  We generalize the recent result of Erdo{\u g}an, Goldberg and Green on the
$L^p$-boundedness of wave operators for two dimensional Schr\"odinger operators
and prove that they are bounded in $L^p(\R^2)$ for all $1<p<\infty$ if and only
if the Schr\"odinger operator possesses no $p$-wave threshold resonances, viz.
Schr\"odinger equation $(-\lap + V(x))u(x)=0$ possesses no solutions which
satisfy $u(x)= (a_1x_1+a_2 x_2)|x|^{-2}+ o(|x|^{-1})$ as $|x|\to \infty$ for an
$(a_1, a_2) \in \R^2\setminus \{(0,0)\}$ and, otherwise, they are bounded in
$L^p(\R^2)$ for $1<p\leq 2$ and unbounded for $2<p<\infty$. We present also a
new proof for the known part of the result.
