International Journal of Mathematics and Mathematical Sciences
Volume 30 (2002), Issue 9, Pages 533-540

Hardy-Littlewood type inequalities for Laguerre series

Chin-Cheng Lin and Shu-Huey Lin

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.


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)|rdtCj=0|cj|rj¯1r/2 for certain r1, where f is the limit function of cj𝔏ja. Moreover, we show that if f(x)cj𝔏ja is in Lr, r1, we have the converse Hardy-Littlewood type inequality j=0|cj|rj¯βC0|f(t)|rdt for r1 and β<r/2.