4 PREFACE

are valid. Moreover, we will show, among other things, that C0

1(ω

) is the closure

of polynomials in

C1(ω

).

Chapter 6 offers a complete description of those analytic symbols g in D for

which the integral operator Tg belongs to the Schatten p-class Sp(Aω),

2

where ω ∈

I ∪ R. If p 1, then Tg ∈ Sp(Aω)

2

if and only if g belongs to the analytic Besov

space Bp, and if 0 p ≤ 1, then Tg ∈ Sp(Aω)

2

if and only if g is constant. It is

appropriate to mention that these results are by no means unexpected. This is due

to the fact that the operators Tg in both Sp(H2) and Sp(Aα) 2 are also characterized

by the condition g ∈ Bp, provided p 1. What makes this chapter interesting

is the proofs which are carried over in a much more general setting. Namely,

we will study the Toeplitz operator, induced by a complex Borel measure and a

reproducing kernel, in certain Dirichlet type spaces that are induced by ω , in the

spirit of Luecking [59]. Our principal findings on this operator are gathered in a

single theorem at the end of Chapter 6.

In Chapter 7 we will study linear differential equations with solutions in either

the weighted Bergman space Aω

p

or the Bergman-Nevalinna class BNω. Our primary

interest is to relate the growth of coeﬃcients to the growth and the zero distribution

of solutions. In Section 7.1 we will show how results and techniques developed in

the presiding chapters can be used to find a set of suﬃcient conditions for the

analytic coeﬃcients of a linear differential equation of order k forcing all solutions

to the weighted Bergman space Aω.

p

Since the zero distribution of functions in

Aωp

is studied in Chapter 3, we will also obtain new information on the oscillation of

solutions. In Section 7.2 we will see that it is natural to measure the growth of the

coeﬃcients by the containment in the weighted Bergman spaces depending on ω,

when all solutions belong to the Bergman-Nevalinna class BNω. In particular, we

will establish a one-to-one correspondence between the growth of coeﬃcients, the

growth of solutions and the zero distribution of solutions whenever ω is regular. In

this discussion results from the Nevanlinna value distribution theory are explicitly or

implicitly present in many instances. Apart from tools commonly used in the theory

of complex differential equations in the unit disc, Chapter 7 also relies strongly on

results and techniques from Chapters 1–5, and is therefore unfortunately a bit hard

to read independently.

Chapter 8 is devoted to further discussion on topics that this monograph does

not cover. We will briefly discuss q-Carleson measures for Aω p when q p, gener-

alized area operators as well as questions related to differential equations and the

zero distribution of functions in Aω. p We include few open problems that are partic-

ularly related to the special features of the weighted Bergman spaces Aω

p

induced

by rapidly increasing weights.

Jos´ e

´

Angel Pel´ aez (M´alaga)

Jouni R¨ atty¨ a (Joensuu)