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 H¨ ormander [46], plays a role on Aω p similar to that of the Hardy-

Littlewood maximal function on the Hardy space Hp. Analogously, the conven-

tional norm in Aω

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 Aα

p

and those of Aω

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 Aα

p

to

Hp

through Aω

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 Aα

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 ∈ Aω 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 : Aω

p

→ Aω

q

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