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Sharpness in Young's Inequalityfor Convolution Products

Published online by Cambridge University Press:  20 November 2018

Ole A. Nielsen
Ole A. Nielsen*
Affiliation:
Queen's University Kingston, OntarioK7L 3N6
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Abstract

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Suppose that Gis a locally compact group with modular function Δ and that p, q, r are three numbers in the interval (l,∞) satisfying . If cp,q(G) is the smallest constant c such that for all functions f, g ∈ Cc(G) (here the convolution product is with respect to left Haar measure and is the exponent which is conjugate to p) then Young's inequality asserts that cp,q(G) ≤ 1. This paper contains three results about these constants. Firstly, if G contains a compact open subgroup then cp,q(G) = 1 and, as an extension of an earlier result of J. J. F. Fournier, it is shown that there is a constant cp,q < 1 such that if G does not contain a compact open subgroup then c<(G) ≤ c≤. Secondly, Beckner's calculation of is used to obtain the value of cp,q(G) for all simply-connected solvable Lie groups and all nilpotent Lie groups. And thirdly, it is shown that for a nilpotent Lie group the set is not contained in the union of the spaces Ls(G), .

Keywords

22E2522D0543A15
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 46 , Issue 6 , 01 December 1994 , pp. 1287 - 1298
Copyright
Copyright © Canadian Mathematical Society 1994

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