An Inequality for the Convolutions on Unimodular Locally Compact Groups and the Optimal Constant of Young's Inequality

Let mu be the Haar measure of a unimodular locally compact group G and m(G) as the infimum of the volumes of all open subgroups of G. The main result of this paper is that integral(G) f circle (phi(1) * phi(2)) (g) dg <= integral(R) f circle (phi(1)* * phi(2)*)(x) dx holds for any measurable functions phi(1), phi(2): G -> R->= 0 with mu(supp phi(1)) + mu(supp phi(2)) <= m(G) and any convex function f : R->= 0 -> R with f (0) = 0. Here phi* is the rearrangement of phi. Let Y-O(P, G) and Y-R(P, G) denote the optimal constants of Young's and the reverse Young's inequality, respectively, under the assumption mu(supp phi(1)) + mu(supp phi(2)) <= m(G). Then we have Y-O(P, G) <= Y-O(P, R) and Y-R(P, G) >= Y-R(P, R) as a corollary. Thus, we obtain that m(G) = infinity if and only if H(p, G) <= H(p, R) in the case of p ' := p/(p - 1) is an element of 2Z, where H(p, G) is the optimal constant of the Hausdorff-Young inequality.