tailieunhanh - Báo cáo hóa học: " Research Article Littlewood-Paley g-Functions and Multipliers for the Laguerre Hypergroup"

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 Littlewood-Paley g-Functions and Multipliers for the Laguerre Hypergroup | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2011 Article ID 741095 13 pages doi 2011 741095 Research Article Littlewood-Paley g-Functions and Multipliers for the Laguerre Hypergroup Jizheng Huang1 2 1 College of Sciences North China University of Technology Beijing 100144 China 2 CEMA Central University of Finance and Economics Beijing 100081 China Correspondence should be addressed to Jizheng Huang hjzheng@ Received 4 November 2010 Accepted 13 January 2011 Academic Editor Shusen Ding Copyright 2011 Jizheng Huang. 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. Let L - d2 dx2 2a 1 x 9 9x x2 92 9t2 x t e 0 TO X R where a 0. Then L can generate a hypergroup which is called Laguerre hypergroup and we denote this hypergroup by K. In this paper we will consider the Littlewood-Paley g-functions on K and then we use it to prove the Holmander multipliers on K. 1. Introduction and Preliminaries In 1 the authors investigated Littlewood-Paley g-functions for the Laguerre semigroup. Let d d2 d La xA- a 1 - xf L- Ế1 dx2 àx where a a1 . ad xi 0 then define the following Littlewood-Paley function Ga by Gaf x TO tVaPf x 2d . 0 t where Va õt -ựx1dx1 . ựxddxd and Pta is the Poisson semigroup associated to La. In 1 the authors prove that Ga is bounded on Lp pa for 1 p TO. In this paper we consider the following differential operator L - dx2 2 xdẽ x t e 0 to X R 2a 1 d x dx 2 Journal of Inequalities and Applications where a 0. It is well known that it can generate a hypergroup cf. 2 3 or 4 . We will define Littlewood-Paley g-functions associated to L and prove that they are bounded on Lp K for 1 p TO. As an application we use it to prove the Homander multiplier theorem on K. Let K 0 to X R equipped with the measure 1 2a 1 dma x t - x2a 1dxdt a 0. n r a 1 We .