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 P± 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,