Local compactness and nonvanishing for weakly singular nonlocal quadratic forms

In this work we study a class of nonlocal quadratic forms given by $$\mathcal{E}_j(u,v)=\frac{1}{2}\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}(u(x)-u(y))(v(x)-v(y))j(x-y)\ dxdy,$$ where $j:\mathbb{R}^N\to[0,\infty]$ is a measurable even function with $\min\{1,|\cdot|^2\}j\in L^1(\mathbb{R}^N)$. Assuming merely $j\notin L^1(\mathbb{R}^N)$, we show local compactness of the embedding $\mathcal{D}^j(\mathbb{R}^N)\hookrightarrow L^2(\mathbb{R}^N)$, where $\mathcal{D}^j(\mathbb{R}^N)$ denotes the space of functions $u\in L^2(\mathbb{R}^N)$ with $\mathcal{E}_j(u,u)<\infty$. Using this local compactness, we establish an alternative which allows to distinguish vanishing and nonvanishing of bounded sequences in $\mathcal{D}^j(\mathbb{R}^N)$. As an application, we show the existence of maximizers for a class of integral functionals defined on the unit sphere in $\mathcal{D}^j(\mathbb{R}^N)$. Our main results extend to cylindrical unbounded sets of the type $\Omega = U \times \mathbb{R}^k$, where $U \subset \mathbb{R}^{N-k}$ is open and bounded. Finally, we note that a Poincar\'e inequality associated with $\mathcal{E}_j$ holds for unbounded domains of this type, thereby extending previously known results for bounded domains.