The $L^p$ boundedness of wave operators for Schr\"odinger Operators with threshold singularities

Let $H=-\mathrm{\Delta }+V$ be a Schr"odinger operator on $L^2({\mathbb{R}}^n)$ with real-valued potential $V$ for $n>4$ and let $H_0=-\mathrm{\Delta }$. If $V$ decays sufficiently, the wave operators $W_{\pm}=s-\lim_{t\to \pm\infty} e^{itH}e^{-itH_0}$areknowntobeboundedon$L^p(\mathbb R^n)$forall$1\leq p\leq \infty$ifzeroisnotaneigenvalue,andon$1<p<\frac{n}{2}$ if zero is an eigenvalue. We show that these wave operators are also bounded on $L^1({\mathbb{R}}^n)$ by direct examination of the integral kernel of the leading term. Furthermore, if ${\int }_{{\mathbb{R}}^n}V(x)\varphi (x)dx=0$ for all eigenfunctions $\varphi$, then the wave operators are $L^p$ bounded for $1\leq p<n$\mathrm{.}If,inaddition$\int_{\mathbb R^n} xV(x) \phi(x) , dx=0$, then the wave operators are bounded for $1\le p<\mathrm{\infty }$.