### Abstract

The well-known duality between the Sobolev inequality and the Hardy–Littlewood–Sobolev inequality suggests that the Nash inequality could also have an interesting dual form, even though the Nash inequality relates three convex functionals instead of two. We provide such a dual form here with sharp constants. This dual inequality relates the L
^{2}
norm to the infimal convolution of the L∞ and H
^{-1}
norms. The computation of this infimal convolution is a minimization problem, which we solve explicitly, thus providing a new proof of the sharp Nash inequality itself. This proof, via duality, also yields the sharp form of some new, weighted generalizations of the Nash inequality as well as the dual of these weighted variants.

Original language | English (US) |
---|---|

Pages (from-to) | 129-144 |

Number of pages | 16 |

Journal | Bulletin of the London Mathematical Society |

Volume | 51 |

Issue number | 1 |

DOIs | |

State | Published - Feb 1 2019 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Keywords

- 26A51
- 46E35 (primary)
- 46N10 (secondary)

### Cite this

*Bulletin of the London Mathematical Society*,

*51*(1), 129-144. https://doi.org/10.1112/blms.12220

}

*Bulletin of the London Mathematical Society*, vol. 51, no. 1, pp. 129-144. https://doi.org/10.1112/blms.12220

**A dual form of the sharp Nash inequality and its weighted generalization.** / Carlen, Eric; Lieb, Elliott H.

Research output: Contribution to journal › Article

TY - JOUR

T1 - A dual form of the sharp Nash inequality and its weighted generalization

AU - Carlen, Eric

AU - Lieb, Elliott H.

PY - 2019/2/1

Y1 - 2019/2/1

N2 - The well-known duality between the Sobolev inequality and the Hardy–Littlewood–Sobolev inequality suggests that the Nash inequality could also have an interesting dual form, even though the Nash inequality relates three convex functionals instead of two. We provide such a dual form here with sharp constants. This dual inequality relates the L 2 norm to the infimal convolution of the L∞ and H -1 norms. The computation of this infimal convolution is a minimization problem, which we solve explicitly, thus providing a new proof of the sharp Nash inequality itself. This proof, via duality, also yields the sharp form of some new, weighted generalizations of the Nash inequality as well as the dual of these weighted variants.

AB - The well-known duality between the Sobolev inequality and the Hardy–Littlewood–Sobolev inequality suggests that the Nash inequality could also have an interesting dual form, even though the Nash inequality relates three convex functionals instead of two. We provide such a dual form here with sharp constants. This dual inequality relates the L 2 norm to the infimal convolution of the L∞ and H -1 norms. The computation of this infimal convolution is a minimization problem, which we solve explicitly, thus providing a new proof of the sharp Nash inequality itself. This proof, via duality, also yields the sharp form of some new, weighted generalizations of the Nash inequality as well as the dual of these weighted variants.

KW - 26A51

KW - 46E35 (primary)

KW - 46N10 (secondary)

UR - http://www.scopus.com/inward/record.url?scp=85056705442&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85056705442&partnerID=8YFLogxK

U2 - 10.1112/blms.12220

DO - 10.1112/blms.12220

M3 - Article

VL - 51

SP - 129

EP - 144

JO - Bulletin of the London Mathematical Society

JF - Bulletin of the London Mathematical Society

SN - 0024-6093

IS - 1

ER -