On the $L^p$ boundedness of wave operators for four-dimensional Schr\"odinger Operators with a threshold eigenvalue

Let $H=-\Delta+V$ be a Schr\"odinger operator on $L^2(\mathbb R^4)$ with real-valued potential $V$, and let $H_0=-\Delta$. If $V$ has sufficient pointwise decay, the wave operators $W_{\pm}=s-\lim_{t\to \pm\infty} e^{itH}e^{-itH_0}$ are known to be bounded on $L^p(\mathbb R^4)$ for all $1\leq p\leq \infty$ if zero is not an eigenvalue or resonance, and on $\frac43<p<4$ if zero is an eigenvalue but not a resonance. We show that in the latter case, the wave operators are also bounded on $L^p(\mathbb R^4)$ for $1\leq p\leq \frac43$ by direct examination of the integral kernel of the leading terms. Furthermore, if $\int_{\mathbb R^4} xV(x) \psi(x) \, dx=0$ for all zero energy eigenfunctions $\psi$, then the wave operators are bounded on $L^p$ for $1 \leq p<\infty$.