Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Potential Operators in Variable Exponent Lebesgue Spaces: Two-Weight Estimates | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010 Article ID 329571 27 pages doi 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 1 Department of Mathematical Analysis A. Razmadze Mathematical Institute 1. M. Aleksidze Street 0193 Tbilisi Georgia 2 Faculty of Exact and Natural Sciences Ivane Javakhishvili Tbilisi State University 2 University Street 0143 Tbilisi Georgia 3 Department of Mathematics Faculty of Informatics and Control Systems Georgian Technical University 77 Kostava Street 0175 Tbilisi Georgia 4 Abdus Salam School of Mathematical Sciences GC University 68-B New Muslim Town Lahore 54600 Pakistan Correspondence should be addressed to Alexander Meskhi alex72meskhi@ 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. 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 p. 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. 1. Introduction We study the two-weight problem for Hardy-type and potential operators in Lebesgue spaces with nonstandard growth defined on quasimetric measure spaces X d ỳ . In particular our aim is to derive easily verifiable sufficient conditions for the boundedness of the
