Extension and trace for nonlocal operators

TL;DR

This paper establishes optimal extension and trace theorems for Sobolev spaces associated with nonlocal operators, using Poisson integrals to solve nonlocal Dirichlet problems and providing a Douglas-type formula.

Contribution

It introduces a new framework for extension and trace theorems in nonlocal Sobolev spaces, including explicit formulas for the quadratic form of the Poisson extension.

Findings

Proved an optimal extension theorem for nonlocal Sobolev spaces.

Developed a Poisson integral-based extension method.

Derived a Douglas-type formula for the quadratic form.

Abstract

We prove an optimal extension and trace theorem for Sobolev spaces of nonlocal operators. The extension is given by a suitable Poisson integral and solves the corresponding nonlocal Dirichlet problem. We give a Douglas-type formula for the quadratic form of the Poisson extension.