1.2. REGULAR AND RAPIDLY INCREASING WEIGHTS 7
if ω I, then supr≤t ψω(r)/(1 r) can grow arbitrarily fast as t
1−.
See the
weight defined in (1.10) and Lemma 1.5 in Section 1.4.
In this study we are particularly interested in the weighted Bergman space
Aωp
induced by a rapidly increasing weight ω, although most of the results are obtained
under the hypotheses “ω I R”. However, there are proofs in which more
regularity is required for ω I. This is due to the fact that rapidly increasing
weights may admit a strong oscillatory behavior. Indeed, consider the weight
(1.9) ω(r) = sin log
1
1 r
vα(r) + 1, 1 α ∞.
It is clear that ω is continuous and 1 ω(r) vα(r)+1 for all r [0, 1). Moreover,
if
1
e−(
π
4
+nπ)
r 1
e−(

4
+nπ),
n N {0},
then log(1 r) [
π
4
+ nπ,

4
+ nπ], and thus ω(r) vα(r) in there. Let now
r (0, 1), and fix N = N(r) such that 1 e−(N−1)π r 1 e−Nπ. Then
1
r
ω(s) ds

n=N
1−e−(

4
+nπ)
1−e−(
π
4
+nπ)
vα(s) ds =

n=N
1
(
π
4
+
)α−1

1
(

4
+ nπ)α−1

n=N
1


N
dx

1
N α−1
1
log
1
1−r
α−1
1
r
vα(s) ds,
and it follows that ω I. However,
ω(1
e−nπ− π
2
)
ω(1 e−nπ)
= vα(1
e−nπ−
π
2
) + 1 ∞, n ∞,
yet
1 (1
e−nπ−
π
2
)
1 (1 e−nπ)
=
e− π
2
(0, 1)
for all n N. Therefore ω does not satisfy (1.2). Another bad-looking example in
the sense of oscillation is
(1.10) ω(r) = sin log
1
1 r
vα(r) +
1
ee
1
1−r
, 1 α ∞,
which belongs to I, but does not satisfy (1.2) by the reasoning above. Moreover,
by passing through the zeros of the sin function, we see that
lim inf
r→1−
ω(r)ee
1
1−r
= 1.
Our last example on oscillatory weights is
(1.11) ω(r) = sin log
1
1 r
(1
r)α
+ (1
r)β,
where −1 α β ∞. Obviously, (1 r)β ω(r) (1 r)α, so p p
Aβ.p
However, ω I because the limit in (1.7) does not exist, and ω R because ω
neither satisfies (1.2) nor (1.4), yet ω obeys (1.3).
In the case when more local regularity is required for ω I in the proof, we
will consider the class I of those ω I that satisfy (1.2).
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