Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-17T15:08:41.061Z Has data issue: false hasContentIssue false
  • English
  • Français

Mellin multipliers and radially symmetric Riesz potentials

Published online by Cambridge University Press:  20 January 2009

W. Lamb  and
S. E. Schiavone
W. Lamb
Affiliation:
Department of Mathematics, University of Strathclyde, Livingstone Tower, 26 Richmond Street, Glasgow G1 1XH, Scotland
S. E. Schiavone
Affiliation:
Department of Mathematics, University of Alberta, Edmonton, Alberta T6G 2G1, Canada
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Riesz potentials with radially symmetric densities are examined from the standpoint of Mellin multipliers. Various results are deduced from the underlying multipliers, including a decomposition of the potential into a product of Erdélyi-Kober fractional integrals. Distributional versions of these results are also produced and shown to be valid under less severe restrictions on the parameters than those required in a weighted Lp setting.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 37 , Issue 3 , October 1994 , pp. 493 - 507
Copyright
Copyright © Edinburgh Mathematical Society 1994

References

REFERENCES

1.Erdélyi, A. et al. , Tables of integral transforms I (McGraw-Hill, New York, 1954).Google Scholar
2.Erdélyi, A. et al. , Higher transcendental functions I (McGraw-Hill, New York, 1953).Google Scholar
3.Lamb, W., Fourier multipliers on spaces of distributions, Proc. Edinburgh Math. Soc. 29(2) (1986), 309–327.Google Scholar
4.McBride, A. C., Fractional calculus and integral transforms of generalised functions (Pitman, London, 1979).Google Scholar
5.McBride, A. C., Fractional powers of a class of Mellin multiplier transforms I/II/III. Appl. Anal. 21 (1986), 89–173.CrossRefGoogle Scholar
6.McBride, A. C., Connections between fractional calculus and some Mellin multiplier transforms, in Univalent functions, fractional calculus and their applications (Srivastava, H. M. and Owa, S. (editors), Ellis Horwood, Chichester, 1989).Google Scholar
7.McBride, A. C. and Spratt, W. J., A class of Mellin multipliers, Canad. Math. Bull. 35 (1992) 252–260.Google Scholar
8.Okikiolu, G. O., Aspects of the theory of bounded integral operators in Lp-spaces (Academic Press, London, 1971).Google Scholar
9.Rooney, P. G., On the ranges of certain fractional integrals, Canad. J. Math. 24 (1972), 1198–1216.Google Scholar
10.Rooney, P. G., A technique for studying the boundedness and extendability of certain types of operators. Canad. J. Math. 25 (1973), 1090–1102.CrossRefGoogle Scholar
11.Rubin, B. S., One-dimensional representation, inversion, and certain properties of the Riesz potentials of radial functions Mat. Zametki 34 (1983), 521–533. (English translation: Math. Notes 34 (1983), 751–757).Google Scholar
12.Schiavone, S. E., Bilateral Laplace multipliers on spaces of distributions, Proc. Edinburgh Math. Soc 33(2) (1990), 461–474.Google Scholar