Counterexamples and weak (1,1) estimates of wave operators for fourth-order Schr\"odinger operators in dimension three

This paper is dedicated to investigating the $L^p$-bounds of wave operators $W_\pm(H,\Delta^2)$ associated with fourth-order Schr\"odinger operators $H=\Delta^2+V$ on $\mathbb{R}^3$. We consider that real potentials satisfy $|V(x)|\lesssim \langle x\rangle^{-\mu}$ for some $\mu>0$. A recent work by Goldberg and Green \cite{GoGr21} 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 this paper, we aim to further establish endpoint estimates for $W_\pm(H,\Delta^2)$ in two significant ways. First, we provide counterexamples that illustrate the unboundedness of $W_\pm(H,\Delta^2)$ on the endpoint spaces $L^1(\mathbb{R}^3)$ and $L^\infty(\mathbb{R}^3)$, even 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 $W_\pm(H,\Delta^2)^*$ in the case where zero is a regular point and $\mu>11$. These estimates depend critically on the singular integral theory of Calder\'on-Zygmund on a homogeneous space $(X,d\omega)$ with a doubling measure $d\omega$.