On the $L^p$ boundedness of wave operators for two-dimensional Schr\"odinger operators with threshold obstructions

Let $H=-\mathrm{\Delta }+V$ be a Schr"odinger operator on $L^2({\mathbb{R}}^2)$ with real-valued potential $V$, and let $H_0=-\mathrm{\Delta }$. If $V$ has sufficient pointwise decay, the wave operators $W_{\pm}=s-\lim_{t\to \pm\infty} e^{itH}e^{-itH_0}$areknowntobeboundedon$L^p(\mathbb R^2)$forall$1< p< \infty$ if zero is not an eigenvalue or resonance. We show that if there is an s-wave resonance or an eigenvalue only at zero, then the wave operators are bounded on $L^p({\mathbb{R}}^2)$ for $1<p<\mathrm{\infty }$. This result stands in contrast to results in higher dimensions, where the presence of zero energy obstructions is known to shrink the range of valid exponents $p$.