  Let $H=-\Delta+V$ be a Schr\"odinger operator on $L^2(\mathbb R^2)$ 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 $L^p(\mathbb R^2)$ for all $1< p<
\infty$ 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 $1 < p<\infty$. 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 $p$.
