$L^p$-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 Schrödinger operator ${\mathcal{H}}_a=-\hat{I}{”}_x-\hat{I}{”}_y+a/|x{|}^2$ on ${\mathbb{R}}^{2+n}$. Using spectral analysis, we establish the $L^p$-boundedness of the wave propagator $(1+\sqrt{{\mathcal{H}}_a}{)}^{-{\hat{I}}^3}{e}^{it\sqrt{{\mathcal{H}}_a}}$ for a range of exponents ${\hat{I}}^3$ and $p$ satisfying $|1/p-1/2|<{\hat{I}}^3/(n+1)$. The key ingredients are the spectral measure kernel of the partial inverse-square operator ${\mathcal{H}}_a$ and the complex interpolation argument.