Abstract:Let $N\in\mathbb{N}\cap[2,\infty),$ $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^N$, and $X(\Omega)$ be a ball Banach function space on $\Omega.$ In this article, under some mild assumptions, we establish a compactness theorem for Sobolev spaces associated with $X(\Omega)$. Different from the fractional Sobolev space, our proof is based on an elaborate decomposition of bounded Lipschitz domains and its corresponding weighted fractional Poincaré inequality on each piece. As applications, we obtain the fractional Poincaré inequality in $X(\Omega)$ that, for any $s\in (s_0,1)$ and $f\in X(\Omega)$, \begin{align*} \|f-f_\Omega\|_{X(\Omega)} \lesssim(1-s)^{\frac 1q} \left\|\left[\int_\Omega \frac{|f(\cdot)-f(y)|^q}{|\cdot-y|^{N+sq}}\,dy \right]^{\frac{1}{q}} \right\|_{X(\Omega)}, \end{align*} where $s_0$ is a given positive constant and the implicit positive constant is independent of $s$ and $f$. Using this, we further establish the well-posedness of a weighted Triebel--Lizorkin type nonlocal variational problem. These results are of wide generality and, even when they are applied to Morrey spaces, weighted Lebesgue spaces, mixed-norm Lebesgue spaces, variable Lebesgue spaces, Orlicz spaces, and Orlicz-slice spaces, the obtained results are also new.
From: Dachun Yang [view email]
[v1]
Mon, 22 Jun 2026 10:28:43 UTC (30 KB)
此内容由惯性聚合(RSS阅读器)自动聚合整理，仅供阅读参考。 原文来自 — 版权归原作者所有。