Nonlocal operators of order near zero

TL;DR

This paper investigates nonlocal operators with very weak singularities, exploring their functional spaces and boundary value problems, and establishing key inequalities and compactness properties.

Contribution

It introduces new properties of Sobolev-type spaces for nonlocal operators with near-zero order kernels and applies these to boundary value problems.

Findings

Properties of Sobolev spaces for weakly singular kernels

Hardy inequalities and symmetrization estimates established

Analysis of Dirichlet and Neumann problems for these operators

Abstract

We study Dirichlet forms defined by nonintegrable L\'evy kernels whose singularity at the origin can be weaker than that of any fractional Laplacian. We show some properties of the associated Sobolev type spaces in a bounded domain, such as symmetrization estimates, Hardy inequalities, compact inclusion in $L^2$ or the inclusion in some Lorentz space. We then apply those properties to study the associated nonlocal operator $\mathfrak{L}$ and the Dirichlet and Neumann problems related to the equations $\mathfrak{L}u=f(x)$ and $\mathfrak{L}u=f(u)$ in $\Omega$.