The best possible constant Dpqt in the inequality |∫∫ dx dy f(x)g(x - y) h(y)| ≤ Dpqt | 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.
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