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
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 : 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 : p 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 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
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