Vol. 72, No. 2, 1977

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Sharpness in Young’s inequality for convolution

John J. F. Fournier

Vol. 72 (1977), No. 2, 383–397

Let p and q be indices in the open interval (1,) such that pq < p + q; let r = pq∕(p + q pq). It is shown here that there is a constant Cp,q < 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

∥g ∗f∥r ≦ Cp,q∥g∥p∥f∥q;

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 Kunze’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.

Mathematical Subject Classification 2000
Primary: 43A15
Received: 27 January 1977
Published: 1 October 1977
John J. F. Fournier