6 1. BASIC NOTATION AND INTRODUCTION TO WEIGHTS

1.2. Regular and rapidly increasing weights

The distortion function of a radial weight ω : [0, 1) → (0, ∞) is defined by

ψω(r) =

1

ω(r)

1

r

ω(s) ds, 0 ≤ r 1,

and was introduced by Siskakis in [83]. A radial weight ω is called regular, if ω is

continuous and its distortion function satisfies

(1.1) ψω(r) (1 − r), 0 ≤ r 1.

The class of all regular weights is denoted by R. We will show in Section 1.4 that

if ω ∈ R, then for each s ∈ [0, 1) there exists a constant C = C(s, ω) 1 such that

(1.2)

C−1ω(t)

≤ ω(r) ≤ Cω(t), 0 ≤ r ≤ t ≤ r + s(1 − r) 1.

It is easy to see that (1.2) implies

(1.3) ψω(r) ≥ C(1 − r), 0 ≤ r 1,

for some constant C = C(ω) 0. However, (1.2) does not imply the existence of

C = C(ω) 0 such that

(1.4) ψω(r) ≤ C(1 − r), 0 ≤ r 1,

as is seen by considering the weights

(1.5) vα(r) = (1 − r) log

e

1 − r

α −1

, 1 α ∞.

Putting the above observations together, we deduce that the regularity of a radial

continuous weight ω is equivalently characterized by the conditions (1.2) and (1.4).

As to concrete examples, we mention that every ω(r) = (1 −

r)α,

−1 α ∞, as

well as all the weights in [11, (4.4)–(4.6)] are regular.

The good behavior of regular weights can be broken in different ways. On one

hand, the condition (1.2) implies, in particular, that ω can not decrease very fast.

For example, the exponential type weights

(1.6) ωγ,α(r) = (1 −

r)γ

exp

−c

(1 − r)α

, γ ≥ 0, α 0, c 0,

satisfy neither (1.2) nor (1.3). Meanwhile the weights ωγ,α are monotone near 1,

the condition (1.2) clearly also requires local smoothness and therefore the regular

weights can not oscillate too much. On the other hand, we will say that a radial

weight ω is rapidly increasing, denoted by ω ∈ I, if it is continuous and

(1.7) lim

r→1−

ψω(r)

1 − r

= ∞.

It is easy to see that if ω is a rapidly increasing weight, then Aω

p

⊂ Aβ

p

for any

β −1, see Section 1.4. Typical examples of rapidly increasing weights are vα,

defined in (1.5), and

(1.8) ω(r) = (1 − r)

N

n=1

logn

expn 0

1 − r

logN+1

expN+1 0

1 − r

α

−1

for all 1 α ∞ and N ∈ N = {1, 2,...}. Here, as usual, logn x = log(logn−1 x),

log1 x = log x, expn x = exp(expn−1 x) and exp1 x =

ex.

It is worth noticing that