Best constants in Young's inequality, its converse, and its generalization to more than three functions

The best possible constant D_pqt in the inequality |∫∫ dx dy f(x)g(x - y) h(y)| ≤ D_pqt | f |_p | g |_q | h |_t, p, q, t ≥ 1, 1 p + 1 q + 1 t = 2, is determined; the equality is reached if f, g, and h are appropriate Gaussians. The same is shown to be true for the converse inequality (0 < p, q < 1, t < 0), in which case the inequality is reversed. Furthermore, an analogous property is proved for an integral of k functions over n variables, each function depending on a linear combination of the n variables; some of the functions may be taken to be fixed Gaussians. Two applications are given, one of which is a proof of Nelson's hypercontractive inequality.