A Pólya-Szegö inequality in a Fermi metric

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• 作者：I. McGillivray ; ^{maiemg@bristol.ac.uk" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Pó ; lya&ndash ; Szegö ; inequality ; Steiner symmetrisation
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2016
• 出版时间：1 December 2016
• 年：2016
• 卷：444
• 期：1
• 页码：390-432
• 全文大小：731 K

文摘

Let n,k∈N$n,k\in \mathbb{N}$ with n≥2$n\ge 2$ and 1≤k<n$1\le k<n$. Given a positive function γ∈C^∞(R^n−k)$\gamma \in C^{\infty }({\mathbb{R}}^{n-k})$ we form the Riemannian metric $\tilde{g}$ on Rⁿ${\mathbb{R}}^n$ associated to the differential expression ds²=|dx^′|²+γ(x^′)²|dy|²$d{s}^2=|d{x}^{\prime }{|}^2+\gamma {(x^{\prime })}^2|dy{|}^2$ where we write Rⁿ∋x=(x^′,y)${\mathbb{R}}^n\ni x=(x^{\prime },y)$ with x^′∈R^n−k$x^{\prime }\in {\mathbb{R}}^{n-k}$ and 24455" title="Click to view the MathML source">y∈R^k$y\in {\mathbb{R}}^k$. Let ν   be a log-convex measure on R^k${\mathbb{R}}^k$ with smooth density and μ   the product measure μ:=ρL^n−k⊗ν$\mu :=\rho {\mathcal{L}}^{n-k}\otimes \nu$ on Rⁿ${\mathbb{R}}^n$ where ρ∈C(R^n−k)$\rho \in C({\mathbb{R}}^{n-k})$ is a positive function. We obtain a Pólya–Szegö inequality of the form for Sobolev functions u   where the operation ⋅^s${\cdot }^s$ refers to the (k,n)$(k,n)$-Steiner symmetrisation with respect to ν  . The gradient operator ${\mathrm{\nabla }}_{\tilde{g}}$ is associated to the metric $\tilde{g}$ and the mapping j may be seen as interpolating between the tangent space at x   and Rⁿ${\mathbb{R}}^n$. 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 the above form.