The $L^p$-boundedness of wave operators for fourth order Schr\"odinger operators on ${\mathbb R}^4$

We prove that the wave operators of scattering theory for the fourth order Schr\"odinger operators $\Delta^2 + V(x)$ in ${\mathbb R}^4$ are bounded in $L^p({\mathbb R}^4)$ for the set of $p$'s of $(1,\infty)$ depending on the kind of spectral singularities of $H$ at zero which can be described by the space of bounded solutions of $(\Delta^2 + V(x))u(x)=0$.