'
mj+l,v 9
m* . We consider the class of smooth-
~* = mjv u ~j+l,~ ' where either . Since
~(p*,r)
adic
(half-open) intervals,
with
x
is a disjoint covering of
lying in its middle half. We define
~*(x)
splitting
is called
~(p*,r) .
~jv E ~(p*,r)
or u*
with dy-
there exists at least one such interval
satisfying the conditions above, for which val
~c=~, c,.* and
satisfies (14.2).
belong to the middle half of
ing intervals
.y .
i8 any dyadic interval, such that
4m(~') < m(~ ~) , then
Let
2-2"
~*(x) m(~*)
~*
as that interval
~*
is maximal. This inter-
the central interval with respect to
x
and the
89 The central interval
Lemma 14.2.
splitting
~(pS, r) , where
middle half of
u*(x)
2m(uSCx))
(14.2)
x
< mCus)
x
x
and the
belongs to the
.
(14.8)
us(x)
(14.9)
If
Proof.
As the biggest interval from
belongs to the middle half of
msCx) ;
is a union of intervals from
u ~ ~(pS, r)
and
condition
~(pS, r) ; I dist(x;u) ~ ~ m ( u )
u ~u~\u~(x) , then
~(pS,r)
is the union of one interval from
adic interval,
and
u s , satisfies the following conditions:
(14.6)
uS(x)
with respect to
pS = ($,(n;us),us)
has measure
~(pS,r)
.
~ ~ m(u s)
and
and a neighbouring dy-
(14.6) is trivial. Furthermore,
(14.7) follows
from the definition. Suppose that
uS(x) = uju u~j+l,~
uS(x)
u"
Then
u' ~ ~(pS,r)
uS(x)
would lie in the middle of ~
O(pS,r)
. Then
suppose that
dist(x,u)
< ~I m(u)
for which
u' , for which
u' uu" = ~s , thus
, which is a contradiction
xEm'
u e
Let
. Then
~(p~,r) u'
x
must
x
uj~ =cu" . would lie
to the maxlmality of
tradicting the assumption,
Lemma 14.3.
the splitting
Let
us(x)
and
u 9u~\uS(x)
, while
be the neighbouring dyadic interval for would lie in the middle half of
and due to the maximality of
uS(x)
we conclude that
and the lemma is proved.
u ,
u' v u = ~s ,
u c u s ~us(x)
, con-
D
be the central interval with respect to
x
and
~(p~,r# .
urn(x) = uj~ UUj+l, ~ , where at least one of the intervals
u j+ 1,~
Uj+l, u
9 Hence we have proved (14.8).
Finally,
If
' say , and suppose
, because of our dyadic partition.
be the neighbouring dyadic interval for
in the middle half of uS(x)
~jv ~ ~S(Ps'r)
is not a union of intervals from
be contained in an interval Let
" where
be longs to
~(p*,r)
then
~
and
90
(14.10) max I C~(n;~j)(~ju;XF), C~(n;~j+1,u) (~'j+I,u;•176} < 2.2-ry . If m(~*(x)) > 2 . 2~ .2 -N
.
(14.11)
Proof.
then 9
0
C~,(n;~,(x))(~ (x),xF) >_ 2 .2-ry .
We may assume that
m.J~ E ~(p*,r) . Then by construction
C~(n;~j~)(mjv;X~) < 2 .2-ry .
By (14.8), the other interval Let
~' r
satisfy
mj+l,~
~' ~ j + l , v
is a union of intervals from [ =c~* , and
~(p*,r).
4m(~j+ I,~) =<m(~*)] .
Then by (14.5) C~(n;~j+l,~)(mj+l,~;X~) < 2 9 2-ry , and (14.10) follows. Finally, if
m(m*(x)) > 2 9 2~ 9 2 -N
do not belong to level
and
m*(x) =ml um2 ' then
N . We may assume that
we infer that there exists a dyadic interval 2m(~') =m(~l)
~I
and
m2
ml ~ ~(p*,r) . Using (14.4) ~' c ~ I , such that
, and such that C~(n;m,)(~';X$) ~ 2 9 2-ry .
As
m'
is one of the four dyadic subintervals of
4m(m') =m(~*(x))
,
m*(x)
for which
, we infer that
o
C~.(n;~.(x))(~ (x);x F) = max >= 2 . 2
I C~(n; -r
,,)(~";XF) J~"c~*(x)
y,
and we have proved (14.11) and thus the lemma.
D
4m(~") =m(~*(x))
1
91
w 15.
Construction of the sets
T*
and
U*
and
In this section we shall construct the exceptional set
EN
existence of a constant
and
Cp a R+ , independent of m(E N) < = Cp p 9 y-P .m(F)
The set Y*
EN
N ,y
T*
and
and prove the F , such that
.
is the union of the previously defined sets
together with two other sets
EN="
S* , V~ , X*
and
U* , which we shall define in
the sequel.
Let
rr
. Suppose that
p* =(~*(n;w*),~0*)EG(r)
, where
is a smoothing interval held fixed in the following. Let splitting of If
~*
m. r 3
i.e.
with respect to , let
t. 3
n
and
XF
defined in w 14
denote the midpoint of
~.j >= 2~ 9 2 -N . For each
x E~*
n EN
w. J
w*
be the
Q
and let
we define a subset
and
O
a(p*,r)
fl(x)
6j =m(mj) of
~(p*,r)
by
(15.1)
a(x) = ~j Ea(p*,r) I V t ~ e j -
Using this subset for
x E ~*
(15.2)
~(x)
of disjoint intervals we define a function
= A(x;~)
(x-tj)2+6~j
X ~ R+ ,
40 [ A(x) > x}) < T exp(- 4~) 9 m(~*) 9
t. is the midpoint of 3 lows from (15.1) for all m. 9 J (15.3)
eg(x)
We have the following estimate for all
m({x 9
Proof.
A(x)
=
~j
Lemma 15. i.
Ix-tl ~ ~
by
A(x)
As
Ix-tjl
~ ~j
m. and 3 that for
1 dist(tj;Cwj) = ~ 6 j
~.
J
E ~(x)
,
,
it fol-
92 from which we derive that
Ix-tl < Ix-tjJ +ltj-tl
(15.4)
< 2lx-tjl
for
t ~ ~. . -I
If
gn(x) =
i
X
8.
dt ,
~j9 (x-t)2+~J
~jcn(x) we get, using (15.4)
82 i
~j
6j
dt > i
.
i
~. E ~ (x)
4
so we have (15.5)
Let
~
{xr
I ~(x) _>X} =c {xE0~* [ g~(x) _> ~ } = FX.
be any non-negative function, for which
I~(x) 9 log + ~ ( x ) d x < + ~
dxdt =
""
=
~.~n(x)
J
~.e~(X) J
~.
exp
9
~. ~* (x-t)2+6~ J 3
=
J
< 2m(~*) + 8 P|
If we choose
=<
[ ~je~(X)
~j
(P6 ~)(t)dt j
~*
~(x)log +~(x)dx
~(x) = exp(~o) 9 Xrk(X )
9
and
. Then by (5.4), (5.15) and theorem 2.3 =
J~*
supp(~) ~m*
it follows from (15.6) that
(x)ga(x)dx <= 2m(~*) + 8 o~-~ ~ exp
9
,
from which we get by a rearrangement, using (15.5) ,
m((xe~o* ,A(x)>X})<= m(rx) <= -40 f-
9
exp
( -4 ~ )
9
m(~*)
.
