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A note on the Stieltjes transformation

Published online by Cambridge University Press:  14 November 2011

Gordon Sinnamon
Affiliation:
Department of Pure Mathematics, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, Canada, N2L 3G1

Synopsis

A necessary and sufficient condition for the boundedness of the generalised Stieltjes transformation on weighted Lebesgue spaces is obtained and applied to extend the Hilbert double series theorem.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 1988

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References

1Andersen, K. F.. Weighted inequalities for the Stieltjes transformation and Hilbert's double series. Proc. Roy. Soc. Edinburgh Sect. A 86 (1980), 7584.CrossRefGoogle Scholar
2Andersen, K. F. and Muckenhoupt, B.. Weighted weak type Hardy inequalities with applications to Hilbert transforms and maximal functions. Studia Math. 72 (1982), 926.CrossRefGoogle Scholar
3Bradley, J. S.. Hardy inequalities with mixed norms. Canad. Math. Bull. 21 (1978), 405408.CrossRefGoogle Scholar
4Erdeélyi, A.. The Stieltjes transformation on weighted L p spaces. Appl. Anal. 7 (1978), 213219.CrossRefGoogle Scholar
5Maz'ja, V.. Einbettungssätze für Sobolewsche Räume I, II. Teubner-Texte zur Mathematik, Bd. 21, 28 (Leipzig: Teubner, 1979, 1981); translated in V. Maz'ja, Sobolev Spaces (Berlin: Springer, 1985).Google Scholar
6Sinnamon, G.. Operators on Lebesgue Spaces with General Measures (Doctoral thesis, McMaster University, 1987).Google Scholar
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