$L^p$-estimates for the wave equation with partial inverse-square potentials

Abstract

This paper investigates $L^p$-estimates for solutions to the wave equation perturbed by a scaling-critical partial inverse-square potential. We study a model in which the singularity of the potential appears only in a subset of the variables, corresponding to the Schr\"{o}dinger operator $\mathcal{H}_a = -\Delta_x - \Delta_y + a/|x|^2$ on $\mathbb{R}^{2+n}$. Using spectral analysis, we establish the $L^p$-boundedness of the wave propagator $(1+\sqrt{\mathcal{H}_a})^{-\gamma} e^{it\sqrt{\mathcal{H}_a}}$ for a range of exponents $\gamma$ and $p$ satisfying $|1/p -1/2| < \gamma/(n+1)$. The key ingredients are the spectral measure kernel of the partial inverse-square operator $\mathcal{H}_a$ and the complex interpolation argument.