Table of Contents Author Guidelines Submit a Manuscript
Journal of Function Spaces and Applications
Volume 2012, Article ID 474681, 26 pages
http://dx.doi.org/10.1155/2012/474681
Research Article

Potential Operators on Cones of Nonincreasing Functions

1A. Razmadze Mathematical Institute, Ivane Javakhishvili Tbilisi State University, 2 University Street, 0143 Tbilisi, Georgia
2Faculty of Informatics and Control Systems, Georgian Technical University, 77 Kostava Street, Tbilisi, Georgia
3Abdus Salam School of Mathematical Sciences, GC University, 68-B New Muslim Town, Lahore, Pakistan

Received 16 May 2011; Accepted 15 June 2011

Academic Editor: V. M. Kokilashvili

Copyright Β© 2012 Alexander Meskhi and Ghulam Murtaza. 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

Necessary and sufficient conditions on weight pairs guaranteeing the two-weight inequalities for the potential operators (πΌπ›Όβˆ«π‘“)(π‘₯)=∞0(𝑓(𝑑)/|π‘₯βˆ’π‘‘|1βˆ’π›Ό)𝑑𝑑 and (ℐ𝛼1,𝛼2βˆ«π‘“)(π‘₯,𝑦)=∞0∫∞0(𝑓(𝑑,𝜏)/|π‘₯βˆ’π‘‘|1βˆ’π›Ό1|π‘¦βˆ’πœ|1βˆ’π›Ό2)π‘‘π‘‘π‘‘πœ on the cone of nonincreasing functions are derived. In the case of ℐ𝛼1,𝛼2, we assume that the right-hand side weight is of product type. The same problem for other mixed-type double potential operators is also studied. Exponents of the Lebesgue spaces are assumed to be between 1 and ∞.