## Abstract

Let n, k ∈ N with n ≥ 2 and 1 ≤ k < n. Given a positive function γ ∈ C

^{∞}(R^{n−k}) we form the Riemannian metric ~_{g}on R^{n}associated to the differential expression ds^{2}= |dx'|^{2}+γ(x')^{2}|dy|^{2}where we write R^{n}∋ x = (x', y) with x' ∈ R^{n−k }and y ∈ R^{k}. Let*ν*be a log-convex measure on R^{k }with smooth density and*µ*the product measure*µ*:= ρL^{n−k}⊗ν on R^{n}where ρ ∈ C(R^{n−k})is a positive function. We obtain a Pólya–Szegö inequality of the form*∫f(u, j(*∇*~*∇_{g}u)) dµ ≥ ∫ f(u^{s}, j(*~*_{g}us)) dµ*Rn Rn*

for Sobolev functions*u*where the operation ·^{s}refers to the (k, n)-Steiner symmetrisation withrespect to*ν*. The gradient operator ∇~_{g}is associated to the metric ~_{g}and the mapping*j*maybe seen as interpolating between the tangent space at*x*and R^{n}. The nonnegative integrand*f*is continuous and convex in the gradient variable and satisfies some additional hypotheses. As an application we derive a Pólya–Szegö inequality in the hyperbolic plane that takes theabove formOriginal language | English |
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Pages (from-to) | 390-432 |

Number of pages | 43 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 444 |

Issue number | 1 |

Early online date | 28 Jun 2016 |

DOIs | |

Publication status | Published - 1 Dec 2016 |

## Keywords

- Pólya–Szegö inequality
- Steiner symmetrisation