  We show that wave operators for three dimensional Schr\"odinger operators
$H=-\Delta + V$ with threshold singularities are bounded in $L^1({\mathbb
R}^3)$ if and only if zero energy resonances are absent from $H$ and the
existence of zero energy eigenfunctions does not destroy the $L^1$-boundedness
of wave operators for $H$ with the regular threshold behavior. We also show in
this case that they are bounded in $L^p({\mathbb R}^3)$ for all $1\leq p \leq
\infty$ if all zero energy eigenfunctions $\phi(x)$ have vanishing first three
moments: $\int_{{\mathbb R}^3} x^\alpha V(x)\phi(x)dx=0$, $|\alpha|=0,1,2$.
