Journal of Inequalities and Applications
Volume 2011 (2011), Article ID 741095, 13 pages
doi:10.1155/2011/741095
Research Article

Littlewood-Paley ๐‘” -Functions and Multipliers for the Laguerre Hypergroup

1College of Sciences, North China University of Technology, Beijing 100144, China
2CEMA, Central University of Finance and Economics, Beijing 100081, China

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.

Abstract

Let ๐ฟ = โˆ’ ( ๐œ• 2 / ๐œ• ๐‘ฅ 2 + ( 2 ๐›ผ + 1 / ๐‘ฅ ) ( ๐œ• / ๐œ• ๐‘ฅ ) + ๐‘ฅ 2 ( ๐œ• 2 / ๐œ• ๐‘ก 2 ) ) ; ( ๐‘ฅ , ๐‘ก ) โˆˆ ( 0 , + โˆž ) ร— โ„ , where ๐›ผ โ‰ฅ 0 . Then ๐ฟ 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 ๐‘” -functions on K and then we use it to prove the Hölmander multipliers on K.