A weighted estimate for two dimensional Schrödinger, matrix Schrödinger, and wave equations with resonance of the first kind at zero energy

Abstract

We study the two dimensional Schrödinger operator, $H=-\Delta +V$, in the weighted $L^1({\mathbb{R}}^2)\to L^{\infty }({\mathbb{R}}^2)$ setting when there is a resonance of the first kind at zero energy. In particular, we show that if $|V(x)|\lesssim ⟨x{⟩}^{-4-}$ and there is only s-wave resonance at zero of $H$, then

with $w(x)={\log }^2(2+|x|)$. Here $Ff=-\frac{1}{4}\psi ⟨\psi ,f⟩$, where $\psi$ is an s-wave resonance function. We also extend this result to wave and matrix Schrödinger equations with potentials under similar conditions.