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Journal of Function Spaces and Applications
Volume 2012, Article ID 454820, 20 pages
http://dx.doi.org/10.1155/2012/454820
Research Article

Weighted Composition Operators from Hardy Spaces into Logarithmic Bloch Spaces

1Department of Mathematical Sciences, George Mason University, Fairfax, VA 22030, USA
2Department of Mathematics, Jiaying University, Guangdong, Meizhou 514015, China

Received 2 February 2012; Accepted 18 March 2012

Academic Editor: Ruhan Zhao

Copyright © 2012 Flavia Colonna and Songxiao Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The logarithmic Bloch space Blog is the Banach space of analytic functions on the open unit disk &#x1D53B; whose elements f satisfy the condition f=supz&#x1D53B;(1-|z|2)log  (2/(1-|z|2))|f'(z)|<. In this work we characterize the bounded and the compact weighted composition operators from the Hardy space Hp (with 1p) into the logarithmic Bloch space. We also provide boundedness and compactness criteria for the weighted composition operator mapping Hp into the little logarithmic Bloch space defined as the subspace of Blog consisting of the functions f such that lim|z|1(1-|z|2)log  (2/(1-|z|2))|f'(z)|=0.