An integro-differential equation without continuous solutions

TL;DR

This paper presents an example of a non-symmetric integro-differential equation of order alpha where H"older estimates fail despite kernels resembling the fractional Laplacian, challenging existing regularity assumptions.

Contribution

It provides a counterexample demonstrating that H"older regularity may not hold for certain non-symmetric integro-differential equations, even with comparable kernels.

Findings

H"older estimates do not hold for the constructed example.

The example involves a non-symmetric kernel comparable to the fractional Laplacian.

Regularity results for symmetric kernels do not necessarily extend to non-symmetric cases.

Abstract

We show an example of a non-symmetric integro-differential equation of order $\alpha$, for $\alpha \in (0,1)$, for which H\"older estimates do not hold even though the kernels are comparable to the fractional Laplacian.