Hardy-Littlewood type inequalities for Laguerre series

Lin, Chin-Cheng; Lin, Shu-Huey

doi:https://doi.org/10.1155/S0161171202108234

International Journal of Mathematics and Mathematical Sciences

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Volume 30 |Article ID 736831 | https://doi.org/10.1155/S0161171202108234

Chin-Cheng Lin, Shu-Huey Lin, "Hardy-Littlewood type inequalities for Laguerre series", International Journal of Mathematics and Mathematical Sciences, vol. 30, Article ID 736831, 8 pages, 2002. https://doi.org/10.1155/S0161171202108234

Hardy-Littlewood type inequalities for Laguerre series

Chin-Cheng Lin¹ and Shu-Huey Lin¹

¹Department of Mathematics, National Central University, Chung-Li, China

Received30 Aug 2001

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.

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Copyright © 2002 Hindawi Publishing Corporation. 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.

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