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Báo cáo hóa học: " Research Article Boundedness of the Maximal, Potential and Singular Operators in the Generalized Morrey Spaces"

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 Boundedness of the Maximal, Potential and Singular Operators in the Generalized Morrey Spaces | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009 Article ID 503948 20 pages doi 10.1155 2009 503948 Research Article Boundedness of the Maximal Potential and Singular Operators in the Generalized Morrey Spaces Vagif S. Guliyev1 2 1 Department of Mathematics Ahi Evran University Kirsehir Turkey 2 Institute of Mathematics and Mechanics Baku Azerbaijan Correspondence should be addressed to Vagif S. Guliyev vagif@guliyev.com Received 12 July 2009 Accepted 22 October 2009 Recommended by Shusen Ding We consider generalized Morrey spaces Mp fRn with a general function w x r defining the Morrey-type norm. We find the conditions on the pair 1 2 which ensures the boundedness of the maximal operator and Calderon-Zygmund singular integral operators from one generalized Morrey space Mp i Rn to another Mp Rn 1 p ro and from the space M1 Rn to the weak space WM . Rn . We also prove a Sobolev-Adams type Mp Rn Mq Rn -theorem for the potential operators Ia. In all the cases the conditions for the boundedness are given it terms of Zygmund-type integral inequalities on 1 2 which do not assume any assumption on monotonicity of 1 2 in r. As applications we establish the boundedness of some Schrodinger type operators on generalized Morrey spaces related to certain nonnegative potentials belonging to the reverse Holder class. As an another application we prove the boundedness of various operators on generalized Morrey spaces which are estimated by Riesz potentials. Copyright 2009 Vagif S. Guliyev. 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. 1. Introduction For x e Rn and r 0 let B x r denote the open ball centered at x of radius r and cB x r denote its complement. Let f e L1oc Rn . The maximal operator M fractional maximal operator Ma and the Riesz potential Ia are defined by Mf x sup B x f