Two-weight norm inequalities for Cesàro means of Laguerre expansions

Muckenhoupt, Benjamin; Webb, David

doi:10.1090/S0002-9947-00-02729-X

Two-weight norm inequalities for Cesàro means of Laguerre expansions
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by Benjamin Muckenhoupt and David W. Webb

Trans. Amer. Math. Soc. 353 (2001), 1119-1149

DOI: https://doi.org/10.1090/S0002-9947-00-02729-X

Published electronically: November 14, 2000

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Abstract:

Two-weight $L^{p}$ norm inequalities are proved for Cesàro means of Laguerre polynomial series and for the supremum of these means. These extend known norm inequalities, even in the single power weight and “unweighted” cases, by including all values of $p\geq 1$ for all positive orders of the Cesàro summation and all values of the Laguerre parameter $\alpha >-1$. Almost everywhere convergence results are obtained as a corollary. For the Cesàro means the hypothesized conditions are shown to be necessary for the norm inequalities. Necessity results are also obtained for the norm inequalities with the supremum of the Cesàro means; in particular, for the single power weight case the conditions are necessary and sufficient for summation of order greater than one sixth.

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