Hardy-Littlewood type inequalities for Laguerre series

Chin-Cheng Lin and Shu-Huey Lin

International Journal of Mathematics and Mathematical Sciences, 2002, vol. 30, 1-8

Abstract:

Let { c j } be a null sequence of bounded variation. We give appreciate smoothness and growth conditions on { c j } to obtain the pointwise convergence as well as L r -convergence of Laguerre series ∑ c j 𝔏 j a . Then, we prove a Hardy-Littlewood type inequality ∫ 0 ∞ | f ( t ) | r d t ≤ C ∑ j = 0 ∞ | c j | r j ¯ 1 − r / 2 for certain r ≤ 1 , where f is the limit function of ∑ c j 𝔏 j a . Moreover, we show that if f ( x ) ∼ ∑ c j 𝔏 j a is in L r , r ≥ 1 , we have the converse Hardy-Littlewood type inequality ∑ j = 0 ∞ | c j | r j ¯ β ≤ C ∫ 0 ∞ | f ( t ) | r d t for r ≥ 1 and β < − r / 2 .

Date: 2002
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:736831

DOI: 10.1155/S0161171202108234

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