D
93
Using that
fl(p*,r)
is a disjoint covering of
1
fn (t) = m(~(t))
(15.7) where
m(t) E ~(p*,r)
It is obvious that mate of
D
]
we define
o
w(t) XF(Y) e - l n Y d y ,
t E m*
,
is uniquely determined by the condition
t ~ m(t) 9
llfn[l~ < I. We shall in lemma 15.2 prove another esti-
[[fni[~ , which will be needed in the following. We note that since
the intervals
~ E ~(p*,r)
are disjoint and satisfy the inequality
m(m) > 2 7 . 2 -N , we have at most
2N
elements in
a step function subordinated the splitting
IfnCt)] ~ c s .y .2 -r
llfnlI| ~ c 6 . y . 2 -r . Here
~(p*,r)
~(p*,r)
We have the following estimate for
Lemma 15.2.
i.e.
m*
for
of
, so
fn(t)
is
m* .
fn(t) , tE~*
c 6 = 2 c 3 , where
c3
is the constant in-
troduced in lemma 9.5.
Proof.
Let t Em. where m. =m(t) ~fl(p*,r) belongs to level ~ . Using 3~ jv the definition (9.3) of the generalized Fourier coefficients, lemma 9.5
and condition (14.2), which is satisfied for intervals from
~(p*,r), we
get Ifn(t) I = ICn.2_v(mj~)l ~ c 3 -C@(n;~)(w;• ~) < 2c 3 , 2 -r .y .
Remark 15.3. lenmm 9.4
D
A careful analysis which uses remark 9.3 and the proofs of
and lemma 9.5 shows that we may choose
c 6 = 2880 .
Let
(15.8) where 8.5
C = 20-1og2"
c2 and
max
I
40,
is the positive constant occurring in the exponent in corollary c3
is the constant from lemma 9.5. Using the function
A(x)
94
given by (15.2) we define (15.9)
U*(p*) = { x E m *
where
L =L(p)
[A(x) > CLr} ,
is the constant introduced in (11.6). We note that
depends on the choice of
A(x)
p* .
Let ^ ^ I I hn(X) = Hm. fn(X) = sup I ~ (pv)
(15.10)
O
f~ x-------~dt I , O
X
X
A
where
H ,
is
the
modified
Hilbert
transform
with
respect
to
m*
intro-
in (8.9), i.e. the supremum is taken over all smoothing intervals
duced c m*
X---
for
which
x
belongs
to
the
middle
half
of
~
X
. We d e f i n e
A
(15.11)
Let
T*(p*) = { X E ~ *
[hn(X) > 2 CLr. 2-ry}
e 7 = max { 2 0 1 o1g 2 ' Cl } ' where the constant
Cl
was introduced
in
corollary 8.5.
Lemma 15.4.
The sets
m(T*~p*))
Proof.
T~(p *)
and
~ c 7 . 2 -20Lr 9 m(~*)
We first consider the set v , x2-r - c2 " ~' n
U~(p *) ,
satisfy m(U~(p*))~
c 7 9 2 -20Lr 9 m(~*) .
T*(p*). From lemma 15,2 follows that c2 " 2Cer ~ - 2c3
" 2CLr
hence according to corollary 8.5 and the definition
m(T*(p*))
the estimates
< c I -m(w*) .exp
c I .m(~*) 9 exp
(-c 2 9 ~ i
(15.8) of
9 2CLr2-ry
C ,
)
c2 ) = " 2 -20Lr - ~3CLr < e7 m(m*)
95
Next, using lemma 15.1 we get for
U*(p*)
40 (~r) m(Um(p*)) __< C---~exp - ~
The sets
T*
and
T* =
U
U p* E G(r)
m(P*uV*)
.
D
are now defined by
r=l
Theorem 1 5 . 5 . Proof.
U*
9 m(w*) __< c 7 9 2 - 2 0 L r 9 m(w*)
T*(p*)
,
U* =
~ e 8 9 w-Pm(F)
By the definition (14.1) of
,
G(r)
U
U
r:l
p* E G(r)
where
e8 =
we have
U*(p*)
1000 9 e?
.
.
G(r) c G 9 for every = rL p* = (~*(n;w*),~ *)
r ~ N . Hence it follows from lemma 13.5 that if we put for short, then
m(m*) ~
p* ~ G(r)
~
m(m*) < 500 9 2 1 9 r L y - p m ( F )
.
p* E G*
rL
From lemma 15.4 follows that
m(
u
(T*(p*) uU*(p*)})<
2c 7 9 2 - 2 0 L r .
p* ~ ~ ( r )
~
re(m*)
p* ~ ~ ( r ) 2 9 e 7 9 2 -20Lr 9 500 9 219Lr y-Pm(F)
= I000 .e 7 .2 -Lr .y-P m(F)
,
SO
m(T* u ~ )
~ i 0 0 0 9 e 7 9 y-Pm(F)
Finally we define the exceptional set (15.12)
~ 2 -rL __< i000. e 7. y-P. re(F) . r=l
EN
by
EN = S* u T * ulY* uV~ u X ~ u Y ~ .
96
Theorem 15.6.
There exists a constant
Cp 9 R+ , such that
m(E N) ~ ~pp y-P m(F) .
Proof.
The theorem follows in~nediately from theorem II.i, theorem 11.5,
theorem 12.5, theorem 12.6 and theorem 15.5.
For
x ~T*(p*) uU*(p*)
pared with
D
we shall need an estimate of
[S~(x;x$;m*(x))I
, where
[Sn(X,XF,mo ) .. o. , I
com-
e*(x) c= m*o =c m* . More precisely we
have
Theorem
15.7.
Let
x 9 ] - 7, 7]
and
x ~ T*Cp*) u U*(p*)
the central interval with respect to let
m* o i)
iii)
, let
and the splitting
m*(x)
be
~(p*,r) , and
be any smoothing interval satisfying the following conditions: x
ii)
x
belongs to the middle half of
~*o J9
~*(x) c=~ =c ~* ; ~\~*(x)
is a union of intervals from
Then there exists a constcmt (15.13)
Proof.
~(p*,r) .
c 9 9 R+ , such that
[ [S~(x; XF, o. mo)[ , - I ~ n ( X ; X Fo; ~ , (x))I [ < c 9 .Lr . 2 - r Y
The estimate (15.13) is trivial if
m* = m*(x) , so we may in the o
following
assume
that
m*(x)
cta*
o
and
m*(x)
# t~* . Now o
[
(15.14)[,.o., , o o) -S*n(x;XF;m*(x))l ISn(X,XF,%) I - lS*(X;XF;~*(x)) I < ISn(X;XF;t0* -int o,~, =--
dt
~ . \ ~ , t x ~~. ~ o
[cf. (10.3)], so we shall only estimate e
I~*\~*(x) o
-into.. XF[t)
x -t
dt = hn(X) + rn(X)
x-t
97 where hn(X ) =
I
%*\~*(x)
fn(t ) I fn(t) dr (pv) ~* x-t dt - (pv) ~*(x) - -x-t
fn(t) x-t a t =
O
and
415.15)
Here we h a v e u s e d
vals of
i
rn(X) =
e
~\~*(x)
fn(t)
-into.. XF~t) - fn(t) x- t dt
defined in (15.7).
Ox-type [cf. (15.10)]
As
~*
O
and
we is~nediately get, since
~*(x)
are inter-
x ~T*(p*) ,
A
415.16)
lhn(X) i =< 2hn(X) =< 4 .CLr -2 -ry
Let us turn to ~j e R(p*,r) ~\m*(x)
rn(X)
and let
defined in (15.15). As above we let tj
denote the midpoint of
is a union of intervals from ~\~*(x)
.
o
u
~.
6j =m(~j)
for
~j . By assumption
R(p*,r) ,
O(x) =~(p*,r)
,
.
