2 PREFACE
function
Mω(ϕ)(z) = sup
I: z∈S(I)
1
ω (S(I))
S(I)
|ϕ(ξ)|ω(ξ) dA(ξ), z D,
introduced by ormander [46], plays a role on p similar to that of the Hardy-
Littlewood maximal function on the Hardy space Hp. Analogously, the conven-
tional norm in
p
is equivalent to a norm expressed in terms of certain square area
functions. These results illustrate in a very concrete manner the significant differ-
ence between the function-theoretic properties of the classical weighted Bergman
space
p
and those of
p
induced by a rapidly increasing weight ω.
We will put an important part of our attention to the integral operator
Tg(f)(z) =
z
0
f(ζ) g (ζ) dζ, z D,
induced by an analytic function g on D. This operator will allow us to further
underscore the transition phenomena from
p
to
Hp
through
p
with ω I.
The choice g(z) = z gives the usual Volterra operator and the Ces` aro operator
is obtained when g(z) = log(1 z). The study of integral operators on spaces
of analytic functions merges successfully with other areas of mathematics, such
as the theory of univalent functions, factorization theorems, harmonic analysis and
differential equations. Pommerenke was probably one of the first authors to consider
the operator Tg [71]. However, an extensive study of this operator was initiated
by the seminal works by Aleman, Cima and Siskakis [7, 10, 11]. In all these
works classical function spaces such as BMOA and the Bloch space B arise in a
natural way. It is known that embedding-type theorems and equivalent norms
in terms of the first derivative have been key tools in the study of the integral
operator. Therefore the study of Tg has also lead developments that evidently have
many applications in other branches of the operator theory on spaces of analytic
functions. Recently, the spectrum of Tg on the Hardy space Hp [9] and the classical
weighted Bergman space
p
[8] has been studied. The approach used in these works
reveals, in particular, a strong connection between a description of the resolvent
set of Tg on
Hp
and the classical theory of the Muckenhoupt weights. In the case
of the classical weighted Bergman space Aα,
p
the Bekoll´ e-Bonami weights take the
role of Muckenhoupt weights.
The approach we take to the study of the boundedness of the integral opera-
tor Tg requires, among other things, a factorization of Aω-functions.
p
In Chapter 3
we establish the required factorization by using a probabilistic method introduced
by Horowitz [48]. We prove that if ω is a weight (not necessarily radial) such that
(£) ω(z) ω(ζ), z Δ(ζ, r), ζ D,
where Δ(ζ, r) denotes a pseudohyperbolic disc, and polynomials are dense in Aω,p
then each f p can be represented in the form f = f1 · f2, where f1 Aω1 p ,
f2 Aω2 p and
1
p1
+
1
p2
=
1
p
, and the following norm estimates hold
(§) f1
p
Aω1
p
· f2
p
Aω2
p

p
p1
f1
p1
Aω1
p
+
p
p2
f2
p2
Aω2
p
C(p1,p2,ω) f
p
Aωp
.
These estimates achieve particular importance when we recognize that under certain
additional hypothesis on the parameters p, p1, and p2, the constant in (§) only
depends on p1. This allows us to describe those analytic symbols g such that
Tg :
p

q
is bounded, provided 0 q p and ω I satisfies (£).
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