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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: 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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  • 1. R. Askey and S. Wainger. Mean Convergence of Expansions in Laguerre and Hermite Series, Amer. J. of Math., 87 (1965), 695-707. MR 32:316
  • 2. A. Bonami and J.-L. Clerc, Sommes de Cesàro et multiplicateurs des développements en harmoniques sphériques, Trans. Amer. Math. Soc. 183 (1973), 223-263. MR 49:3461
  • 3. B. Muckenhoupt. Asymptotic Forms for Laguerre Polynomials, Proc. Amer. Math. Soc., Vol. 24 (1970), 288-292. MR 40:4503
  • 4. B. Muckenhoupt and D. Webb, Two-weight norm inequalities for Cesàro means of Laguerre expansions, Trans. Amer. Math. Soc. 353 (2001), 1119-1149. MR 2001m:42051
  • 5. E. Poiani. Mean Cesàro Summability of Laguerre and Hermite Series, Trans. Amer. Math. Soc., 173 (1972), 1-31. MR 46:9635
  • 6. G. Szegö. Orthogonal Polynomials, fourth ed., Amer. Math. Soc. Colloq. Publ. vol. 23, Amer. Math. Soc., Providence, RI, 1975. MR 51:8724
  • 7. S. Thangavelu. Lectures on Hermite and Laguerre Expansions, Princeton Univ. Press (1993). MR 94i:42001
  • 8. D. Webb. Pointwise estimates of the moduli of Cesàro-Laguerre kernels, to appear.

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Additional Information

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

David W. Webb
Affiliation: Department of Mathematics, DePaul University, Chicago, Illinois 60614-7807

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

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