基于外推法的退化Sobolev不等式与Poincaré不等式
Degenerate Sobolev and Poincaré inequalities via extrapolation
浏览论文内容
中文总结 AI 辅助
本文利用Rubio de Francia外推技术，在满足特定条件的权重与矩阵函数下，证明退化Sobolev与Poincaré不等式，并在John域等示例中验证相关结果。
中文摘要 AI 辅助
本文利用Rubio de Francia外推理论衍生的技术，证明带矩阵权重的Sobolev不等式与Poincaré不等式。给定定义在欧氏空间ℝⁿ的连通开子集Ω上的权重w、v及对称非负定矩阵值函数Q，满足下椭圆性条件：w(x)^p ≤ |√Q(x)ξ|^p（ξ∈ℝⁿ），本文给出权重w、v的Lebesgue可积性条件，确保存在τ≥1，使得对光滑函数u，形如(∫_Ω |u|^{τp} v dx)^{1/(τp)} ≤ C(v,w)(∫_Ω |√Q∇u|^p dx)^{1/p}的Sobolev不等式，以及形如(∫_Ω |u - ⟨u⟩_{Ω,v}|^{τp} v dx)^{1/(τp)} ≤ C(v,w)(∫_Ω |√Q∇u|^p dx)^{1/p}的Poincaré不等式成立。本文还在John域、Heisenberg群、CR流形等多个示例的语境下，探讨了这些结果及相关结论。
英文摘要
In this paper we prove matrix weighted Sobolev and Poincaré inequalities using techniques derived from the theory of Rubio de Francia extrapolation. Given weights $w,\,v$ and a symmetric non-negative definite matrix valued function $Q$ defined on a connected open subset $Ω$ of $\mathbb{f}R^n$ that satisfies the lower ellipticity condition \[ w(x)^p \leq |\sqrt{Q(x)}ξ|^p,\quad ξ\in \mathbb{R}^n, \] we give Lebesgue integrability conditions on the weights $w,v$ that ensure there exists $τ\geq 1$ so that Sobolev and Poincaré inequalities of the form \[\bigg(\int_Ω|u|^{τp} \,vdx\bigg)^{\frac{1}{τp}} \leq C(v,w) \bigg(\int_Ω|\sqrt{Q}\nabla u|^p\,dx\bigg)^{\frac{1}{ p}},\textrm{ and}\] \[\bigg(\int_Ω|u-\langle u\rangle_{Ω,v}|^{τp} \,v dx\bigg)^\frac{1}{τp} \leq C(v,w)\bigg(\int_Ω|\sqrt{Q}\nabla u|^p \, dx\bigg)^{\frac{1}{p}}\] hold for smooth $u$. We explore these and related results in the context of several examples that include John domains, the Heisenberg group, and CR manifolds.
