tailieunhanh - Báo cáo hóa học: "A UNIFYING APPROACH FOR CERTAIN CLASS OF MAXIMAL FUNCTIONS"

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: A UNIFYING APPROACH FOR CERTAIN CLASS OF MAXIMAL FUNCTIONS | A UNIFYING APPROACH FOR CERTAIN CLASS OF MAXIMAL FUNCTIONS AHMAD AL-SALMAN Received 16 January 2006 Revised 12 April 2006 Accepted 13 April 2006 We establish Lp estimates for certain class of maximal functions with kernels in Lq Sn-1 . As a consequence of such Lp estimates we obtain the Lp boundedness of our maximal functions when their kernels are in L logL 1 2 Sn-1 or in the block space B0 -1 2 Sn-1 q 1. Several applications of our results are also presented. Copyright 2006 Ahmad Al-Salman. 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 and statement of results Let Rn n 2 be the n-dimensional Euclidean space and let Sn-1 be the unit sphere in Rn equipped with the normalized Lebesgue measure dơ. For nonzero y e Rn we will let y I y -1 y. Let o be an integrable function on Sn-1 that is homogeneous of degree zero on Rn and satisfies the cancelation property o y dơ y 0. Sn-1 Consider the maximal function o Mo f x sup f x - y I y -nh y o y dy heU Rn I where U is the class of all h e L2 R r-1dr with h L2 K r-1dr 1. The operator Mn was introduced by Chen and Lin 7 . They showed that Mn is bounded on Lp Rn for all p 2n 2n 1 provided that o e Sn-1 . Recently we have been able to show that the Lp Rn boundedness of Mn still holds for all p 2 if the condition o e Sn-1 is replaced by the more natural and weaker condition o e L log L 1 2 Sn-1 2 . Moreover we showed that if the condition o e L log L 1 2 Sn-1 is replaced by any condition in the form o e L log L r Sn-1 for some r 1 2 then Mo might fail to be bounded on L2. Hindawi Publishing Corporation Journal ofInequalities and Applications Volume 2006 Article ID 56272 Pages 1-17 DOI JIA 2006 56272 2 A unifying approach for certain class of maximal functions On the other hand when Q lies in B0 -1 2 Sn-1 s 1 which is a special class .