J Using (14.9) in leu~na 14.2 we get so the index set
~(x)
dlst(x;~j) ~ ~1 ~j
for each
~j ~ O(x) ,
above is identical with the index set introduced
in (15.1). Especially, it follows from lemma 15.2 that (15.17)
Ifn(t) I <= c 6 9 2-r y
for all
t ~*
9
Next we note that _1_I=i__/__+ x - t X - t. J
t-t,
(x - t)(x - t.) J
and as " 1 I XF~Y)e o . . -iny9oy .m(~j) =0 (t)dt - m~-~j) I~.{ e-intx~F (t) -fn(t)}dt = I e-lntx~ J 3 3
98 we infer that i
e
-into.. XF[t) -fn(t) dt = 0 x - t4
.
J
J Hence (15.18)
rn(X )
=
I
e ~'~*'x'~\ ~ )
m. e~(x) J
-into.. XFkt) - fn(t) x- t
~. J
x-tj
dt
(x-tl(x-tj)J I e
XF~t) -
i t-t. J I , -int XF(t)d t ~.e~(x) m. (x-t)(x-tj) e J J
I ~. ~ ~(x)
~.
J
Since
Ix-t] => ~26 j
for all
t-t.. (x-t) (xJ-tj) fn (t)dt " J
t c ~j , we get
Ix - tl ~ Ix - tjl - yI ~j
and so by (15.3) (15.19)
l(x-t)(x-tj)l => (x-tj) 2- 162jlx-tj I I 2 + ~} __> (x-tj) 2 - l(x-tj)2 = l(x-tj)2__> ~{(x-tj)
from which we derive for t-tj (x-~-tj)
(15.2o)
t E mj , I
1 __<-~ 8j 9
1 ~{(x-t$)2+6~}
26.
J
(x-tj)2+ 6~j
Using this inequality together with (15.17) we get for the last term in (15.18) as x~U*(p*) ,
(15.21)
m
~ I t-tj fn(t)dt j ~ ~(x) mj (x-t)(x-tj)
26~ < c 6. 2-ry ~ = ~j ~ ~(x) (x-tj)2+ 6~ 3 < 2c 6" 2-ryA(x) __<2c 6. CLr 9 2-ry .
9g Finally, we shall prove an estimate of the same type for the first term in (15.18), i.e. for
(15.22)
mj r
I
i t-t.~ . XF(t)d t -int mJ (x-t)(x-tj) e
Consider the function t-t~ exp (-i{n - ~ 2~ (15.23) ~j(t) = (x-t)(x-tj) where mj ~ ~(x) . For convenience, let 2~
= 2~ . As x ~ j
~(n;mj)}t ) ,
t~mj ,
6j = m(~j) = 2~ 9 2-~ , i.e.
, it is obvious that ~ C 2 ( ~ j )
, so using lemma 9.4
we i n f e r t h a t ~0j(t) =
I ~j ~Z
e x p ( - i . 2v.~-t)
tern. 3 '
'
where (i + ~2) i~j~I < c 2 " {max l~j[+ 2-2u max I~o'~1} j mj J
(15.24)
9
From (15.23) and (15.20) follows (15.25)
l~j(t) l ~
and using that
26. $ 6~ (x-tj)2+ 3
In-2 v~(n;mj) I~ 2u we get after a small computation
l~(t) l ~ 22v l~j(t) I + 2. 2v.
+2 i
+2. i Ix-tjllx-tl
i
l+2t
(x-t)2 (x-tj) SO using (15.19), (15.20), (15.25) and that
2
'
I< (2+ 4+ 4+4+_ =
~
~
~2
I
t-t.
3 (x-t)2 (x-tj)
2v
t (X-t) 3 (x-tj) I '
Ix-tl > ~i6 j = ~ 9 2-v we get 6,
~
<6"
(x_tj)2+ 62
3
(x-tj)2+ 6~3
100 Substituting this inequality and (15.25) into (15.24) we get 6.
(l+~2)IYj~l ~ clO"
J (x-tj)2+~2
Using this inequality we get the following estimate of (15.22) 9 , t-t~ e_int XF(t)dt lm.E~(X) IV (x-t)(x--tj) 3 3 t-t.
~.E~(X) In. (x-t)(x!tj) exp(-i{n-2~(n;wj)}t)" exp(- i 2~@(n;mj)t) 9 XF(t)dt J J = I~.(~(x)lw ~oj(t). exp(-i 2~(n;~j)t) 9 XF(t)dt J J = c0jE~(x).I~j, ~(Z ~ ~j~ exp(-i2~{~(n;mJ)+~}t)-XF(t)dt
--
.E~(x) ~eZ J .~n(~) p~z
0
yj~ m(wj) 9 C~(n;wj)+ ~ (wj;XF)
~JP J l+p2
~(n;~j)+~ (~j;x~)
J
< Cl0 9 m.cR(x)j ~(x-tj)2+6~
=
<: I0 9 elO 9
~(Z 1+~2
~(n,~j)+~ (~j;XF) ' 9
J
mjc~(x) (x_tj)2+S~ " C*(n;mj) (wj;X$) 6~
< 10.Cl0-2.2-ry. ~ J mjC~(x) (x-tj)2+~
=
20-Cl0-2-ryA(x) J 20.Cl0.2-r.cLr'y ,
where we have used that C~(n;~j)(~j;X~) < 2 9 2-ry by (14.2) and that x~U*(p*) , so A(x) ~ C .L .r . Thus, by means of (15.14), (15.16), (15.21) and the estimate above we get XF;~o) l- IS~(x;XF;W o . (x))] = c9 .CLr -2-ry
D
101
Remark 15.8.
By using all the previously determined
that we may choose
c9=32
w 16.
constants we can prove
.
Estimation
We have defined the exceptional
for elements
set
EN
.
in (15.12) and proved an estimate
for this set in theorem 15.5. Now, consider long to the middle half of a smoothing
p* ~ G ; L
a point
interval
x~E N
and let
x
be-
~* , where O
m(~;) = 2 9 2~ ~ -~
and
v < N-I
. Let
n
=
with
,.,* 0
='
i.e. there exists an
the extra factor
2
~*(no;~)
tions
PrL
general, if
0
E N o , such that
of two dyadic
= m O . If
defined
O
m
n
0
= m
0
9 2 v+l, where
comes from the fact that we are considering
ing interval consisting that
be any frequency associated
intervals
(mo'~;) ( G *rL
from level
for some
in w 12 . This condition
v . This means
r , we can use our func-
is of course not satisfied
so in this section we shall prove two theorems,
po~ = (mo;~o) ~ G r9L
a smooth-
then we can find an element
in
which ensure that
~ * E G*rE ' which in some Po9 ~ G*rL " It is
sense does not lie "too far away" from the given element here essential
that we in (11.6) have defined
Theorem 16.1.
Let
x ~ EN
of a smoothing interval no = m 0 9 2 v+1
p~ = ( m o , ~ )
J moeN o
and suppose that ~o* " where
m(~)
L r N , such that
x
L > I00 .
belongs to the middle half
= 2 .2~ .2 -~ " ~ =< N - ]
and assume that there exists an
rEN
. Let
, such that
~ G*rL and , o 2 - r y =
(16.1)
0
Then there exist middle half,
satisfying
a smoothing interval
m(~*) = 2 .2~ .2 -~ , ~ v m = ~*(~;~*) , i.e.
~~* ~ ~o*
containing
, and two integers
~ = m . 2 ~+1 , such that
x
in its
~
and
p* = (m,~*) E G~L
and
(is.~
l**(~;~) -~*(no;~I = I[~" 2~-'] -mol ~ so. 2r
Furthermore,
if
c~e.s)
Ir
Po ~* = (~*(~;~), ~ )
, then for any
n
satisfying
-**(no;~$)[ = I~*(n;~) -mol ~ 120 9 22r
102
we have (16.4)
II~o
Proof.
