Local compactness and nonvanishing for weakly singular nonlocal quadratic forms

Abstract

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 Poincaré inequality associated with ${\mathcal{E}}_j$ holds for unbounded domains of this type, thereby extending previously known results for bounded domains.