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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 $j\notinL^1(\mathbb{R}^N)$, 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 $u\inL^2(\mathbb{R}^N)$ with $\mathcal{E}_j(u,u)<\infty$. Using this localcompactness, we establish an alternative which allows to distinguish vanishingand nonvanishing of bounded sequences in $\mathcal{D}^j(\mathbb{R}^N)$. As anapplication, we show the existence of maximizers for a class of integralfunctionals defined on the unit sphere in $\mathcal{D}^j(\mathbb{R}^N)$. Ourmain 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 knownresults for bounded domains.

Local compactness and nonvanishing for weakly singular nonlocal quadratic forms