As
~ = ; xo / % ) l,
- I s ,,* ( = ; x ~,,
x ~ E N ' we have
, 4m(~ O) =m(m~)
m' c ~ O = O
r sN . , be one of the subintervals of m*o ' for which
%(no;~ ~ ) (%'• ,.o and let If
P
O
(~;• + 2.2-ry},
{ c*,~,('~;~)
2oo.
=
~*o ~S* , so by lermna 11.3 we get that (16.1)
is in fact satisfied for some Let
I<
~
:C*m ~ (~*;XF) ,
~' c~* be any other subinterval of o P denote the functions
e~
for which
O
4m(~') = m(~*o) .
and
Po = P r L ( ' ; ~ )
and
P = PrL(.;~')
introduced in w 12 , it follows from lemma 13.8 that
p=P
,
O
It follows from remark 12.1 that o 2-rL yp/2 ICm(m';X F - P ) 1 5
(16.5) Since
~*o # S*
yp/2 ~ 2L r/ 4y
and
2 -r Y ~ Cm(mo,XF ) ,.. o
for all
m .
it follows from lemma 11.4 that
.
It is obvious from the construction of
P
use lernma 9.6. From (16.5) follows that any positive integer. We choose
that
B = 2 - r L y p/2
for
p=2
i
, so we may
and that
M = 210rL . In order to find
9.6 we shall use (11.3) in len~na 11.2
I~
X~ - P e L2(m')
A
M
in lemma
. We get I
}{
I
I
7 I
so using (13.2) in len~na 13.1 [note that
may be
12 x7 P
m#~
as
I { -,-I , I ~, IXF-PI2d ~ x}~ _< y + 2 2 r L y p / 2
x ~ E N]
= A
we get
.
103
An application of lemma 9.6 now gives for all
n
[cf. (16.5)]
(16.6)
< 9 - { 2 - f L y p/2. lOrL. l o g 2 + 2 - 5 r L y + 2
_< 9 .{I0 l o g 2 . r L .
-SrL. 22rLy p/2}
2-rL+2-3rL} 9 2rL/4y + 9 9 2 -5rL y
< 2 -rL/2 y , =
where we in the latter estimate have used that ,. o = C* C~(no; ~ ~ )(~o,XF) . o => 2-r y mo (Wo;XF)
Since
C~(no;~)(w~;F o)
+ 2-rL/2
L ~ I00 and
and
P =Po
rEN .
we get
o Y ~ C~(no;W~)(w~;eo) + C@(no;~)(m';XF-P) ~)( Wo,XF) ,. o ~ 2-r Y , > C~ = (no;~
and hence by rearrangemeDt (16.7)
C~(nolW~)(~;eo)
~ (2-r-2-rL/2)y
We shall now prove the existence of a frequency
1
in
.
P
, such that O
I~(I;~) -~(no;~) I < 25rL " Suppose that this is not the case, i.e. suppose that every
1
[~(I.,wo) ' -~(no;~)I
satisfies
~ 25re . Using lemma
9.7 and the second inequality in (13.2) we get, applying once more that yp/2 < 2Lr/4 =
Y '
C~(no;m~)(w~;Fo)
~ 2-SrL 9 ~ lanl ~ 2-5re .22rLyp/2
~ 2-2rey ,
which is a contradiction to (16.7), so we have proved that for some occurring as a frequency in
P
we have O
I~(%;w~) -~(no;W~)l We choose one such in w 12 follows that
1
and denote it by 1
< 25rE
n . From the construction of
must be associated to some interval
~' me' = O
PrL '
104 such that (~(~;~'), ~')e GrL . Let ~* be a smoothing interval, such that 4m(~') =m(~*) and such that x (e ~') is contained in the middle half of
~* . Then m(~*)
0
and
4
=
"
2n
9
2 -~ ,
~*(n;~*) . . = .m
. and.
p* = (m,~*)~ G* We shall prove (16.2) for this element rL "
Po* ~ G*rL
Since
n = m. 2 ~ , p* = (m,~*) .
we may write N
P = pe where
Ql(X)
frequencies
inx
+ QoCX) +Ql(X)
contains all the terms from %'
Q~(x)
for which the corresponding
satisfy
l*(no;~~)-*(x' and where
P
,
'~o) ' I ~ 210rE
by lemma 13.7 satisfies the estimate (since we already
have removed the term
p e Inx)
IQ~(x) I ~ 23 9 2 -8rLyp/2 _< 4 . 2 -TrL
where we once more have applied that C~(n;~,)(~'lP-p e i~x) ~ 4. 2 -TrLy where
~'
yp/2 =< 2rL/4 Y . Hence for
is any of the four subintervals of
l~(n;~') -~(no;~') I ~ 29rE ~* . From (16.6) then folO
lows that .N
(16.8)
, O einX) , o(~,;p - P elnX) C~(n;~,)(~ ;XF - p ~ C~(n;~,)(~ ;XF P) +C@(n;~, ) =< 2 -rL/2y+4 9 2 -TrLy =< 2- 2 -rL/2y
for
I~(n;~') - ~ ( n o ; ~ ' ) I ~ 2 9rL , so using lemma 13.8 and (16.7) we get
(16.9)
C~(no;m~)(m~," pe Inx) :> C~(no;m~)(~;Po) -C~(no;m~)(m~;P-p
> (2 -r-2-rL/2)y-4 for
l~(n;~') -~(no;~')I
~ 29rL .
-2 -TrLy > I
e i~x)
2-r Y
105 Now,
I~I
(16.10)
= lO. % ( ~ ; ~ , ) ( ~ , ; pe i ~ ) I0 9
{%(~;~,)(~, i~.)+ C~(~;m, )(~';x~)} ;• e
=< IO'{2"2-rL/2y+C$(~;~)(~;X~)}
,
where we have used that (16.8) especially is valid for
n=n
.
Using (11.4) in lerm~a 11.2 we immediately get (16.11)
IPl < I 0 . { 2 . 2-rL/2y+y}
Using (16.9) and l e m m 9.7 [note that here
<30y
.
f(x) =O e IL'~] ~ we get
I
~
--2" 2-ry =< %(no;%)('';o ~ ei~x) -< IPl "l*(n o,'o)' -*(n;%)i -1 so using (16.11) above we get
I*(%;%)-~(n;%)I
<= 2 . I~I " : "
!y --< 60. 2 r
proving (16.2).
Suppose that
n
satisfies the condition
l**(n;~;) -**(no;~%)l ~ 120. ==r We have einX ~ . ~. o Sn(X,~o,XF) - e
(16.12)
in x
o S~ ~. o n (X;~o,X F) O
< [einx S*tv.m*.v ~ n ~s
=
§
n
o
OsAF
"~ inoX , o i~x) I 0e xnx) -e S*n (X;mo;XF-0e o
: % I9
o
~~
J
where the latter two terms are estimated in the following way by lemma 10.9
106
and (16.10) [note again that
f(x) =p e i~x]
s* (x;~*; peinX) l + Sn~o(x;~0:; p e i~zx) I 2< . : u
IPl " I0
< 200{2.2-rL/2y+C , ~ ~ ~ o } = ~*(n;m) (m;XF) 9 The first term on the right hand side of (16.12) is estimated by means of lemma 10.7, applied at most
120 9 22r
times, using (16.8) each time. Hence,
einXs*(x;m*;X~ - p einX) -ein~ n o ~
S* . ,. o i p e i~x) 1 n (X,mo,XF O
<
=
120
9 22r
9 2 9
2 -rL/2
"Y'C 5 ,
so from (16.12) e inx
S~(x;~;X;)- e
in~ S*
, o I (X;~o;XF)
n o
=< 240 9 c 5 9 22r-Lr/2 y + 400 9 2 -Lr/2 y + 200 9 C*~.(n;m)(m;XF ) ~~ ~ o and (16.4) follows, because 85,000 .
