tailieunhanh - Báo cáo hóa học: "ON WEIGHTED INEQUALITIES FOR PARAMETRIC MARCINKIEWICZ INTEGRALS"

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: ON WEIGHTED INEQUALITIES FOR PARAMETRIC MARCINKIEWICZ INTEGRALS | ON WEIGHTED INEQUALITIES FOR PARAMETRIC MARCINKIEWICZ INTEGRALS H. M. AL-QASSEM Received 25 February 2005 Revised 30 May 2005 Accepted 3 July 2005 We establish a weighted Lp boundedness of a parametric Marcinkiewicz integral operator Mpnh if Q is allowed to be in the block space Bq0 1 2 Sn-1 for some q 1 and h satisfies a mild integrability condition. We apply this conclusion to obtain the weighted Lp boundedness for a class of the parametric Marcinkiewicz integral operators . ị and .ÁdQ h S related to the Littlewood-Paley g -function and the area integral S respectively. It is known that the condition Q e Bq0 1 2 Sn-1 is optimal for the L2 boundedness of Q 1- Copyright 2006 H. M. Al-Qassem. 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 Suppose that Sn-1 is the unit sphere of Rn n 2 equipped with the normalized Lebesgue measure dơ dơ . Let Q be a function defined on Sn-1 with Q e L1 Sn-1 and satisfies the vanishing condition Q x dơ x 0. Sn-1 For y 1 let Ay R denote the set of all measurable functions h on R such that 1 c R sup h t Ydt co. R 0 R 0 It is easy to see that the following inclusions hold and are proper Lo R c Ap R c A R for a p. p a Throughout this paper we let x denote x ỊxỊ for x e Rn 0 and p denote the conjugate index of p that is 1 p 1 p 1. Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2006 Article ID 91541 Pages 1-17 DOI JIA 2006 91541 2 Weighted marcinkiewicz integrals Suppose that r t is a strictly monotonic c1 function on R and h R - C is a measurable function. Define the parametric Marcinkiewicz integral operator MQ r h by ưTO 1 2 p 2 d t 3 FQ ỉ hf t x I where FQ rhf t x x - r lwl w h u du t u t u F p ơ ÍT ơ T e R with Ơ 0 f e y R the space of Schwartz functions. For the sake of simplicity we denote MQ r