Transactions of the American Mathematical Society

Author(s): Benjamin Muckenhoupt; David W. Webb
Journal: Trans. Amer. Math. Soc. 353 (2001), 1119-1149.
MSC (1991): Primary 42C10
Posted: November 14, 2000
Retrieve article in: PDF DVI PostScript
This article is available free of charge

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.

1.: Richard Askey and Stephen Wainger, Mean convergence of expansions in Laguerre and Hermite series, Amer. J. Math. 87 (1965), 695-708. MR 32:316
2.: Aline Bonami and Jean-Louis Clerc, Sommes de Cesàro et multiplicateurs des développements en harmoniques sphériques, Trans. Amer. Math. Soc. 183 (1973), 223-263. MR 49:3461
3.: Clemens Markett, Norm estimates for Cesàro means of Laguerre expansions, approximatioins and function spaces, in Approximation and Function Spaces, North Holland, Amsterdam-New York (1981), 419-435. MR 83f:42018
4.: -, Mean Cesàro summability of Laguerre expansions and norm estimates with shifted parameter, Analysis Math. 8 (1982), 19-37. MR 83j:40004
5.: B. Muckenhoupt, Asymptotic forms for Laguerre polynomials, Proc. Amer. Math. Soc. 24 (1970), 288-292. MR 40:4503
6.: -, Mean convergence of Hermite and Laguerre series II, Trans. Amer. Math. Soc. 147 (1970), 433-460. MR 41:711
7.: Eileen Poiani, Mean Cesàro summability of Laguerre and Hermite series, Trans. Amer. Math. Soc. 173 (1972), 1-31. MR 46:9635
8.: Elias Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press (1970). MR 44:7280
9.: Elias Stein and Guido Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press (1971). MR 46:4102
10.: Krzysztof Stempak, Almost everywhere summability of Laguerre series II, Stud. Math. 103 (3) (1992), 317-327. MR 92a:42053
11.: Gabor Szegö, Orthogonal Polynomials, fourth ed., Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, R.I., 1975. MR 51:8724
12.: Sundaram Thangavelu, Lectures on Hermite and Laguerre Expansions, Princeton Univ. Press (1993). MR 94i:42001
13.: -, Summability of Laguerre expansions, Analysis Math. 16 (1990), 303-315. MR 92h:33021
14.: David Webb, Pointwise estimates of the moduli of Cesàro-Laguerre kernels, to appear.

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 42C10

Benjamin Muckenhoupt
Affiliation: Department of Mathematics, Rutgers University, Piscataway, New Jersey 08854-8019
Email: muckenho@math.rutgers.edu

David W. Webb
Affiliation: Department of Mathematics, DePaul University, Chicago, Illinois 60614-3504
Email: dwebb@condor.depaul.edu

DOI: 10.1090/S0002-9947-00-02729-X
PII: S 0002-9947(00)02729-X
Keywords: Ces\`aro means, Laguerre expansions, Laguerre polynomials, two-weight norm inequalities, weighted norm inequalities
Received by editor(s): May 28, 1999
Posted: November 14, 2000
Copyright of article: Copyright 2000, American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6850(e) ISSN 0002-9947(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2008, American Mathematical Society Privacy Statement		Search the AMS