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Journal of Function Spaces and Applications
Volume 1, Issue 1, Pages 45-59
http://dx.doi.org/10.1155/2003/932158

Singular integrals and potentials in some Banach function spaces with variable exponent

1A. Razmadze Mathematical Institute, M. Aleksidze St., 1, 380093 Tbilisi, Georgia
2Universidade do Algarve, Faro 8000, Portugal

Received 1 December 2002

Academic Editor: Alois Kufner

Copyright © 2003 Hindawi Publishing Corporation. 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

We introduce a new Banach function space - a Lorentz type space with variable exponent. In this space the boundedness of singular integral and potential type operators is established, including the weighted case. The variable exponent p(t) is assumed to satisfy the logarithmic Dini condition and the exponent β of the power weight ω(t)=|t|β is related only to the value p(0). The mapping properties of Cauchy singular integrals defined on Lyapunov curves and on curves of bounded rotation are also investigated within the framework of the introduced spaces.