The $L^p$-continuity of wave operators for fractional order Schr\"odinger operators

We consider fractional Schrödinger operators $H=(-\Delta)^\alpha+V(x)$ in $n$ dimensions with real-valued potential $V$ when $n>2\alpha$, $\alpha>1$.

We show that the wave operators extend to bounded operators on $L^p(\mathbb R^n)$ for all $1\leq p\leq\infty$ under conditions on the potential that depend on $n$ and $\alpha$ analogously to the case when $\alpha\in \mathbb N$.

As a consequence, we deduce a family of dispersive and Strichartz estimates for the perturbed fractional Schrödinger operator.