### Abstract

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.

Original language | English (US) |
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Pages (from-to) | 151-173 |

Number of pages | 23 |

Journal | Advances in Mathematics |

Volume | 20 |

Issue number | 2 |

DOIs | |

State | Published - May 1976 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

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## Cite this

*Advances in Mathematics*,

*20*(2), 151-173. https://doi.org/10.1016/0001-8708(76)90184-5