TY - JOUR

T1 - Best constants in Young's inequality, its converse, and its generalization to more than three functions

AU - Brascamp, Herm Jan

AU - Lieb, Elliott H.

PY - 1976/5

Y1 - 1976/5

N2 - 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.

AB - 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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U2 - 10.1016/0001-8708(76)90184-5

DO - 10.1016/0001-8708(76)90184-5

M3 - Article

AN - SCOPUS:0001126703

VL - 20

SP - 151

EP - 173

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

IS - 2

ER -