Two-weight norm inequalities for the Cesàro means of Hermite expansions

Authors:
Benjamin Muckenhoupt and David W. Webb

Journal:
Trans. Amer. Math. Soc. **354** (2002), 4525-4537

MSC (2000):
Primary 42C10

Published electronically:
July 2, 2002

MathSciNet review:
1926887

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Abstract | References | Similar Articles | Additional Information

Abstract: An accurate estimate is obtained of the Cesàro kernel for Hermite expansions. This is used to prove two-weight norm inequalities for Cesàro means of Hermite polynomial series and for the supremum of these means. These extend known norm inequalities, even in the single power weight and ``unweighted'' cases. An almost everywhere convergence result is obtained as a corollary. It is also shown that the conditions used to prove norm boundedness of the means and most of the conditions used to prove the boundedness of the Cesàro supremum of the means are necessary.

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Additional Information

**Benjamin Muckenhoupt**

Affiliation:
Department of Mathematics, Rutgers University, 110 Frelinghuysen Road, Piscataway, New Jersey 08854-8019

Email:
muckenho@math.rutgers.edu

**David W. Webb**

Affiliation:
Department of Mathematics, DePaul University, Chicago, Illinois 60614-7807

Email:
dwebb@condor.depaul.edu

DOI:
http://dx.doi.org/10.1090/S0002-9947-02-03093-3

Keywords:
Ces\`{a}ro means,
Hermite expansions,
Hermite polynomials,
two-weight norm inequalities,
weighted norm inequalities

Received by editor(s):
September 28, 2001

Received by editor(s) in revised form:
October 2, 2001

Published electronically:
July 2, 2002

Additional Notes:
The second author was supported by a URC grant at DePaul University.

Article copyright:
© Copyright 2002
American Mathematical Society