8 AHMED
I.
ZAYED
for an L
1
function
cp
and a measurable set R with IRI
-/=-
0. Let D = Dq be the
dyadic lattice rooted in a cube
Q
C
IRn.
For each
I
E
D
and every x
E
I,
define the
dyadic cone
ri(x)
to be
(0.3) f1(x) ={JED: J
c;;_
I,J
3
x}.
Take any
cp
E
L
1
(
Q)
and for a cube
I
E
D,
let
h, h, ... , hn
be its immediate dyadic
offspring, i.e. the 2n subcubes of its first dyadic generation. Let
x
1
= (cp)I and

= (cp)
h.
The authors define the s-function of
cp
on
Q
to be
( )
1/2
Scp(x)
= " max
{lx
1
-
x£n
L...
1kzn
IErq(x) - -
(0.4)
In a paper by Chang, Wilson, and Wolff, it was shown that, for a function
cp
whose s-function
Scp
is bounded on a cube
Q
C
IRn,
one has
l{s
E
Q:
cp(s)-
(cp)q -\}I:::; IQiexp(--\
2
/(2IIScpll~)).
This immediately implies the integrability of
ef3(cp-(cp)q)
2
for an appropriate range
of {3. In fact, they showed that if
cp
is a real-valued function on an interval
I
and
Scp
is the corresponding s-function, then
(0.5)
(e'P-('P)r)I:::;
(e~(Scp)z)I.
L. Slavin and A. Volberg extend this result to an arbitrary measurable bounded
set
E
C
IRn.
Namely, they show that
(0.6)
.2_ {
ef3(cp-(cp)E)z
2R{J \1{3
E
[o
_!__)
lEI
J
E -
1- 2R(J' '2R '
where
R
= IIScpllioo(E) and the s-function is defined on any cube containing
E.
Moreover, they showed that if
E
C
IRn
is a bounded measurable set with lEI
-!=-
0
and
Q
is a cube in
!Rn
containing
E
with the corresponding dyadic latticeD= Dq,
then
1~1
k
et(cp-(cp)E) :::;
exp
(~t21IScpiiioo(E))
'\It
E IR,
where
cp
E
L
1
(Q),
Scp
is its s-function and

E
L
00
(E).
In closing, I hope that I have succeeded in giving the reader a glimpse of the
contents of this volume and making clear that indeed there are strong connections
between harmonic analysis and ergodic theory. But what is particularly interesting
about this interplay is the extent of it, the amount and diversity of past work and
the current level of research activities. Some chapters demonstrate this interplay
more clearly than others. These chapters constitute only a part of a larger picture
in modern day mathematics, techniques and ideas from one field being used in other
fields to make significant progress in solving important and difficult problems.
DEPARTMENT OF MATHEMATICAL SCIENCES, DEPAUL UNIVERSITY, CHICAGO,
IL 60614
E-mail address:
azayed!Dmath. depaul. edu
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