## Abstract

We prove a new inequality which improves on the classical Hardy inequality in the sense that a nonlinear integral quantity with super-quadratic growth, which is computed with respect to an inverse square weight, is controlled by the energy. This inequality differs from standard logarithmic Sobolev inequalities in the sense that the measure is neither Lebesgue's measure nor a probability measure. All terms are scale invariant. After an Emden-Fowler transformation, the inequality can be rewritten as an optimal inequality of logarithmic Sobolev type on the cylinder. Explicit expressions of the sharp constant, as well as minimizers, are established in the radial case. However, when no symmetry is imposed, the sharp constants are not achieved by radial functions, in some range of the parameters.

Original language | English |
---|---|

Pages (from-to) | 2045-2072 |

Number of pages | 28 |

Journal | Journal of Functional Analysis |

Volume | 259 |

Issue number | 8 |

DOIs | |

Publication status | Published - 15 Oct 2010 |

## Keywords

- Caffarelli-Kohn-Nirenberg inequalities
- Emden-Fowler transformation
- Hardy inequality
- Hardy-Sobolev inequalities
- Interpolation
- Logarithmic Sobolev inequality
- Radial symmetry
- Scale invariance
- Sobolev inequality
- Symmetry breaking

## ASJC Scopus subject areas

- Analysis