The Hardy-Littlewood theorem on fractional integration for Laguerre series

Kanjin, Yūichi; Sato, Enji

doi:10.1090/S0002-9939-1995-1257113-2

The Hardy-Littlewood theorem on fractional integration for Laguerre series
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by Yūichi Kanjin and Enji Sato

Proc. Amer. Math. Soc. 123 (1995), 2165-2171

DOI: https://doi.org/10.1090/S0002-9939-1995-1257113-2

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Abstract:

The Hardy-Littlewood theorem on fractional integration for Fourier series says that if ${I_\sigma }g \sim \sum \nolimits _{n \ne 0} {|n{|^{ - \sigma }}\hat g} (n){e^{\operatorname {int} }}$, 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 $\left \{ {{{\left ( {\frac {{n!}}{{\Gamma (n + \alpha + 1)}}} \right )}^{\frac {1}{2}}}L_n^\alpha (x){e^{ - \frac {x}{2}}}{x^{\frac {\alpha }{2}}}} \right \}_{n = 0}^\infty$.

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