Best Constants in Young’s Inequality, Its Converse, and Its Generalization to More than Three Functions

The best possible constant Dmt in the inequality | ∬ dx dyf(x)g(x —y) h(y)|< D_pgt||f||p||g||Q||h||t, p,q,t>|, 1/p + llq+ 1/t = 2, is determined; the equality is reached if /, 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 pr∞f of Nelson’s hypercontractive inequality.