L~IO0
, and
c5
,
may be chosen equal to
0
In the next theorem we improve the results above, such that we shall also get an estimate for ~(p*;r)
C*(p*) , and we shall be able to involve the splitting
for the constructed element
p* . The assumptions are the same as
in theorem 16.1. We shall, however, repeat all these conditions once more in the formulation of the theorem.
Theorem 16.2.
Let
of a smoothing
interval
x tE N
n o = m o 9 2 ~+I , m o e N o
a n d suppose ~*o " where
, a n d assume
that
m(~)
x
belongs
to the m i d d l e h a l f
= 2 " 2~ " 2 -~ " ~ =< N - 1
that there exists ~
. Let
r 9 N , such that
Po*= (mo;~) ~ C*rL and 2 - r y < C ~ (p~ ) = C* m
.. o (~o'XF)
< 2
9 2-r
y .
0
T h e n one can f i n d a s m o o t h i n g i n t e r v a l middle
half,
containing
x
in its
m(~*) = 2 . 2 ~ 9 2 -~ , ~_
~* ~ = ~
n
and
m
107
=~*(n;~*) , i.e.
satisfying
kE{1,2 ..... r} , such that
n = m 9 2 ~+I , and another integer p* = (m,~*) C G*kL .
If p~ = (~,~) e ~ L
is given by theorem 16.1 , then
(18.1s)
, 0 ~:~:) = c~Cgo;XF; <
2-(k-l)
y.
Furthermore, (18.14)
6~(p *) = 6~ (co-*', XE)~ < 2-(k-1) Y
and the splitting
n(p*;k)
of
~*
tral interval with respect to
x
is defined. If and
~*(x)
~(p*;k) , then
denotes the cen-
~*(x) c9" =
~*(x) ~9"0
if x
defining the middle half of from ~(p*;k) . Proof.
By
smoothing
E
O
and "
is not an endpoint for one of the two dyadic intervals
we denote
interval
and
9"o " and
~\~*(x)
the set of triples s r {1,2, ... , r}
is a union of intervals
(n,w*,%)
, n E N , 9"
, satisfying
a
the following
four
conditions: i)
9" m g * =
o
m(9*)
and
x
belongs
= 4 ,2w 9 2 -T
n = m . 2 T , i.e. ii)
C*(p;)
to the middle
then there exists ~*(n;9*) = m
= C*~ (9;;X;)
< 2-(s
m
half of
9"
an integer
, and if m ~ N , such that
. y
'
where
~*
is defined
Po
by theorem
16.1.
l~*(n;9*)-~*(no;9;) I < 60
iii) iv)
(~*(n;9*), 9*) ~ G;L
Z #@
c*(~)
2-(r-l)y
=
Suppose find
m that
s
o
* (9o;XF)
<
C*(p~)
. Let
~
~*
9 2r
and
be given by theorem
, then it follows
=> 2 -(r-l) y
{1,2, ... , r - l }
120
.
First we claim that
c~
r ~ 2J< j=l
at once
that
. From lemma 11.3 follows
16.1.
(~,]*,r) E E
that one can
, such that
2-~ y =< c*(~*o) < 2-(~-I) y If
~*EG~ PO
we define
n' -
4.2~ m(9~)
9 ~*(n;w;)
If
, and it is easy to check
9
108
that
(n',to~,r ~ E .
We are therefore left with the possibility that of theorem 16.1 with
n
and
r
replaced by
Po~*~ G;L " An application ~'
and
~
gives a new
O
couple
(nl,to~) , and we claim that the triple
(nl,to~,~)
belongs to
E .
As the conditions i), ii) and iv) are trivial, we only have to check iii). If we insert
~*(n';to:)
and then use the triangle inequality, we get
]~*(~l"too )* -~*(no;to:)l ~ l~*(~l;to:) -~*(~';to;)i + 1~*(~';to:) -~*(no;to:)l
< l~*(nl;~I) -**(n' ;~I) I + 1**(n'",%)*-** (no ;to*)L where we in the latter estimate have used that
to~ mto* O
l~*(nl;to:)-~*(n';to: )' = t[ 1~ ~
since
m(~T) =
m(-~o)
2s
for some
change the amount of
~*
.n
~
i"
m(
) -
~-~
s E N , and the entire part
nm o: ] [-] can at most
by one unit. Now, by (16.2) in theorem 16.1
l~*(n';t0:) -t~*(no;~0:) I < 60. 2r where
SO
8-~ *--'~ 1
[1
=<
'
and
l~*(nl;to~) -k0*(n';m;)[ < 60. 2 s
< r , so a rough estimate gives r
I**(;1;~;) -~*(no;~;)l ~
2 j < 120" 2 r ,
60.
j=l proving iii) , and we have shown that
If we choose
(n,~*,k) e E such that
E #@ .
k
is the smallest possible integer,
we only have to prove the inequality (16.13) and the claims concerning
fl(p*;k) .
109
Suppose that (16.13) is not fulfilled,
i.e. assume that C~(p ~) ~ 2 -k-l) y.
Using lennna 11.3 once more we can find an integer
~ e {1,2, ... , k-l}
such
that 2=s If
p* e G~L
of
k , so we must have
~ C*(p*) < 2-(~-l)y
we conclude that
once more with
(n,~*,s
p* ~G~L
~o~,no,%
~ Z , contradicting
the definition
. But in that case we use theorem 16.1
replaced by
~*,n,~
. This gives us a new couple
(nl,0~~) , such that [cf. the construction above] < k-i
.
, this is again contradicting
(nl,~,s
the definition of
EZ
, and since
k . Hence we have
proved (16.13).
From the construction above follows that the splitting introduced in w 14 respect to
x
and
is defined. Let ~(p~;k) ~*(x)
Then o f c o u r s e
and if
x
belongs
~(p~;k)
of
~
be the central interval with
and let = (~*(~;~*(x))
.
,~*(x))
to the middle half
of both
~* 0
and
~*(x)
,
is not an endpoint of one of the two dyadic intervals in the
middle half of w 14
x
~(x)
~*o ' then
~*(x) ~ w ~
this means that either
Suppose that
~*(x) =~*
. By the construction of
~(x) ~
. Then
or
m(~(x))
~*(x) = ~
~*(x)
(strictly)
in
.
> 2 9 2~ 9 2 -N , which by (14.11)
O
in lemma 14.3 implies that
c*(~*(x)) = c~,(~,(x)(~*(x);• Rence (lemma 11.3) there exists an integer
~ 2-(k-1)y
.
% c {1,2, ..., k-I}
, such that
2-~y ~ c*(~*(x)) < 2-(~-l)y . If
p*(x) ~G~L
minimal, with
so
~,no,r
, then
(n,~*(x),~) ~Z
, contradicting
replaced by
~*(x),n,~
is
, constructing another element
(n,~*,~) E Z , which again is contradicting that
k
p~(x) ~ G~L . But in that case we use theorem 16.1 once more
AA
conclude
the fact that
~ ( x ) r ~* O
(strictly)
.
the mlnimality of
k . Hence we
110
Finally,
since
~(x)
to concern ourselves
is a union of intervals from that
~
~(p~,k)
we only have
is a union of intervals from
~(p*;k) . This
o follows from the construction of
~*(x), from (16.13) and from the fact
that
= @ .
g~(x) r o ' i.e.
~(x)\~
[]
w 17. Final estimate of
S~(xj_x~;~I)=.
In this section we shall prove the estimate of x ~ ]-~, ~[\E N
and
Inl =< N
. o. Sn(X,XF,~_I)
, which was claimed in the introduction
chapter. The proof, which depends heavily on the preceeding very technical,
for to this
sections,
is
and it uses an iterative procedure explained in lemma 17.3
below. Once this lemma has been proved the rest is easy.
