Trudinger不等式对$M^{1,s}$空间成立

中文摘要 AI 辅助

我们在度量测度空间上证明了$M^{1,s}$ Sobolev空间的Trudinger不等式，其指数为$\frac{s}{s-1}$，仅需满足测度增长假设$\mu(B(x,r))\ge br^s$。这改进了先前已知的幂次为1的指数可积性。我们的论证还给出了当$p\uparrow s$时Sobolev常数的渐近精确界。在倍测度空间上，我们进一步去除了与$(1,p)$-Poincaré不等式相关的Trudinger不等式在$p<s$时的连通性假设。一个Ahlfors正则反例表明，在临界指数$p=s$时该推广不成立。

英文摘要

We prove the Trudinger inequality with exponent $\frac{s}{s-1}$ for $M^{1,s}$ Sobolev spaces on metric measure spaces under the sole measure growth assumption $μ(B(x,r))\ge br^s$. This improves the previously known exponential integrability with power one. Our argument also yields sharp asymptotic bounds for the Sobolev constants as $p\uparrow s$. On doubling spaces, we further remove the connectedness assumption from the Trudinger inequality associated with a $(1,p)$-Poincaré inequality when $p<s$. An Ahlfors regular counterexample shows that this extension fails at the critical exponent $p=s$.

发表机构

• Amherst College（阿默斯特学院）
• University of Pittsburgh（匹兹堡大学）

机构由 AI 辅助整理，请以论文原文为准。