An annulus multiplier and applications to the limiting absorption principle for Helmholtz equations with a step potential

We consider the Helmholtz equation $-\Delta u + Vu - \lambda u = f$ on $\mathbb{R}^n$ where the potential $V : \mathbb{R}^n\to\mathbb{R}$ is constant on each of the half-spaces $\mathbb{R}^{n−1}\times(−\infty, 0)$ and $\mathbb{R}^{n−1}\times(0,\infty)$. We prove an $L^p − L^q$-Limiting Absorption Principle for frequencies $\lambda > \text{max }V$ with the aid of Fourier Restriction Theory and derive the existence of nontrivial solutions of linear and
nonlinear Helmholtz equations. As a main analytical tool we develop new $L^p − L^q$ estimates for a singular Fourier multiplier supported in an annulus.