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 p Ap β 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 exp N+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
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