Magnetic fractional Poincar\'e inequality in punctured domains

We study Poincar\'e-Wirtinger type inequalities in the framework of magnetic fractional Sobolev spaces. In the local case, Lieb-Seiringer-Yngvason [E. Lieb, R. Seiringer, and J. Yngvason, Poincar\'e inequalities in punctured domains, Ann. of Math., 2003] showed that, if a bounded domain $\Omega$ is the union of two disjoint sets $\Gamma$ and $\Lambda$, then the $L^p$-norm of a function calculated on $\Omega$ is dominated by the sum of magnetic seminorms of the function, calculated on $\Gamma$ and $\Lambda$ separately. We show that the straightforward generalisation of their result to nonlocal setup does not hold true in general. We provide an alternative formulation of the problem for the nonlocal case. As an auxiliary result, we also show that the set of eigenvalues of the magnetic fractional Laplacian is discrete.