Our preliminary
goal is to define a finite sequence
of smoothing intervals and corresponding
~ i ' ~o'
finite sequences of non-negatlve
integers n=n_l
,no ,n I . . . . . n j =
m_l>mo>ml
> ...>mj
0,
0 < n j =
,
s u c h that
0 ~j) S ~ (X;XF;~ nj
=
9 O. ~..) + 0 S~ (x,x~,~ nj+ I ~ 3+I Z
Since the n-sequence
is not necessarily
*Y
decreasing,
,
we shall for technical
reasons prove the existence of another finite sequence consisting of non-negative
integers,
kj+ I < mj ~ kj
nj+ I ~ (I + 2 - k j ) n j ,
and
Let us first examine one particular ing integer. If and
n ~N
o
k_l , ko, k I , ... , kj
such that
~
j =-I,0,i ..... J- I .
and let n. ~ N be the correspond3 3 o is composed of two dyadic intervals from level 9 ,
~ 3 satisfies the condition ~*(n;~)
then of course
j = - 1,0 ~i, ... , J-i 9
= nj .2
111
(17.1)
0 ~ n-n
3 < 2 v+l
If on the other hand
n EN
~*(n;m~) = ~*(nj,~j)
is given, we simply define 4.2~ nj = m ( ~ )
(17.2)
and
and
(n,nj,m~)
~ ~*(n;~)
n.
.
by
,
has the properties described above in (17.1).
If n. E N is given and we want (17.1) to be fulfilled for n = n. , then j o 3 of course ~*(n-;m~)jj defines a level number 9 E N o , such that m~ is J 2-v- 1 composed of two dyadic intervals from level v , and ~*(n~;~).. = nj. Especially,
nj = 0
if and only if
~*(ni;m ~) = 0 . Using that
x
should
belong to the middle half of a given
n.. J
Let
Lermua 17.1.
Proof.
Let
Since
m~ we have only got two choices of ~ J J be any one of these.
m~ J
N => ? . I f
nj ~0
, then
m(~) J
> 8 9 2~ 9 2 - N
[ log(l+2 -i) < [ 2 -i = i , we have i=l i=l
N >= 7
, then the conditions
kj+ I < kj
and
so if
.
J)nj
imply for
8~
9
~*(nj,mj)" ~ ~ ~I , and as
=<
4*(nj,mj)" * E No
to the remarks above this means that
n_l=n
and
m*-i = ]-4~,4~]
there exists an integer (17.3)
=< ~~i 9 2 N
,
m(m~) =< 8 .27 9 2 -N , then
=
If
nj+ I __< (1+2
that nj __< ~ (l+2-i)N < e N i=l
so
@
E (1+2 -i) < e . If i=l -k
n_l = n
for
9 8 9 2~ 9 2 -N
we get
9 2 N+I _< 4nj < ~
~*(nj;~) =0
. According
n. = 0 , and the lemma follows. J
are given, we get from lemma 11.3 that
k ~ N , such that ,
,
. o. < 2 - ( k - l )
2 - k y __< Cn(~_I,XF)
y .
,
D
112 Lemma 17.2.
If
k~N
i s d e f i n e d by (17.3) and
] - 2 ~ , 2~] r
, then
(~*(n_1;~!1), ~ 1 ) ~ a~c 9 Proof.
In w 9 we defined
~*(n;~*l) = n , so we might as well write
(n i,~*i) cGEL . Let us suppose that -
(~*(n
-
be any number, such that
-I
;~* ), ~* )~ G* -I
-i
(m;~rO) ~ GkL , and we get from the definition (12.1) of (17.4) Since
kL
"
Let
.
men
Im-n_l I = Im-n I ffi Im-~*(n;~*l) l <210k~
iCm(~r0;X~)l ]- 2~, 2~] r S* , we have
.Then
Gk(~r0) that
< 2-kLyp/2
~r0 r S , and it follows from (11.2) that i
i
Now, (17.5) and (17.4) are exactly the assumptions of l e n a 9.6, so if we choose
Affiyp/2 and
B = 2 -kLyp/2
and
M=210kL
, we get from that le~na
o < 9{ 2 -kL y p / 2 lO kL 9 log 2 +yp/2 9 2-5kL } < 2-kL/2 y < 2-ky , Cn(t0rO;XF) where we have used that
L =L(p) ~ i00 , cf. (11.6). But this inequality
is contradicting the definition (17.3) of
k .
D
We shall now turn to the crucial lemma, in which we shall make precise the roles of the finite sequences mentioned above.
Let
Lemma 17.3.
F == ]- ~, ~] set
N >=7 , I
and
y r R§
be given constants, and let
be a given measurable set. Then there exists another measurable
E== ]-4~, 4~] , such that
re(E) <= ~p y-Pro(F) , where the set To each
E
furthermore satisfies the following condition:
xE]-~,~]\E
and each
n~{0,1, ..., N} one can find a finite
sequence WWl
"
~*o "
9"I..... ~
of smoothing intervals, and three corresponding sequences
113
n =~_1,~o,~1, ... , n j = O ,
k_l,ko, kl, ... ,kj ,
m_l,mo, ml, ... ,mj ,
of non-negative integers, such that 4.27
-k.
9 ~*(~;~)
,
rid+1 ~ (1 +2
kj+ I <mj~kj ,
#)nj ,
and 8u~h t h a t 0
(~?.61
~
=
9 O.
l~(x;xF;~J) l < l~j+l(X'x~md+l)l +c129 L , m .o.
y
and .0.~
(1 ?. ?) Here,
L = L(p) ~ 100
is the constant introduced in (11.6), and
consto~t, which is independent of prove that
Proof.
c12
keN
N, p, y
may be chosen equal to
We may assume that
and let
=
o~d
c12
2 9 104 .)
m(F) >0 . Let
n l=n
and
~[I = ]-47'47] ,
he defined by (17.3), i.e. 2 -k y < C*
(~*,;X~) < 2-(k-l) Y 9
By lemma 17.2 we have
(~*(n_l;~l), ~*i ) E GEL , so the splitting
~((~*(n;~*l), ~[i); k)
of
We choose
E
~11
is well-defined.
as the exceptional set
EN
defined by (15.12). Since
it follows from theorem 15.7 that (17.8)
+c 9. Lk.
2-(k-l)
y
.
We define k_l=m_l=k If
~*(x)
,
~*o=~*(x)
and
4.27 no= ~ 9 *(n_l;~ ~1
is composed of two dyadic intervals from level
v, ~*o =~*(x) =~.j~ u~j+l,~ '
is a
F . (It is possible to
it follows from (17.2) and (17.I) that
0 = < n - l - n o <2 ~+I , so we may use lemma 10.3 to get
114
S$_I (X;• o , I
=< S~o(XlXF;mo)
+ c 4 .max C~(no;~j~ )
Now, by definition of and
~j+l,~
belongs
1
<
o
,
IS*n
(X;•
+c 4"
2-(k-l) y
o *o) I + 1 0 -2
c4"Lk
o . S* (x;XF;m_l) n-I
no=0
" 2-(k-l)
y
,
o
. When this inequality is substituted
(17.9)
If
m.Jv so by (14.10) in len~na 14.3 we get
<= iS* n (X;XF;~ o) o
=
Lk~100
(no;~j+l,v)
m*(x) in w 14, at least one of the intervals to the splitting,
o o) . S* (x;XF;m n- 1
as
,
< =
o . S~ (X;XF;~0 o) O
the process stops. If
no#0
into (17.8) we find
+c13 .Lk 9
2-(k-l)
y
.
it follows from lemma 17.1 that
m(m~) > 8 9 27 9 2 -N , so we infer from (14.4) in lermma 14.1 that either
.
