Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Two-weight norm inequalities for Cesàro means of Laguerre expansions

Authors: Benjamin Muckenhoupt and David W. Webb
Journal: Trans. Amer. Math. Soc. 353 (2001), 1119-1149
MSC (1991): Primary 42C10
Published electronically: November 14, 2000
MathSciNet review: 1804415
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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 $p\geq1$ 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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Additional Information

Benjamin Muckenhoupt
Affiliation: Department of Mathematics, Rutgers University, Piscataway, New Jersey 08854-8019

David W. Webb
Affiliation: Department of Mathematics, DePaul University, Chicago, Illinois 60614-3504

Keywords: Ces\`aro means, Laguerre expansions, Laguerre polynomials, two-weight norm inequalities, weighted norm inequalities
Received by editor(s): May 28, 1999
Published electronically: November 14, 2000
Article copyright: © Copyright 2000 American Mathematical Society