Counterexamples and weak (1,1) estimates of wave operators for fourth-order Schrödinger operators in dimension three

Abstract

This paper is dedicated to investigating the $L^p$-bounds of wave operators $W_{\pm }(H,{\Delta }^2)$ associated with fourth-order Schrödinger operators $H={\Delta }^2+V$ on ${\mathbb{R}}^3$ with real potentials satisfying $|V(x)|\lesssim ⟨x{⟩}^{-\mu }$ for some $\mu >0$. A recent work by Goldberg and Green (2021) has demonstrated that wave operators $W_{\pm }(H,{\Delta }^2)$ are bounded on $L^p({\mathbb{R}}^3)$ for all $1<p<\infty$ under the condition that $\mu >9$ and zero is a regular point of $H$. In the paper, we aim to further establish endpoint estimates for $W_{\pm }(H,{\Delta }^2)$ in two significant ways. First, we provide counterexamples to illustrate the unboundedness of $W_{\pm }(H,{\Delta }^2)$ on the endpoint spaces $L^1({\mathbb{R}}^3)$ and $L^{\infty }({\mathbb{R}}^3)$ for non-zero compactly supported potentials $V$. Second, we establish weak $(1,1)$ estimates for the wave operators $W_{\pm }(H,{\Delta }^2)$ and their dual operators $W_{\pm }(H,{\Delta }^2{)}^{\ast }$ in the case where zero is a regular point and $\mu >11$. These estimates depend critically on the singular integral theory of Calderón–Zygmund on a homogeneous space $(X,d\omega )$ with a doubling measure $d\omega$.