Pablo Raúl Stinga's <A HREF="https://arxiv.org/abs/1808.05159">User’s guide to the fractional Laplacian and the method of semigroups</A> (2018) may provide a helpful entry point to the literature. The semigroup connection is expressed by:

> The fractional Laplacian $L^s=(-\Delta)^s$, $0<s<1$ can be expressed
> in terms of the heat diffusion semigroup $v=e^{-tL}u$ generated by $L$
> acting on $u$ through the integral formula $$L^s
> u=\frac{1}{\Gamma(-s)}\int_0^\infty\left(e^{-tL}u-u\right)\frac{dt}{t^{1+s}}.$$
> The solution to $L^s u=f$ can then be written as
> $$u=\frac{1}{\Gamma(s)}\int_0^\infty e^{-tL}f\frac{dt}{t^{1-s}}.$$

This connection forms the starting point of the regularity study reviewed by Stinga, see in particular theorems 13-15 (Schauder–Hölder–Zygmund estimates).