The $L^p$ boundedness of wave operators for Schr\"odinger operators with threshold singularities II. Even dimensional case

In this paper we consider the wave operators $W_{\pm}$ for a Schr\"odinger operator $H$ in ${\bf{R}}^n$ with $n\geq 4$ even and we discuss the $L^p$ boundedness of $W_{\pm}$ assuming a suitable decay at infinity of the potential $V$. The analysis heavily depends on the singularities of the resolvent for small energy, that is if 0-energy eigenstates exist. If such eigenstates do not exist $W_{\pm}: L^p \to L^p$ are bounded for $1 \leq p \leq \infty$ otherwise this is true for $\frac{n}{n-2} < p < \frac{n}{2}$. The extension to Sobolev space is discussed.