International Journal of Mathematics and Mathematical Sciences
Volume 30 (2002), Issue 9, Pages 533-540
doi:10.1155/S0161171202108234
Hardy-Littlewood type inequalities for Laguerre series
Department of Mathematics, National Central University, Chung-Li, 320, Taiwan, China
Received 30 August 2001
Copyright © 2002 Chin-Cheng Lin and Shu-Huey Lin. 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 {cj} be a null sequence of bounded variation. We give appreciate smoothness and growth conditions on {cj} to obtain the pointwise convergence as well as Lr-convergence of Laguerre series ∑cj𝔏ja. Then, we prove a Hardy-Littlewood type inequality ∫0∞|f(t)|rdt≤C∑j=0∞|cj|rj¯1−r/2 for certain r≤1, where f is the limit function of ∑cj𝔏ja. Moreover, we show that if f(x)∼∑cj𝔏ja is in Lr, r≥1, we have the converse Hardy-Littlewood type inequality ∑j=0∞|cj|rj¯β≤C∫0∞|f(t)|rdt for r≥1 and β<−r/2.