Regularity estimates for nonlocal space-time master equations in bounded domains

TL;DR

This paper establishes sharp regularity estimates for solutions to nonlocal space-time master equations in bounded domains, detailing boundary behavior and characterizing intermediate parabolic Hölder spaces.

Contribution

It provides the first sharp Schauder estimates for nonlocal space-time equations with boundary conditions and characterizes relevant Hölder spaces.

Findings

Sharp interior and global Schauder estimates are derived.

The boundary behavior of solutions is precisely characterized.

A new characterization of intermediate parabolic Hölder spaces is introduced.

Abstract

We obtain sharp parabolic interior and global Schauder estimates for solutions to nonlocal space-time master equations $(\partial_t +L)^su = f$ in $\mathbb{R} \times \Omega$, where $L$ is an elliptic operator in divergence form, subject to homogeneous Dirichlet and Neumann boundary conditions. In particular, we establish the precise behavior of solutions near the boundary. Along the way, we prove a characterization of the correct intermediate parabolic H\"older spaces in the spirit of Sergio Campanato.