Fractional Gaussian estimates and holomorphy of semigroups

Let $\Omega\subset\R^N$ be an arbitrary open set and denote by $(e^{-t(-\Delta)_{\RR^N}^s})_{t\ge 0}$ (where $0<s<1$) the semigroup on $L^2(\RR^N)$ generated by the fractional Laplace operator. In the first part of the paper we show that if $T$ is a self-adjoint semigroup on $L^2(\Omega)$ satisfying a fractional Gaussian estimate in the sense that $|T(t)f|\le e^{-t(-\Delta)_{\RR^N}^s}|f|$, $0\le t \le 1$, $f\in L^2(\Omega)$, then $T$ defines a bounded holomorphic semigroup of angle $\frac{\pi}{2}$ that interpolates on $L^p(\Omega)$, $1\le p<\infty$. Using a duality argument we prove that the same result also holds on the space of continuous functions. In the second part, we apply the above results to realization of fractional order operators with the exterior Dirichlet conditions.