Magnetic fractional Poincar\'e inequality in punctured domains

TL;DR

This paper investigates magnetic fractional Poincaré inequalities in punctured domains, showing that direct generalizations from local to nonlocal cases fail and providing alternative formulations, along with spectral properties of the magnetic fractional Laplacian.

Contribution

It demonstrates the failure of straightforward nonlocal generalizations of local inequalities and proposes an alternative approach, also establishing the discreteness of the magnetic fractional Laplacian's eigenvalues.

Findings

Straightforward nonlocal generalization does not hold.

An alternative formulation for the nonlocal case is provided.

Eigenvalues of the magnetic fractional Laplacian are discrete.

Abstract

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.