## The $L^p$ boundedness of wave operators for Schr\"odinger operators with threshold singuralities I. The odd dimensional case

J. Math. Sci. Univ. Tokyo
Vol. 13 (2006), No. 1, Page 43--93.

Yajima, Kenji
The $L^p$ boundedness of wave operators for Schr\"odinger operators with threshold singuralities I. The odd dimensional case
Let $H=-\lap +V(x)$ be an odd $m$-dimensional Schr\"odinger operator, $m \geq 3$, $H_0=-\lap$, and let ${\ds W_\pm=\lim_{t\to \pm \infty} e^{itH}e^{-itH_0}}$ be the wave operators for the pair $(H, H_0)$. We say $H$ is of generic type if $0$ is not an eigenvalue nor a resonance of $H$ and of exceptional type if otherwise. We assume that $V$ satisfies $\Fg(\ax^{-2\s}V) \in L^{m_\ast}$ for some $\s>\frac{1}{m_\ast}$, $m_\ast=\frac{m-1}{m-2}$. We show that $W_\pm$ are bounded in $L^p(\R^m)$ for all $1\leq p \leq \infty$ if $V$ satisfies in addition $|V(x)|\leq C \ax^{-m-2-\ep}$ for some $\ep>0$ and if $H$ is of generic type; and that $W_\pm$ are bounded in $L^p(\R^m)$ for all $p$ between $\frac{m}{m-2}$ and $\frac{m}{2}$ but not for $p$ outside the closed interval $[\frac{m}{m-2}, \frac{m}{2}]$ if $V$ satisfies $|V(x)|\leq C \ax^{-m-3-\ep}$ and if $H$ is of exceptional type. This in particular implies that the continuous part of the propagator satisfies the $L^p$-$L^q$ estimates $\|e^{-itH}P_c(H)u \|_p \leq C |t|^{\frac{1}{m}\left(\frac12-\frac{1}{q}\right)}\|u\|_q$ for the dual exponents $\frac{1}{p}+\frac1{q}=1$ such that $1\leq q\leq 2 \leq p\leq \infty$ if $H$ is of generic type, and for $\frac{m}{m-2}< q\leq 2 \leq p < \frac{m}{2}$, $m \geq 5$, or \$\frac32