CHAPTER 1

Basic Notation and Introduction to Weights

In this chapter we first define the weighted Bergman spaces and the classical

Hardy spaces of the unit disc and fix the basic notation. Then we introduce the

classes of radial and non-radial weights that are considered in the monograph, show

relations between them, and prove several lemmas on weights that are instrumental

for the rest of the monograph.

1.1. Basic notation

Let H(D) denote the algebra of all analytic functions in the unit disc D =

{z : |z| 1} of the complex plane C. Let T be the boundary of D, and let

D(a, r) = {z : |z − a| r} denote the Euclidean disc of center a ∈ C and radius

r ∈ (0, ∞). A function ω : D → (0, ∞), integrable over D, is called a weight function

or simply a weight. It is radial if ω(z) = ω(|z|) for all z ∈ D. For 0 p ∞ and a

weight ω, the weighted Bergman space Aω

p

consists of those f ∈ H(D) for which

f

p

Aωp

=

D

|f(z)|pω(z)

dA(z) ∞,

where dA(z) =

dx dy

π

is the normalized Lebesgue area measure on D. As usual, we

write Aα p for the classical weighted Bergman space induced by the standard radial

weight ω(z) = (1 − |z|2)α, −1 α ∞. For 0 p ≤ ∞, the Hardy space Hp

consists of those f ∈ H(D) for which

f

Hp

= lim

r→1−

Mp(r, f) ∞,

where

Mp(r, f) =

1

2π

2π

0

|f(reiθ)|p

dθ

1

p

, 0 p ∞,

and

M∞(r, f) = max

0≤θ≤2π

|f(reiθ)|.

For the theory of the Hardy and the classical weighted Bergman spaces, see [28,

29, 31, 42, 87].

Throughout the monograph, the letter C = C(·), with a subscript if needed,

will denote an absolute constant whose value depends on the parameters indicated

in the parenthesis, and may change from one occurrence to another. We will use

the notation a b if there exists a constant C = C(·) 0 such that a ≤ Cb, and

a b is understood in an analogous manner. In particular, if a b and a b, then

we will write a b.

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