Type: Article
Publication Date: 1994-12-01
Citations: 6
DOI: https://doi.org/10.4153/cjm-1994-073-7
Abstract 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 c p,q (G) is the smallest constant c such that for all functions f, g ∈ C c (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 c p,q (G) ≤ 1. This paper contains three results about these constants. Firstly, if G contains a compact open subgroup then c p,q (G) = 1 and, as an extension of an earlier result of J. J. F. Fournier, it is shown that there is a constant c p,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 c p,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 L s (G), .