Counterexamples and weak (1,1) estimates of wave operators for fourth-order Schrödinger 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 Schrödinger operators H=\Delta^{2}+V on \mathbb{R}^{3} with real potentials satisfying |V(x)|\lesssim \langle x\rangle^{-\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})^{*} 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 .