A note on endpoint $L^p$-continuity of wave operators for classical and higher order Schrödinger operators

We consider the higher order Schrödinger operator $H=(-\mathrm{\Delta }{)}^m+V(x)$ in $n$ dimensions with real-valued potential $V$ when $n>2m$, $m\in \mathbb{N}$. We adapt our recent results for $m>1$ to show that the wave operators are bounded on $L^p({\mathbb{R}}^n)$ for the full the range $1\le p\le \mathrm{\infty }$ in both even and odd dimensions without assuming the potential is small. The approach used works without distinguishing even and odd cases, captures the endpoints $p=1,\mathrm{\infty }$, and somehow simplifies the low energy argument even in the classical case of $m=1$.