# Fractional Gaussian estimates and holomorphy of semigroups

## TL;DR

This paper demonstrates that certain fractional Gaussian estimates imply the holomorphicity of semigroups on various function spaces, and applies these results to fractional operators with Dirichlet boundary conditions.

## Contribution

It establishes that fractional Gaussian estimates ensure semigroup holomorphy and interpolation properties, extending the understanding of fractional Laplacian semigroups.

## Key findings

- Semigroups satisfying fractional Gaussian estimates are bounded holomorphic of angle π/2.
- Holomorphicity extends to spaces of continuous functions via duality.
- Results apply to fractional operators with exterior Dirichlet conditions.

## Abstract

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.