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Weighted inequalities for the Stieltjes transform and the maximal spherical partial sum operator on radial functions*

Published online by Cambridge University Press:  14 November 2011

Kenneth F. Andersen
Kenneth F. Andersen
Affiliation:
Department of Mathematics, University of Alberta, Edmonton, Alberta, T6G-2G1, Canada e-mail: kanderse@vega.math.ualberta.ca
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Abstract

If TRf(x) is the spherical partial sum of the Fourier transform of f and T*f(x) = SUPR > 0 | TRf(x)|, sufficient conditions are given on the non-negative weight function ω(x) which ensure that T* restricted to radial functionsis bounded on the Lorentz space Lp,s(Rn,ω) into Lp,q(Rn, ω) For power weights, these conditions are also necessary. The weight pairs (u,v) for which the generalised Stieltjes transform Sλ is bounded from LP,S(R+, v)into Lp,q(R+, u)are also characterised. These are an essential ingredient for the study of T*.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 125 , Issue 1 , 1995 , pp. 195 - 204
Copyright
Copyright © Royal Society of Edinburgh 1995

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