Sharp Nonlocal Poincar\'e Inequalities and Liminf Lower Bounds for Sobolev and BV Energies on Metric Measure Spaces

arXiv:2610.40743v1 Announce Type: new Abstract: Let $p\in[1,\infty)$, and let $(X,\rho,\mu)$ be a complete doubling $q$-Poincar\'e space for some $q\in[1,p]$ with a lower dimension $d$ and an upper dimension $D$. In this article, we establish Poincar\'e inequalities for two families of nonlocal nonconvex functionals defined using distances or volumes. More precisely, for any ball $B$ with its radius $r_B\in(0,\operatorname{diam} X)$, any $\lambda\in(0,\infty)$, and any measurable function $f:X\to\mathbb R$, \begin{align*} &\frac{1}{\mu(B)^2}\int_B\int_B|f(x)-f(y)|^p\,d\mu(x)\,d\mu(y)\\ &\quad\lesssim\lambda^pr_B^p\phi(B)^{\gamma p}+\frac{\lambda^pr_B^p}{\mu(B)}\mathop{\int_{\Lambda B}\int_{\Lambda B}}\limits_{\genfrac{}{}{}{2}{|f(x)-f(y)|}{\rho(x,y)\phi(x,y)^\gamma}>\lambda}\frac{\phi(x,y)^{\gamma p}}{V(x,y)}\,d\mu(x)\,d\mu(y), \end{align*} where $V(x,y):=\mu(B(x,\rho(x,y)))$ and either $\phi=\rho$ with $\phi(B)=r_B$ and $\gamma\in(-\infty,-\frac{\min\{q,D\}}{p}]$ or $\phi=V$ with $\phi(B)=\mu(B)$ and $\gamma\in(-\infty,-\frac{q}{p\max\{q,d\}}]$. Here both ranges of $\gamma$ are sharp over the class of $q$-Poincar\'e spaces with the prescribed lower and upper dimensions, and the constant $\Lambda\in[1,\infty)$ and the implicit positive constant depend only on $p$, $\gamma$, and the geometric constants of $X$. We also obtain lower bounds for Sobolev energy and total variation in terms of the limit inferior of these functionals, for $\gamma$ in either the above negative ranges or $(0,\infty)$. Our proofs use the curve characterization of Poincar\'e spaces and an adaptive dyadic refinement argument based on medians.

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