TOPICS IN ERGODIC THEORY AND HARMONIC ANALYSIS: AN OVERVIEW
7
then
In terms of almost everywhere convergence, we have the following result: If
{2~
2
Jo
lf(eill)l (log+ lf(eill)l) d() oo,
then limN---+oo SNf(x)
=
f(x) almost everywhere. It has been conjectured that if
rh
Jo
lf(eill)llog+ lf(eill)l
d()
oo,
then limN---+oo SNf(x)
=
f(x) almost everywhere.
In this short chapter , Hagelstein discusses connections between this conjecture
and recent developments in interpolation theory regarding sublinear translation
invariant restricted weak type operators.
Chapter Nine by M. Lacey is concerned with Nehari's theorem on a Hardy
space on the polydisk. The Hardy space H
2
('Jl')
=
Ht('ll') on the unit circle
'Jl'
is by
definition the closed subspace of L
2
('Jl')
generated by
{zn
I
n 2:':
0}. It is natural to
call these functions analytic as f
E
H
2
('ll') admits an analytic extension to the disk
~given
by
F(z)
=
L
f(n)zn.
n;:::O
Functions in H.:_ ('ll')
=
L
2
('ll') 8 H
2
('ll') are referred to as antianalytic.
Let us describe the Hankel operators on H
2
('Jl').
Let be the orthogonal
projection from L
2
('Jl')
onto the subspace HJ:('ll'). A Hankel operator with symbol b
is an operator Hb from Ht ('ll') to Ht ('ll') given by Hbrp
=
P
+ Mb(j5, where Mb is the
operator of multiplication by b. The following theorem is due to Nehari.
THEOREM 0.8 (Nehari). The Hankel operator
Hb
is bounded from Ht('ll') to
Ht('ll') if and only if there is a bounded function (3 with P+b
=
P+f3· Moreover,
IIHbll
=
inf ll/3lloo ·
{3:P+f3=P+b
Going to higher dimensions, Lacey first defines 'little' Hankel operators on the
product Hardy space H
2
(C'£) by
Hbrp
=
PffiMb(j5.
where Pffi is the orthogonal projection from
L2(JR.d)
to H
2(C'£).
He then presents
the proof of Ferguson and Lacey and Lacey and Terwilleger of the equivalence of
the norms
IIHbll '::::' llbiiBMO(C'lJ'
for analytic functions b. Here, BMO(C'£) is the dual space of H
1
(C'£) as discovered
by Chang and
R.
Fefferman. The article begins with the classical Nehari Theorem,
and then proceeds to present the necessary background for the proof of the ex-
tension above. The proof of the extension is an induction on parameters, with a
bootstrapping argument.
The last chapter by L. Slavin and A. Volberg is about the s-function and ex-
ponential integrability. To introduce the s-function, let us first define
(rp)R
=
~~~l
rp,
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