Journal of Inequalities and Applications
Volume 2008 (2008), Article ID 697407, 19 pages
doi:10.1155/2008/697407
Research Article

Weighted Estimates of a Measure of Noncompactness for Maximal and Potential Operators

Muhammad Asif1 and Alexander Meskhi2

1Abdus Salam School of Mathematical Sciences, GC University, c-II, M. M. Alam Road, Gulberg III, Lahore 54660, Pakistan
2A. Razmadze Mathematical Institute, Georgian Academy of Sciences, 1, M. aleksidze Street, 0193 Tbilisi, Georgia

Received 5 April 2008; Accepted 19 June 2008

Academic Editor: Siegfried Carl

Copyright © 2008 Muhammad Asif and Alexander Meskhi. 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

A measure of noncompactness (essential norm) for maximal functions and potential operators defined on homogeneous groups is estimated in terms of weights. Similar problem for partial sums of the Fourier series is studied. In some cases, we conclude that there is no weight pair for which these operators acting between two weighted Lebesgue spaces are compact.