Compactness and Its Applications of Sobolev Spaces Associated with Ball Banach Function Spaces

Let $N\in \mathbb{N}\cap [2,\mathrm{\infty }),$ $\mathrm{\Omega }$ be a bounded Lipschitz domain in ${\mathbb{R}}^N$, and $X(\mathrm{\Omega })$ be a ball Banach function space on $\mathrm{\Omega }\mathrm{.}$ In this article, under some mild assumptions, we establish a compactness theorem for Sobolev spaces associated with $X(\mathrm{\Omega })$. 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 $X(\mathrm{\Omega })$ that, for any $s\in (s_0,1)$ and $f\in X(\mathrm{\Omega })$, \begin{align*} |f-f_Ω|{X(Ω)} \lesssim(1-s)^{\frac 1q} \left|\left[\int_Ω\frac{|f(\cdot)-f(y)|^q}{|\cdot-y|^{N+sq}},dy \right]^{\frac{1}{q}} \right|{X(Ω)}, \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.