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
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 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


0
|f(reiθ)|p

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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