A Pólya–Szegö inequality in a Fermi metric

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Abstract

Let n, k ∈ N with n ≥ 2 and 1 ≤ k < n. Given a positive function γ ∈ C(Rn−k) we form the Riemannian metric ~g on Rn associated to the differential expression ds2 = |dx'|2 +γ(x')2|dy|2 where we write Rn ∋ x = (x', y) with x' ∈ Rn−k and y ∈ Rk. Let ν be a log-convex measure on Rk with smooth density and µ the product measure µ := ρL n−k⊗ν on Rn where ρ ∈ C(Rn−k)is a positive function. We obtain a Pólya–Szegö inequality of the form

  ∫f(u, j(~gu)) dµ ≥  ∫  f(us, j(~gus)) 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 Rn. 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 form
Original languageEnglish
Pages (from-to)390-432
Number of pages43
JournalJournal of Mathematical Analysis and Applications
Volume444
Issue number1
Early online date28 Jun 2016
DOIs
Publication statusPublished - 1 Dec 2016

Keywords

  • Pólya–Szegö inequality
  • Steiner symmetrisation

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