Preface

This work concerns the weighted Bergman space Aω p of the unit disc D that is

induced by a radial continuous weight ω : [0, 1) → (0, ∞) such that

(‡) lim

r→1−

1

r

ω(s) ds

ω(r)(1 − r)

= ∞.

A radial continuous weight ω with the property (‡) is called rapidly increasing and

the class of all such weights is denoted by I. Each Aω p induced by ω ∈ I lies between

the Hardy space

Hp

and every classical weighted Bergman space Aα

p

of D.

In many respects the Hardy space

Hp

is the limit of Aα,

p

as α → −1, but it is

well known to specialists that this is a very rough estimate since none of the finer

function-theoretic properties of the classical weighted Bergman space Aα

p

is carried

over to the Hardy space

Hp

whose (harmonic) analysis is much more delicate. One

of the main motivations for us to write this monograph is to study the spaces

Aω,p

induced by rapidly increasing weights, that indeed lie “closer” to

Hp

than any Aα

p

in

the above sense and explore the change of these finer properties related to harmonic

analysis. We will see that the “transition” phenomena from

Hp

to Aα

p

does appear

in the context of Aω

p

with rapidly increasing weights and raises a number of new

interesting problems, some of which are addressed in this monograph.

Chapter 1 is devoted to proving several basic properties of the rapidly increasing

weights that will be used frequently in the monograph. Also examples are provided

to show that, despite of their name, rapidly increasing weights are by no means

necessarily increasing and may admit a strong oscillatory behavior. Most of the

presented results remain true or have analogues for those weights ω for which the

quotient in the left hand side of (‡) is bounded and bounded away from zero. These

weights are called regular and the class of all such weights is denoted by R. Each

standard weight ω(r) = (1 − r2)α is regular for all −1 α ∞. Chapter 1 is

instrumental for the rest of the monograph.

Many conventional tools used in the theory of the classical Bergman spaces fail

to work in Aω p that is induced by a rapidly increasing weight ω. For example, one can

not find a weight ω = ω (p) such that f

Aω

p

f

Ap

ω

for all analytic functions

f in D with f(0) = 0, because such a Littlewood-Paley type formula does not exist

unless p = 2. Moreover, neither p-Carleson measures for Aω

p

can be characterized

by a simple condition on pseudohyperbolic discs, nor the conjugate operator is

necessarily bounded on Lω

p

(of harmonic functions) if 0 p ≤ 1. However, it is

shown in Chapter 2 that the embedding Aω

p

⊂

Lq(μ)

can be characterized by a

geometric condition on Carleson squares S(I) when ω is rapidly increasing and

0 p ≤ q ∞. In these considerations we will see that the weighted maximal

1