PREFACE 3

By doing this we avoid the use of interpolation theorems and arguments based on

Kinchine’s inequality, that are often employed when solving this kind of problems

on classical spaces of analytic functions on D. The techniques used to establish

the above-mentioned factorization result permit us to show that each subset of an

Aω-zero

p

set is also an Aω-zero

p

set. Moreover, we will show that the Aω-zero

p

sets

depend on p whenever ω ∈ I ∪ R. This will be done by estimating the growth of

the maximum modulus of certain infinite products whose zero distribution depends

on both p and ω. We will also briefly discuss the zero distribution of functions in

the Bergman-Nevanlinna class BNω that consists of those analytic functions in D

for which

D

log+

|f(z)|ω(z) dA(z) ∞.

Results related to this discussion will be used in Chapter 7 when the oscillation of

solutions of linear differential equations in the unit disc is studied.

In Chapter 4 we first equip Aω

p

with several equivalent norms inherited from

different Hp-norms through integration. Those ones that are obtained via the

classical Fefferman-Stein estimate or in terms of a non-tangential maximal function

related to lens type regions with vertexes at points in D, appear to be the most

useful for our purposes. Here, we also prove that it is not possible to establish a

Littlewood-Paley type formula if ω ∈ I unless p = 2. In Section 4.2 we characterize

those analytic symbols g on D such that Tg : Aω p → Aω, q 0 p, q ∞, is bounded

or compact. The case q p does not give big surprises because essentially standard

techniques work yielding a condition on the maximum modulus of g . This is no

longer true if q = p. Indeed, we will see that Tg : Aω p → Aω p is bounded exactly

when g belongs to the space

C1(ω

) that consists of those analytic functions on D

for which

g

2

C1(ω )

=

|g(0)|2

+ sup

I⊂T

S(I)

|g

(z)|2ω

(z) dA(z)

ω (S(I))

∞,

where

ω (z) =

1

|z|

ω(s) log

s

|z|

s ds, z ∈ D \ {0}.

Therefore, as in the case of

Hp

and Aα,

p

the boundedness (and the compactness)

is independent of p. It is also worth noticing that the above

C1(ω

)-norm has

all the flavor of the known Carleson measure characterizations of BMOA and B.

Both of these spaces admit the important and very powerful property of conformal

invariance. In fact, this invariance plays a fundamental role in the proofs of the

descriptions of when Tg is bounded on either

Hp

or Aα.

p

In contrast to BMOA

and B, the space C1(ω ) is not necessarily conformally invariant if ω is rapidly

increasing, and therefore we will employ different techniques. In Section 4.3 we

will show in passing that the methods used are adaptable to the Hardy spaces,

and since they also work for Aω p when ω is regular, we consequently will obtain

as a by-product a unified proof for the classical results on the boundedness and

compactness of Tg on Hp and Aα. p Chapter 5 is devoted to the study of the space

C1(ω ) and its “little oh”counterpart C0 1(ω ). In particular, here we will prove that

if ω ∈ I admits certain regularity, then C1(ω ) is not conformally invariant, and

further, the strict inclusions

BMOA

C1(ω

) B