Remarks on $L^p$-boundedness of wave operators for Schr\"odinger operators with threshold singularities

We consider the continuity property in Lebesgue spaces $L^p(\R^m)$ of wave operators $W_\pm$ of scattering theory for Schr\"odinger operator $H=-\lap + V$ on $\R^m$, $|V(x)|\leq C\ax^{-\delta}$ for some $\delta>2$ when $H$ is of exceptional type, i.e. $\Ng=\{u \in \ax^{-s} L^2(\R^m) \colon (1+ (-\lap)^{-1}V)u=0 \}\not=\{0\}$ for some $1/2<s<\delta-1/2$. It has recently been proved by Goldberg and Green for $m\geq 5$ that $W_\pm$ are bounded in $L^p(\R^m)$ for $1\leq p<m/2$, the same holds for $1\leq p<m$ if all $\f\in \Ng$ satisfy $\int_{\R^m} V\f dx=0$ and, for $1\leq p<\infty$ if in addition $\int_{\R^m} x_i V\f dx=0$, $i=1, \dots, m$. We make the results for $p>m/2$ more precise and prove in particular that these conditions are also necessary for the stated properties of $W_\pm$. We also prove that, for $m=3$, $W_\pm$ are bounded in $L^p(\R^3)$ for $1<p<3$ and that the same holds for $1<p<\infty$ if and only if all $\f\in \Ng$ satisfy $\int_{\R^3}V\f dx=0$ and $\int_{\R^3} x_i V\f dx=0$, $i=1, 2, 3$, simultaneously.