Endpoint Mapping Properties of Wave Operators for Two-Dimensional Schrödinger Operators

Abstract

We establish sharp endpoint mapping properties for the wave operators $W_{\pm }(H,-\Delta )$ of two-dimensional Schrödinger operators $H=-\Delta +V$ with real-valued decaying potentials $V$. Together with the known non-endpoint $L^p$ theory, our results give a complete classification of the $L^p$ mapping properties of the two-dimensional wave operators, and reveal an unexpected reversal of the usual threshold paradigm at the endpoints $p=1$ and $p=\infty$. When zero is a regular point of $H$, the wave operators fail to be bounded on $L^1({\mathbb{R}}^2)$ and on $L^{\infty }({\mathbb{R}}^2)$, but they satisfy the atural substitute estimates of Calderón--Zygmund type: $$L^1({\mathbb{R}}^2)⟶L^{1,\infty }({\mathbb{R}}^2),{\mathcal{H}}^1({\mathbb{R}}^2)⟶L^1({\mathbb{R}}^2),L^{\infty }({\mathbb{R}}^2)⟶\mathrm{BMO}({\mathbb{R}}^2).$$ When zero is instead a threshold singularity of the first kind---an s-wave resonance with no other threshold obstruction, the wave operators are bounded on both endpoint spaces $L^1({\mathbb{R}}^2)$ and $L^{\infty }({\mathbb{R}}^2)$. Thus, in dimension two, an s-wave resonance improves the endpoint behavior of the wave operators, in sharp contrast with dimensions $n\ge 3$, where the only regular case is the favorable one. We also determine the endpoint behavior in the remaining zero-energy spectral configurations of $H$. A p-wave resonance obstructs both the $L^1$- and the $L^{\infty }$-boundedness of the wave operators, while in the zero-eigenvalue case we obtain necessary and sufficient conditions for endpoint boundedness, expressed in terms of the presence of s- and p-wave resonances and of explicit second-order harmonic moment cancellations satisfied by the zero-energy eigenfunctions.

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