Preface
This work concerns the weighted Bergman space 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 p induced by ω I lies between
the Hardy space
Hp
and every classical weighted Bergman space
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
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
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
p
does appear
in the context of
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 p that is induced by a rapidly increasing weight ω. For example, one can
not find a weight ω = ω (p) such that f

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
p
can be characterized
by a simple condition on pseudohyperbolic discs, nor the conjugate operator is
necessarily bounded on
p
(of harmonic functions) if 0 p 1. However, it is
shown in Chapter 2 that the embedding
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
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