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 Endpoint Estimates for a Class of Littlewood-Paley Operators with Nondoubling Measures | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009 Article ID 175230 28 pages doi 2009 175230 Research Article Endpoint Estimates for a Class of Littlewood-Paley Operators with Nondoubling Measures Qingying Xue and Juyang Zhang Laboratory of Mathematics and Complex Systems School of Mathematical Sciences Beijing Normal University Ministry of Education Beijing 100875 China Correspondence should be addressed to Qingying Xue qyxue@ Received 18 April 2009 Accepted 18 August 2009 Recommended by Shusen Ding Let p be a positive Radon measure on Rd which may be nondoubling. The only condition that p satisfies is p B x rf C0rn for all x e Rd r 0 and some fixed constant C0. In this paper we introduce the operator gý p related to such a measure and assume it is bounded on L2 p . We then establish its boundedness respectively from the Lebesgue space L1 p to the weak Lebesgue space L1z p from the Hardy space H1 p to L1 p and from the Lesesgue space LTO p to the space RBLO p . As a corollary we obtain the boundedness of gyp in the Lebesgue space lp p with p e 1 to . Copyright 2009 Q. Xue and J. Zhang. 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 A positive Radon measure p on Rd is said to be doubling if there exists some constant C such that p B x 2r Cp B x r for all x e supp p r 0. It is well known that the doubling condition is an essential assumption in many results of classical Calderon-Zygmund theory. However in the recent years it has been shown that a big part of the classical theory remains valid if the doubling assumption on p is substituted by the growth condition as follows p B x r Corn for all x e Rd where n is some fixed number with 0 n d. For example In 2001 Tolsa in 1 2 investigated the weak 1 1 inequality for singular integrals .
