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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: CONTINUITY OF MULTILINEAR OPERATORS ON TRIEBEL-LIZORKIN SPACES | CONTINUITY OF MULTILINEAR OPERATORS ON TRIEBEL-LIZORKIN SPACES LANZHE LIU Received 4 February 2006 Revised 20 September 2006 Accepted 28 September 2006 The continuity of some multilinear operators related to certain convolution operators on the Triebel-Lizorkin space is obtained. The operators include Littlewood-Paley operator and Marcinkiewicz operator. Copyright 2006 Lanzhe Liu. 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 Let T be the Calderon-Zygmund singular integral operator a well-known result of Coif-man et al. see 6 states that the commutator b T f T bf - bT f where b e BMO is bounded on Lp Rn 1 p oo Chanillo see 1 proves a similar result when T is replaced by the fractional integral operator in 8 9 these results on the Triebel-Lizorkin spaces and the case b e Lip p where Lip p is the homogeneous Lipschitz space are obtained. The main purpose of this paper is to study the continuity of some multilinear operators related to certain convolution operators on the Triebel-Lizorkin spaces. In fact we will obtain the continuity on the Triebel-Lizorkin spaces for the multilinear operators only under certain conditions on the size of the operators. As the applications the continuity of the multilinear operators related to the Littlewood-Paley operator and Marcinkiewicz operator on the Triebel-Lizorkin spaces are obtained. 2. Notations and results Throughout this paper Q will denote a cube of Rn with side parallel to the axes and for a cube Q let fQ IQI-1 Qf x dx and f x supxeQ IQI-1 Jq I f y - fQ dy. For 1 r o and 0 8 n let 1 r 1 r Ms r f x sup ĩQĩ JQlf y I dy 2-1 Hindawi Publishing Corporation Journal ofInequalities and Applications Volume 2006 Article ID 58473 Pages 1-11 DOI 10.1155 JIA 2006 58473 2 Continuity of multilinear operators we denote Mg r f Mr f if 8 0 which is the .