(17.10) or
o
X F "X~. - 0 . In the latter case o
(17.7)
o
become t r i v i a l
1 1 . 3 we c o n c l u d e
that
-k 2
We define
Po9
Po* s
=
,
o ,
and (17.6) and
for
w* , so we may a s s u m e ( 1 7 . 1 0 ) . o there exists a k e N , such that o
Then b y lemma
. o -(k o-I) o Y =< C$.(no;~ ~176 )(~o;xF ) < 2 y , k ~
.
(~*(no;~*) ~*)
L , then the splitting o
y
S*n (x;XF;~o) = 0 o
and it follows at once from (17.10) that
If
2-(k-l)
C$*(no;~)(~o;XF ) =>
~(p~;ko)
of
m*o
is defined,
and using
115
the same procedure as above in the derivation of (17.10)
[applying lemma
10.3 and lemma 14.3] we get (17.11)
, . o. ,o) ]Sno(X,XF,~
~
, . o. ,I) Snl(x,XF,~
+c13 .Lm ~ 9 2 -(mo-l~ y
,
where we have put ~=~(x)
,
mo=k ~
4.2~ nl = m ( ~ )
and
" ~*(no;~)
2k If
Po~G~
L
and
$*(no;~ o) > 120 9 2
o , then choose
~
according to
O
theorem 16.1 and
n ,~* ,r
according to theorem 16.2. E s p e c i a l l y ,
r ~k ~
Using theorem 15.7 we get (17.12)
S~(x;• I n ~
o
<= S~(x,XF,, . o. ~*(x)) k
As
[~*(n;e~)-~*(no;~)
I < 120.2
+c 9 .Lr 9 2 -(r-l) y 2k
o < 120. 2
o 9 it follows from theorem
16.1 that (17.13)
S* (x;XF;~ o) n
< =
n
~
,
o
~
~,. o
.
O
On the other hand we get from theorem 16.2 that *
N
,
O
C~,(n;~)(~olX F) < 2-(r-l) y
so substituting
,
this inequality and (17.12) into (17.13) and using that
L r >= i00 , we get , S~o (x;XF;mo)o
(17.14)
* o <= IS~(x;XF;~(x))
+c14 .Lr 9 2-(r-l) Y
.
In this case we put (17.15)
~
=~*(x)
,
m ~ =m
,
We have not concluded the construction need not be equal to
n,
and
and
4"2z nl = m ( ~ )
9 ~,(~;m~)
in this case, however,
n may be larger than
n
since
nI
. We shall prove O
9
116
Let
Lema 17.4.
nl = ~4"~
9 ~.(~;~)
and
~*(no;~ ~) >1~0 .2
2k o
. Then
-k n1 ~ ~ ~ (1+~
If
Proof.
n
~176 .
, the claim is trivial, so assume that
(~ + i ) -n.-2-
n
O
O
is composed of two intervals from level ( ~ 9 2 -'')+1" < 1 2 0
-n
ko
-k o
9 2
m* O
v , we get by assumption
= 2
2k o 9 120 9 2
-k <2
o ,n
O
9 2 -'v+l'(~ O
-k from which follows by a rearrangement that
n < (I + 2
~176
Using this lemma and (14.10) in lemma 14.3 and corollary 10.6 we get from (17.14), using once more that (17.16)
IS* (x;XF;mo) o , n O
<
Lr >i00 ,
* . o . . ) +Cl4.Lr.2-(r-l)y SE(X,XF,~I
=
<= I*Snl(x'XF'ml) . o. , +c4.max{Cmo(mj`),XF),C m o" o(~J+l'`);XF)}~ +Cl4.Lr.2-(r-l)y
<= ISnl * (x,"XF,~ ~ *I ) I +Cl5"Lr'2-(r-l)Y i.e. we have proved (17.6) in this case.
At last, consider the case where
Po* ~ G~ L O
N
Again, choose
n
as in theorem 16.1 and
Then by theorem 16.1 we get for
(17.17)
l
n=0
~
S*no(X;XF;mo) <= S*(x;x~t
o
and _
k ~*(n o"'~o* ) ~ 120 9 2 o
_
n , m* , r
as in theorem 16.2.
,
l {
+200. C~.(~;m;)(eo;XF) +2 ,o
so using (16.13) in theorem 16.2 we get IS, (x;XF;~o) < O, no Now,
~*=~*(X)o
=
S,(x;xO o t
is an interval of
,)I +200{ 2-(r-l) Y+ 2-(k~ o Ox-type, and as
x~V~
Y}
[ef. (ii.8)],
117 O
i.e.
sup ~I (pv) So o
x
dt
=< L y
, we get
x
S,(x;xO;m,) 0
9
=
0
proving (17.7)
9
[as
In this case we put
i ~(pv)
2"
XF(t)
=
mo=m=l *
sup
I
I ~(pv)
0
~* O
X
may be replaced by
o
S~o ( X ; • o
(17.18)
o x_---u-~dt <
* O
and
n I =0
S ,o ( X ,.X Fo ,. ~ o,)
= <
~
3"
and
whenever
~*o=~
+c12
x ' t dt
~
J
< Ly =
,
is defined].
' and we get
.Lm o 9 2-(m~
y
.
The whole process is then iterated until either (17.18) is obtained, or until we get so small intervals in (17.14) or (17.16) that we by len~na 17.1 conclude that
nj = 0 . One of these possibilities will occur within
a finite number of steps, and lemma 17.3 is proved.
D
Finally we have the following theorem
Theorem 17.5.
set
F == ]- ~, ~[
Let
NeN
, N>7= , p e ]I,+=[ , y e R +
and the measurable
be given. There exists a measurable set
E c_ ]- 4~, 4~] ,
such that
re(E) < ^VCpy-PmCF) , and such that we have for every
n cZ
with
Inl
and for every
x e ]- i,, ~]\E
~Cx;x~162 where Proof.
%
and
L=L(p)
Case i. (n=0).
and
012
:< 5cI2" L Y 9 are the constants given in len~na 17.3.
By lemma 17.3 we have , 0 . ,
ISo(X,XF,~_I) I ~ c12 . L y
118
Case 2. ( 0 < n ! N ) .
By lemma 17.3 we have
-(m_l-1)
l o * II+el2 . L y . m l S~ (x;XF;~o)
2
<
I..
O
< iSnj , (x'xF'mJ) . o. , I +c12 . L y =
c12 . L y + C l 2 . where we have used that
Ly.
J-i [ m.. 2-(mj-l) j=-I J
~ i 9 2 -(i-l) = 5c12 i=l
Ly,
k,+ij < m. < k. , so especially all the j
=
j
m.
are
J
different.
Case 3. ( - N s n <
0).
As
IS*n<X,• . o .,~. )I = I S~n(X;X~;~,) I
this case follows
irmmediately from case 2.
w 18. Proof of theorem 4.2. In w 4 we proved the Carleson-Hunt theorem assuming that theorem 4.2 is true. All the results in the sections following w 4 have been proved in order to be able here to prove theorem 4.2, thus concluding the proof of the famous Carleson-Hunt theorem.
In w i0 we introduced the operator (18.1)
M* :LP([ -~, ~]) + ~
by
M*f(x) = suplS~(x;f;m!l)i = suplS*(x;f;m*l) . . _ I n~l n~N O
Concerning this operator we prove Theorem 18.1.
The operator
M~
is of restricted weak type
p , p ~ ]1,+ ~[ .
