Embedding of analytic function spaces with given mean growth of the derivative

✍ Scribed by Óscar Blasco; Daniel Girela; M. Auxiliadora Márquez

• Publisher: John Wiley and Sons
• Year: 2006
• Tongue: English
• Weight: 156 KB
• Volume: 279
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.200410453

✦ Synopsis

Abstract

If ϕ is a positive function defined in [0, 1) and 0 < p < ∞, we consider the space ℒ︁(p, ϕ) which consists of all functions f analytic in the unit disc 𝔻 for which the integral means of the derivative M ~p~ (r, f ′) = $ \left ({\textstyle {{1} \over {2 \pi} }} \int ^{\pi}_{- \pi} | f' (re^{i \theta}) |^p , d \theta \right) ^{1/p}, $ 0 < r < 1, satisfy M ~p~ (r, f ′) = O(ϕ (r)), as r → 1. In this paper, for any given p ∈ (0, 1), we characterize the functions ϕ, among a certain class of weight functions, to be able to embedd ℒ︁(p, ϕ) into classical function spaces. These results complement other previously obtained by the authors for p ≥ 1. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)