The $L^p$-boundedness of wave operators for four dimensional Schr\"odinger operators with threshold resonances

We prove that the low energy parts of the wave operators $W_\pm$ for Schr\"odinger operators $H = -\lap + V(x)$ on $\R^4$ are bounded in $L^p(\R^4)$ for $1<p\leq 2$ and are unbounded for $2<p\leq \infty$ if $H$ has resonances at the threshold. If $H$ has eigenfunctions only at the threshold, it has recently been proved that they are bounded in $L^p(\R^4)$ for $1\leq p<4$ in general and for $1\leq p<\infty$ if all threshold eigenfunctions $\ph$ satisfy $\int_{\R^4}x_j V(x) \ph(x)dx=0$ for $1\leq j\leq 4$. We prove in this case that they are unbounded in $L^p(\R^4)$ for $4<p<\infty$ unless the latter condition is satisfied. It is long known that the high energy parts are bounded in $L^p(\R^4)$ for all $1\leq p\leq \infty$ and that the same holds for $W_\pm$ if $H$ has no eigenfunctions nor resonances at the threshold.