are valid. Moreover, we will show, among other things, that C0
) is the closure
of polynomials in
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),
where ω
I R. If p 1, then Tg Sp(Aω)
if and only if g belongs to the analytic Besov
space Bp, and if 0 p 1, then Tg Sp(Aω)
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
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.
Since the zero distribution of functions in
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 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
by rapidly increasing weights.
Jos´ e
Angel Pel´ aez (M´alaga)
Jouni atty¨ a (Joensuu)
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