$L^p$-boundedness of wave operators for bi-Schrödinger operators on the line

Abstract

This paper is devoted to establishing several types of $L^p$-boundedness of wave operators $W_{\pm }=W_{\pm }(H,{\Delta }^2)$ associated with the bi-Schrödinger operators $H={\Delta }^2+V(x)$ on the line $\mathbb{R}$. Given suitable decay potentials $V$, we firstly prove that the wave and dual wave operators are bounded on $L^p(\mathbb{R})$ for all $1<p<\infty$: $$\Vert W_{\pm }f{\Vert }_{L^p(\mathbb{R})}+\Vert W_{\pm }^{\ast }f{\Vert }_{L^p(\mathbb{R})}\lesssim \Vert f{\Vert }_{L^p(\mathbb{R})},$$ which are further extended to the $L^p$-boundedness on the weighted spaces $L^p(\mathbb{R},w)$ with general even $A_p$-weights $w$ and to the boundedness on the Sobolev spaces $W^{s,p}(\mathbb{R})$. For the limiting case, we prove that $W_{\pm }$ are bounded from $L^1(\mathbb{R})$ to $L^{1,\infty }(\mathbb{R})$ as well as bounded from the Hardy space $\H^1(\R)$ to $L^1(\mathbb{R})$. These results especially hold whatever the zero energy is a regular point or a resonance of $H$. We also obtain that $W_{\pm }$ are bounded from $L^{\infty }(\mathbb{R})$ to $\BMO(\R)$ if zero is a regular point or a first kind resonance of $H$. Next, we show that $W_{\pm }$ are neither bounded on $L^1(\mathbb{R})$ nor on $L^{\infty }(\mathbb{R})$ even if zero is a regular point of $H$. Moreover, if zero is a second kind resonance of $H$, then $W_{\pm }$ are shown to be even not bounded from $L^{\infty }(\mathbb{R})$ to $\BMO(\R)$ in general. In particular, we remark that our results give a complete picture of the validity of $L^p$-boundedness of the wave operators for all $1\le p\le \infty$ in the regular case. Finally, as applications, we deduce the $L^p$-$L^q$ decay estimates for the propagator $e^{-itH}{P}_{\mathrm{ac}}(H)$ with pairs $(1/p,1/q)$ belonging to a certain region of ${\mathbb{R}}^2$, as well as establish the Hörmander-type $L^p$-boundedness theorem for the spectral multiplier $f(H)$.

57 pages. This is a final version. To appear in Adv. Math., 2024

Cited by in corpus (1)

• The $L^p$-continuity of wave operators for higher order Schrödinger operators