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The Hardy-Littlewood theorem on fractional integration for Laguerre series


Authors: Yūichi Kanjin and Enji Sato
Journal: Proc. Amer. Math. Soc. 123 (1995), 2165-2171
MSC: Primary 42C10; Secondary 42A45
DOI: https://doi.org/10.1090/S0002-9939-1995-1257113-2
MathSciNet review: 1257113
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Abstract: The Hardy-Littlewood theorem on fractional integration for Fourier series says that if $ {I_\sigma }g \sim \sum\nolimits_{n \ne 0} {\vert n{\vert^{ - \sigma }}\hat g} (n){e^{\operatorname{int} }}$, then $ {I_\sigma }$ is bounded from $ {L^p}$ to $ {L^q}$, where $ 1 < p < q < \infty ,\frac{1}{q} = \frac{1}{p} - \sigma $. We shall establish an analogue of this theorem for the Laguerre function system $ \left\{ {{{\left( {\frac{{n!}}{{\Gamma (n + \alpha + 1)}}} \right)}^{\frac{1}{... ...\alpha (x){e^{ - \frac{x}{2}}}{x^{\frac{\alpha }{2}}}} \right\}_{n = 0}^\infty $.


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DOI: https://doi.org/10.1090/S0002-9939-1995-1257113-2
Keywords: Laguerre polynomials, fractional integration, multipliers, transplantation
Article copyright: © Copyright 1995 American Mathematical Society