AI 中文总结
本文为非局部非凸泛函族建立了尖锐Poincaré不等式，确定了成立条件为γ≤-1，推广了Nguyen的结果，并应用于Γ-收敛、Sobolev与BV空间刻画及VMO正则性判据。
AI 中文摘要
设 $N\in\mathbb N$，$p\in[1,\infty)$，且 $Q_0:=(0,1)^N$。在本文中，我们为非局部非凸泛函 $$ \Phi_{\lambda,p}^\gamma(g;Q_0):=\lambda^p\iint_{\{(x,y)\in Q_0^2:\\,x\ne y,\\,|g(x)-g(y)|>\lambda|x-y|^{1+\frac{\gamma}{p}}\}} |x-y|^{\gamma-N}\\,dx\\,dy, $$ 建立尖锐的Poincaré不等式，其中 $g:Q_0\to\mathbb R$ 是可测的，$\lambda\in(0,\infty)$，且 $\gamma\in\mathbb R$。具体地，我们证明估计式 $$ \int_{Q_0}\int_{Q_0}|g(x)-g(y)|^p\\,dx\\,dy \lesssim\lambda^p+\Phi_{\lambda,p}^\gamma(g;Q_0) $$ 对任意这样的 $g$ 和 $\lambda$ 成立，当且仅当 $\gamma\in(-\infty,-1]$，其中隐含的正常数仅依赖于 $N$、$p$ 和 $\gamma$。这推广了Nguyen关于$\gamma=-p$的Poincaré不等式，并为通过这些泛函研究正则性和紧性提供了有用工具。作为应用，我们获得了$\Phi^\gamma_{\lambda,p}$的$\Gamma$-收敛的下界，以及Sobolev和BV空间的相关渐近刻画。此外，我们确定了$\gamma$的最优范围，使得Rellich--Kondrachov定理和涉及固定阈值下$\Phi^\gamma_{\lambda,p}$的VMO正则性判据均成立。这些结果也回答了H.-M. Nguyen在[C. R. Math. Acad. Sci. Paris 363 (2025)]中提出的关于非局部Poincaré不等式的问题。对于$\gamma\in(-\infty,-1]$，它们进一步回答了同一篇文章和H. Brezis等人在[Anal. PDE 17 (2024)]中提出的关于Sobolev和BV刻画的问题。
英文摘要
Let $N\in\mathbb N$, $p\in[1,\infty)$, and $Q_0:=(0,1)^N$. In this article, we establish sharp Poincaré inequalities for the nonlocal nonconvex functionals $$ Φ_{λ,p}^γ(g;Q_0) :=λ^p\iint_{\{(x,y)\in Q_0^2:\,x\ne y,\,|g(x)-g(y)|>λ|x-y|^{1+\fracγ{p}}\}} |x-y|^{γ-N}\,dx\,dy, $$ where $g:Q_0\to\mathbb R$ is measurable, $λ\in(0,\infty)$, and $γ\in\mathbb R$. Specifically, we prove that the estimate $$ \int_{Q_0}\int_{Q_0}|g(x)-g(y)|^p\,dx\,dy \lesssimλ^p+Φ_{λ,p}^γ(g;Q_0) $$ holds for any such $g$ and $λ$ if and only if $γ\in(-\infty,-1]$, where the implicit positive constant depends only on $N$, $p$, and $γ$. This extends Nguyen's Poincaré inequality for $γ=-p$ and provides a useful tool for studying regularity and compactness through these functionals. As applications, we obtain a lower bound for the $Γ$-convergence of $Φ^γ_{λ,p}$ and related asymptotic characterizations for Sobolev and BV spaces. Moreover, we identify the optimal range of $γ$ for which both the Rellich--Kondrachov theorem and a criterion for VMO regularity involving $Φ^γ_{λ,p}$ at a fixed threshold hold. These results also answer the question on nonlocal Poincaré inequalities posed by H.-M. Nguyen in [C. R. Math. Acad. Sci. Paris 363 (2025)]. For $γ\in(-\infty,-1]$, they further answer questions on Sobolev and BV characterizations posed in the same article and by H. Brezis et al. in [Anal. PDE 17 (2024)].
