Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

The Hardy-Littlewood theorem on fractional integration for Laguerre series


Authors: Yūichi Kanjin and Enji Sato
Journal: Proc. Amer. Math. Soc. 123 (1995), 2165-2171
MSC: Primary 42C10; Secondary 42A45
DOI: https://doi.org/10.1090/S0002-9939-1995-1257113-2
MathSciNet review: 1257113
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The Hardy-Littlewood theorem on fractional integration for Fourier series says that if $ {I_\sigma }g \sim \sum\nolimits_{n \ne 0} {\vert n{\vert^{ - \sigma }}\hat g} (n){e^{\operatorname{int} }}$, then $ {I_\sigma }$ is bounded from $ {L^p}$ to $ {L^q}$, where $ 1 < p < q < \infty ,\frac{1}{q} = \frac{1}{p} - \sigma $. We shall establish an analogue of this theorem for the Laguerre function system $ \left\{ {{{\left( {\frac{{n!}}{{\Gamma (n + \alpha + 1)}}} \right)}^{\frac{1}{... ...\alpha (x){e^{ - \frac{x}{2}}}{x^{\frac{\alpha }{2}}}} \right\}_{n = 0}^\infty $.


References [Enhancements On Off] (What's this?)

  • [B] H. Bavinck, A special class of Jacobi series and some applications, J. Math. Anal. Appl. 37 (1972), 767-797. MR 0308475 (46:7589)
  • [D] J. Długosz, $ {L^p}$-multipliers for the Laguerre expansions, Colloq. Math. 54 (1987), 287-293.
  • [GT] G. Gasper and W. Trebels, Jacobi and Hankel multipliers of type $ (p,q),1 < p < q < \infty $, Math. Ann. 237 (1978), 243-251. MR 0510993 (58:23326)
  • [K] Y. Kanjin, A transplantation theorem for Laguerre series, Tôhoku Math. J. 43 (1991), 537-555. MR 1133867 (93a:42011)
  • [MS] B. Muckenhoupt and E. M. Stein, Classical expansions and their relation to conjugate harmonic functions, Trans. Amer. Math. Soc. 118 (1965), 17-92. MR 0199636 (33:7779)
  • [O] R. O'Neil, Convolution operators and $ L(p,q)$ spaces, Duke Math. J. 30 (1963), 129-142. MR 0146673 (26:4193)
  • [S] G. Szegő, Orthogonal polynomials, Amer. Math. Soc. Colloq. Publ., Amer. Math. Soc., Providence, RI, 1975.
  • [ST] K. Stempak and W. Trebels, On weighted transplantation and multipliers for Laguerre expansions, Math. Ann. 300 (1994), 203-219. MR 1299060 (95k:42052)
  • [SW] E. M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Univ. Press, Princeton, NJ, 1971. MR 0304972 (46:4102)
  • [T] S. Thangavelu, Lectures on Hermite and Laguerre expansions, Math. Notes, no. 42, Princeton Univ. Press, Princeton, NJ, 1993. MR 1215939 (94i:42001)
  • [Ti] E. C. Titchmarsh, Introduction to the theory of Fourier integrals, Chelsea, New York, 1986. MR 942661 (89c:42002)
  • [W] G. N. Watson, A treaties on the theory of Bessel functions, Cambridge Univ. Press, London, 1966. MR 1349110 (96i:33010)
  • [Z] A. Zygmund, Trigonometric series, 2nd ed., Cambridge Univ. Press, London and New York, 1968. MR 0236587 (38:4882)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 42C10, 42A45

Retrieve articles in all journals with MSC: 42C10, 42A45


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1995-1257113-2
Keywords: Laguerre polynomials, fractional integration, multipliers, transplantation
Article copyright: © Copyright 1995 American Mathematical Society

American Mathematical Society