119
Proof. stant
We recall from w i that we have to prove the existence of a conAp , such that
(18.2)
m ( { x E ]-7, 7] I~f*XF(X) >Y}) <= A pp y - p m ( F )
for every
y E R+ , every
Let
and
y ,p
F
p ~ ]I,+~[
and every measurable
set
Fr
~] .
be given as above and put N~N
o
,
In[_
Clearly,
E~ +E ~ , and in order to prove (18.2) it therefore suffices to
prove the existence of a constant
A
P
, such that
m(E~) ~ A ~ y - P m ( F ) By the. definition of
E~
y=5
Cl2LYl
< y =
for
to
Yl
and
x E ]-~, ~]\E~
i -I L-I Yl = 5 " el2" y) , where
(i.e.
are the constants from theorem 17.5. If 17.5 corresponding
N EN .
we have
sup [ S ~ ( x ; x ~ ; ~ ) [ [n[~N " ~ -L We now put
for all
N
EN
.
c12
and
denotes the set in theorem
above, we have
]-7, ~]\E~ ~ ]-~, ~]\m N , i.e.
E ~ c E N , and we infer from theorem 17.5 that m(E~) <= m(EN) <= c pP y 1 Pm(F)
which proves the theorem with
= (5c12 9 Cp) p. y-Pm(F)
Ap = 5 c12 9 C
,
[] P
Corollary 18.2.
Proof.
The operator
M*
is of restricted type
p , p c ]1,+ ~[
This follows immediately from theorem 18.1 and lemma 1.8 (inter-
polating between
Po
and
Pl
where
I <po < p
0
L
120
Corollary 18.3.
The operator
More precisely, to each that
IIMXEII_p =< AplIXEII..p
Proof.
Let
M
is of restricted type
p 9 ]~+|
for all measurable sets
u :
in
{0,1,2, ... , N}
(18,3)
be a simple function. When the range of , we call
T
on
Lp
~
a simple function of
by
Tef(x) = S (x)(X;f)
Clearly,
T
of order
N .
Lemma 18.4.
is a linear operator on
Let
~
As
x~
Lp
]-~,~]
.
for every simple function
E r= ]- ~, ~ ] and any lIT= •
Ap
,
p 9 ]I,+|
a
N . Then we have
denote any simple function of order
for any measurable set
Proof.
ISn(x;f) I .
N .
We next define an operator
where
max O
]-~, ~] + {0,1,2, .... N}
is contained
order
E c= ]-~, ~]
be fixed and put MNf(x) =
u
.
Ap , such
This follows immediately from theorem 10.2 and corollary 18.2.
NoN
Let
p, p 9 ]I,+~[
there exists a constant
,
p <_ Ap lIXEIIp ,
is the constant in corollary 18.3. Mf(x) =
sup nE N
ISn(x;f) I we clearly have
IT f(x) l ~ Mr(x)
and
O
hence a l s o
THf p H
<=
lIMfll
P . The r e s u l t
~ow f o n o w s from c o r o l l a r y
18.3.
D Lerma 18.5.
To every
p 9 ]1,+|
for every simple function a
of order
N
we have
there exists a constant
f cLP(]-~, ~])
C' p " such that and for every simple function
lITafllp < c~llfllp .
121
Proof.
This follows immediately from lemma 18.4 and the theorem of Stein-
Weiss in w 3
(interpolating between
Po and
Pl ' where
We note that the constant Marcinkiewicz
Lemma 18.6. order
N
C' only depends on P interpolation theorem.
For every
f c LP(] - % ~])
p
I < Po < p < Pl < + ~) "
and not on
~ , cf. the
and every simple function
~
of
we have
litfllp< IlrIIp i.e.
Ta
p, p c ]1,+~[ . Here
is of type
C' P
is the constant in le~na
18.5.
Proof.
Let
f cLP(] -7, ~]) . It is well-known that there exists a sequence
of simple functions
fk
satisfying
fk(x) + f(x) , Clearly, we also have of order
N
and
Jfk(x) J ~ Jf(x) I
for all
JT fk(x) I + JT f(x) I IIfllp
IIfkIlp
x ~ ]-7, 7] .
for any simple function
by Lebesgue's
theorem on dominated con-
vergence. Using Fatou's lemma and lemma 18.5 we have liT fkJ j < C' JJT=flJp <= liminf k++~ P = p
lim
[Ifkn
k§
P
= C;lJfJJ
D
P
Finally, we are able to prove
Theorem 4.2.
Proof.
Let
The operator
M
is of type
p
p ~ ]1,+ ~[ .
f ~ L P ( ] -7, ~]) . From the definition of the operator
follows that there exists a simple function depends on
a~
of order
N
MN
(where
it eo
f ) , such that
IT~ f(x)J = o
max O
ISn(x;f)J =
Then, using lemma 18.6, we have MNf(x)
for all
is increasing ana
the theorem follows.
MNf(x )
for all
x E l-n, ~3 .
JlMN f]!p <= C~IIfJl P . As the sequence
lim MNP(X ) = M f(x)
for every
x c ]- 7, ~] ,
REFERENCES.
[I]
Carleson, L.
Convergence and growth of partial sums of Fourier
series. Acta Math. 116 (1966), 135-157. [2]
Garsia, A.
Topics in almost everywhere convergence. Markham Publish-
ing Company, Chicago, 1970. [3]
de Guzm~n, M.
Differentiation of Integrals in
Rn . Lecture Notes
in Mathematics 481, Springer, 1975. [4]
Hunt, R.A.
On the convergence of Fourier series. Orthogonal Expan-
sions and Their Continuous Analogues. Proc. Conf. Edwardsville. Ill. (1967), 235-255. Southern Illinois Univ. Press, Carbondale, Ill. (1968). [5]
Loomis.
A note on the Hilbert transform. Bull. Am. Math. Soc. 52
(1946), 1082-1086. [6]
Marsden, J.
Basic complex analysis. Freeman and Company, San Fran-
cisco, 1973. [7]
Mozzochi, Ch.J.
On the Pointwise Convergence of Fourier Series.
Lecture Notes in Mathematics 199, Springer 1971. [8]
Titchmarsh.
Introduction to the theory of Fourier integrals. Claren-
don Press, Oxford, 1962. [9]
Z~gmund, A. Press, 1959.
Trigonometric series, Vol. I-II, Cambridge University
INDEX.
adjoint operator, formally 15
kernel, Dirichlet 62
Carleson-Hunt's theorem 17, 20
Marcinkiewicz' theorem 8
central interval 88
maximal Hilbert transform 24 maximal Hilbert transform with
Dirichlet kernel 62
respect to
m* 47
Dirichlet problem25
maximal operator I0
distribution function 4
maximal operator, Hardy-Littlewood i0
dyadic interval 46
modified maximal Hilbert transform with respect to
~*
48
exponential estimate 40 estimate, exponential 40
operator, formally adjoint 15
estimate of
operator, maximal i0
S~(x;x$,m!l ) II0
estimates of
Pk(X;~) 80
estima,tes of
S~(x;f;~*) 64
estimation of elements
operator, Hardy-Littlewood maximal I0 operator of restricted type
p* ~ GEL
I01
operator of strong type formally adjoint operator 15
operator of type
Fourier coefficients, generalized 52
operator of weak type
function, simple 4
operator, sublinear 4
function, simple of order
N
p
p
5 p
p
restricted weak type
Hardy-Littlewood maximal operator I0
simple function 4
Hilbert transform 24
simple function of order
Hilbert transform, maximal 24
smoothing interval 46
Hilbert transform, with respect
splitting 87
47
p
6
N
120
Stein-Weiss' theorem 15, 17
Hilbert transform, maximal with respect to
5
6
generalized Fourier coefficients 52
m*
5
120 restricted type
to
p
m~
47
strong type
p
5
sublinear operator 4
Hilbert transform, modified maximal with respect to
4"
48
6
operator of restricted weak type p
theorem, Carleson-Hunt's 17, 20 theorem, Marcinkiewicz'8
interval, central 88
theorem, Stein-Weiss' 15, 17
interval, dyadic 46
type
p
5
interval, smoothing 46 weak type
p
5
6
