The semigroup generated by the Dirichlet Laplacian of fractional order

Abstract

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