  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 $\mathbb R^d$.
