Sobolev spaces on arbitrary domains and semigroups generated by fractional Laplacian

TL;DR

This paper develops a framework for Sobolev spaces and semigroups associated with the fractional Laplacian on any domain in , including their properties and inequalities.

Contribution

It introduces a general procedure to define Sobolev spaces and semigroups for fractional Laplacians on arbitrary domains, extending existing theory.

Findings

Sobolev spaces are well-defined on arbitrary domains.

The semigroup generated by the fractional Laplacian is continuous and smoothing.

Key inequalities like Gagliardo-Nirenberg are established for these spaces.

Abstract

We describe a procedure to introduce Sobolev spaces and the semigroup generated by the fractional Dirichlet Laplacian on an arbitrary domain of $\R^d$. In particular, the well-definedness of the spaces of both non-homogeneous and homogeneous type together with their duality properties, embeddings, and Gagliardo-Nirenberg inequalities will be discussed. We also show the continuity and the smoothing property of the semigroup.