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

We consider the Helmholtz equation $-\mathrm{\Delta }u+Vu-\lambda u=f$ on ${\mathbb{R}}^n$ where the potential $V:{\mathbb{R}}^n\to \mathbb{R}$ is constant on each of the half-spaces ${\mathbb{R}}^{n-1}\times (-\mathrm{\infty },0)$ and ${\mathbb{R}}^{n-1}\times (0,\mathrm{\infty })$. We prove an $L^p-L^q$-Limiting Absorption Principle for frequencies $\lambda >\max V$ with the aid of Fourier Restriction Theory and derive the existence of nontrivial solutions of linear and nonlinear Helmholtz equations.