Classical solutions to integral equations with zero order kernels

TL;DR

This paper establishes higher-order regularity estimates and proves the existence of classical solutions for Dirichlet integral equations involving nonintegrable kernels with weak singularities, including the logarithmic Laplacian.

Contribution

It introduces new regularity estimates and existence results for classical solutions to integral equations with weakly singular kernels, extending the theory to logarithmic operators.

Findings

Higher-order log-Hölder regularity estimates for solutions.

Existence of classical solutions for logarithmic Laplacian and Schrödinger operators.

Applicable under mild regularity assumptions on the data.

Abstract

We show global and interior higher-order log-H\"older regularity estimates for solutions of Dirichlet integral equations where the operator has a nonintegrable kernel with a singularity at the origin that is weaker than that of any fractional Laplacian. As a consequence, under mild regularity assumptions on the right hand side, we show the existence of classical solutions of Dirichlet problems involving the logarithmic Laplacian and the logarithmic Schr\"odinger operator.