The $L^p$-boundedness of wave operators for fourth order Schrodinger operator on the line

In this talk we will consider the $L^p$-bounds of wave operators $W(H,\Delta^2)$ associated with bi-Schr\”odinger operators $H=\Delta^2+V(x)$ on R. Under a suitable decay condition on V and the absence of embedded eigenvalues of H, we first prove that the wave and dual wave operators are bounded on $L^p (R)$ for all $1<p<\infty$. This result is further extended to the weighted $L^p$-boundedness with the sharp $A_p$-bounds for general even $A_p$-weights and to the boundedness on the Sobolev spaces $W^{s,p}(R)$. For the limiting case $p=1$, we also obtain several weak-type boundedness, including $W(H,\Delta^2)\in B(L^1,L^\infty)$ and $B(H^1,L^1)$. These results especially hold whatever the zero energy is a regular point or a resonance. Next, for the case that zero is a regular point, we prove that the wave operators are neither bounded on $L^1(R)$ nor on $L^\infty(R)$, and they are even not bounded from$L^\infty(R)$ to BMO(R) if V is compactly supported. Finally, as applications, we can deduce the $L^p$-$L^q$ decay estimates for the propagator $e^{-itH}$ with pairs $(1/p,1/q)$ belonging to certain region of $R^2$, as well as the H\”ormander-type $L^p$-boundedness theorem for the spectral multiplier $f(H)$.