## Abstract

We prove that weakly differentiable weights w which, together with their reciprocals, satisfy certain local integrability conditions, admit a unique associated first-order *p*-Sobolev space, that is *H ^{1,p}(R^{d}, w dx) = V^{1,p}(R^{d}, w dx) = W^{1,p}(R^{d}, w dx)*, where

*d∈N*and

*p∈[1, ∞)*. If w admits a (weak) logarithmic gradient

*∇w/w*which is in

*L*, we propose an alternative definition of the weighted

^{q}_{loc}(w dx;**R**^{d}), q=p/(p-1)*p*-Sobolev space based on an integration by parts formula involving

*∇w/w*. We prove that weights of the form

*exp(-β*are

^{q}-W-V)*p*-admissible, in particular, satisfy a Poincaré inequality, where

*β∈(0, ∞)*,

*W*,

*V*are convex and bounded below such that

*|∇W|*satisfies a growth condition (depending on

*β*and

*q*) and

*V*is bounded. We apply the uniqueness result to weights of this type. The associated nonlinear degenerate evolution equation is also discussed.

Original language | English |
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Pages (from-to) | 3195-3223 |

Number of pages | 29 |

Journal | Journal of Functional Analysis |

Volume | 263 |

Issue number | 10 |

DOIs | |

Publication status | Published - 15 Nov 2012 |

MoE publication type | A1 Journal article-refereed |

## Keywords

- Density of smooth functions
- H=W
- Nonlinear degenerate parabolic equation
- Nonlinear Kolmogorov operator
- p-Laplace operator
- Poincaré inequality
- Smooth approximation
- Weighted p-Laplacian evolution
- Weighted Sobolev spaces