The semigroup generated by the Dirichlet Laplacian of fractional order

TL;DR

This paper investigates the properties of the semigroup generated by the fractional Dirichlet Laplacian on arbitrary open sets, extending known heat semigroup estimates from the whole space to more general domains in Besov spaces.

Contribution

It provides new $L^p$-$L^q$ estimates and regularity results for the fractional Dirichlet Laplacian semigroup on arbitrary open sets, broadening the understanding of fractional PDEs in irregular domains.

Findings

Establishment of $L^p$-$L^q$ estimates for the semigroup

Extension of heat semigroup regularity to arbitrary open sets

Analysis of the semigroup in Besov spaces

Abstract

In the whole space $\mathbb R^d$, linear estimates for heat semi-group in Besov spaces are well established, which are estimates of $L^p$-$L^q$ type, maximal regularity, e.t.c. This paper is concerned with such estimates for semi-group generated by the Dirichlet Laplacian of fractional order in terms of the Besov spaces on an arbitrary open set of $\mathbb R^d$.