tailieunhanh - Báo cáo hóa học: " Research Article On Boundedness of Weighted Hardy Operator in Lp · and Regularity Condition"

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 On Boundedness of Weighted Hardy Operator in Lp · and Regularity Condition | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010 Article ID 837951 14 pages doi 2010 837951 Research Article On Boundedness of Weighted Hardy Operator in Lp l and Regularity Condition Aziz Harman1 and Farman Imran Mamedov1 2 1 Education Faculty Dicle University 21280 Diyarbakir Turkey 2 Institute of Mathematics and Mechanics of National Academy of Science Azerbaijan Correspondence should be addressed to Farman Imran Mamedov Received 22 September 2010 Accepted 26 November 2010 Academic Editor P. J. Y. Wong Copyright 2010 A. Harman and F. I. Mamedov. 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. We give a new proof for power-type weighted Hardy inequality in the norms of generalized Lebesgue spaces L IfR . Assuming the logarithmic conditions of regularity in a neighborhood of zero and at infinity for the exponents p x q x p x necessary and sufficient conditions are proved for the boundedness of the Hardy operator Hf x J y x f ỳ dy from LX tì K into Lq j . -. P 0- q . RN . Also a separate statement on the exactness of logarithmic conditions at zero and at infinity is given. This shows that logarithmic regularity conditions for the functions p p at the origin and infinity are essentially one. 1. Introduction The object of this investigation is the Hardy-type weighted inequality lxlPH- p H- HHf Il. C x P - f IIlp . . Hf x J y s x f M in the norms of generalized Lebesgue spaces U tR . This subject was investigated in the papers 1-7 . For the one-dimensional Hardy operator in 1 the necessary and sufficient condition was obtained for the exponents p p q. We give a new proof for this result in more general settings for the multidimensional Hardy operator. Also we prove that the logarithmic regularity conditions are essential one for such kind .