H^1-ESTIMATES OF LITTLEWOOD-PALEY AND LUSIN FUNCTIONS FOR JACOBI ANALYSIS

作     者：Takeshi Kawazoe 

作者机构：Department of Mathematics Keio University at Fujisawa 5322 EndoFujisawaKanagawa 252-8520 Japan 

出 版 物：《Analysis in Theory and Applications》 (分析理论与应用（英文刊）)

年 卷 期：2009年第25卷第3期

页      面：201-229页

学科分类：07[理学] 070104[理学-应用数学] 0701[理学-数学] 

基　　金：Partly supported by Grant-in-Aid for Scientific Research (C)  No. 20540188  Japan Society for the Promotion of Science 

主　　题：Hardy space Jacobi analysis Littlewood-Paleyfunction Lusinfunction 

摘      要：For a ≥β≥ -1/2 let △（x） = （2shx）^2α＋1 （2chx）2β＋1 denote the weight function on R＋ and L^1 （△） the space of integrable functions on R＋ with respect to △（x）dx, equipped with a convolution structure. For a suitable Ф ∈ L^1 （△）, we put Фt（x） = t^-1 △（x）^-1 △（x/t）Ф（x/t） for t 〉 0 and define the radial maximal operator MФ, as usual manner. We introduce a real Hardy space H^1 （△） as the set of all locally integrable functions f on R＋ whose radial maximal function MФ （f） belongs to L^1 （△）. In this paper we obtain a relation between H^1 （△） and H^1 （R）. Indeed, we characterize H^1 （△） in terms of weighted H^1 Hardy spaces on R via the Abel transform of f. As applications of H^1 （△） and its characterization, we shall consider （H^1 （△）,L^1 （△））-boundedness of some operators associated to the Poisson kernel for Jacobi analysis： the Poisson maximal operator Me, the Littlewood-Paley g-function and the Lusin area function S. They are bounded on L^p（△） for p 〉 1, but not true for p = 1. Instead, Mp, g and a modified Sa,r are bounded from H^1 （△） to L^1 （△）.