Journal of Inequalities and Applications
Volume 2010 (2010), Article ID 329571, 27 pages
doi:10.1155/2010/329571
Research Article

Potential Operators in Variable Exponent Lebesgue Spaces: Two-Weight Estimates

Vakhtang Kokilashvili,1,2 Alexander Meskhi,1,3 and Muhammad Sarwar4

1Department of Mathematical Analysis, A. Razmadze Mathematical Institute, 1. M. Aleksidze Street, 0193 Tbilisi, Georgia
2Faculty of Exact and Natural Sciences, Ivane Javakhishvili Tbilisi State University, 2 University Street, 0143 Tbilisi, Georgia
3Department of Mathematics, Faculty of Informatics and Control Systems, Georgian Technical University, 77 Kostava Street, 0175 Tbilisi, Georgia
4Abdus Salam School of Mathematical Sciences, GC University, 68-B New Muslim Town, Lahore 54600, Pakistan

Received 17 June 2010; Accepted 24 November 2010

Academic Editor: M. Vuorinen

Copyright © 2010 Vakhtang Kokilashvili et al. 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

Two-weighted norm estimates with general weights for Hardy-type transforms and potentials in variable exponent Lebesgue spaces defined on quasimetric measure spaces ( 𝑋 , 𝑑 , 𝜇 ) are established. In particular, we derive integral-type easily verifiable sufficient conditions governing two-weight inequalities for these operators. If exponents of Lebesgue spaces are constants, then most of the derived conditions are simultaneously necessary and sufficient for corresponding inequalities. Appropriate examples of weights are also given.