标题： 马兹亚型界限对于分数Poincaré-Sobolev不等式中的精确常数 显示英文标题

标题： Maz'ya-type bounds for sharp constants in fractional Poincaré-Sobolev inequalities

摘要： 我们证明了与开集相关的分数Poincaré-Sobolev不等式中精确常数的估计，这些估计是基于其内径的非局部容度扩展。 这项工作建立在Maz'ya和Shubin以及第一位作者和Brasco之前在局部情况下获得的结果之上。 我们依赖于一个新的Maz'ya-Poincaré不等式，并且顺便也证明了新的分数Poincaré-Wirtinger型估计。 这些不等式在不同可微分数阶下的极限行为表现出精确性。 作为副产品，我们得到了一个关于齐次Sobolev空间$\mathcal{D}^{s,p}_0(\Omega)$在$L^q(\Omega)$中的嵌入的新准则，在亚临界区域和$p \le q < p^*_s$的情况下有效。 即使对于分数Laplacian的第一个特征值，我们的结果也是新的，并且包含了分数Cheeger常数正性的最优表征。 显示英文摘要

摘要： We prove estimates for the sharp constants in fractional Poincar\'e-Sobolev inequalities associated to an open set, in terms of a nonlocal capacitary extension of its inradius. This work builds upon previous results obtained in the local case by Maz'ya and Shubin and by the first author and Brasco. We rely on a new Maz'ya-Poincar\'e inequality and, incidentally, we also prove new fractional Poincar\'e-Wirtinger-type estimates. These inequalities display sharp limiting behaviours with respect to the fractional order of differentiability. As a byproduct, we obtain a new criterion for the embedding of the homogeneous Sobolev space $\mathcal{D}^{s,p}_0(\Omega)$ in $L^q(\Omega)$, valid in the subcritical regime and for $p \le q < p^*_s$. Our results are new even for the first eigenvalue of the fractional Laplacian and contain an optimal characterization for the positivity of the fractional Cheeger's constant.