Dispersive estimates for fractional order Schr\"odinger operators

We prove dispersive bounds for fractional Schr\"odinger operators on $\mathbb R^n$ of the form $H=(-\Delta)^{\alpha}+V$ with $V$ a real-valued, decaying potential and $\alpha \notin\mathbb N$. We derive pointwise bounds on the resolvent operators for all $0<\alpha<\frac{n}{2}$, a quantitative limiting absorption principle for $\frac12<\alpha<\frac{n}{2}$, and establish global dispersive estimates in dimension $n\geq 2$ for the range $\frac{n+1}{4}\leq \alpha <\frac{n}2$.