Sharpness in young’s inequality for convolution

Abstract

Let p and q be indices in the open interval (1, ∞) such that pq < 1 such that, if G is a locally compact, unimodular group with no compact open subgroups, and if g and f are functions in LP(G) and Lq(G) respectively, then (Formula Presented) here g*f denotes the convolution of g and f. Thus, in this case, Young’s inequality for convolution is not sharp; this result is used to prove a similar statement about sharpness in Eunze’s extension of the Hausdorff-Young inequality. The best constants in these inequalities are known in many special cases; the methods used here do not yield good estimates for these constants, but they do lead to the first proof of nonsharpness for general unimodular groups without compact open subgroups. © 1977 Pacific Journal of Mathematics. All rights reserved.