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

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 ${\mathcal{H}}_a=-{\mathrm{\Delta }}_x-{\mathrm{\Delta }}_y+a\mathrm{/}\mathrm{\mid }x{\mathrm{\mid }}^2$ 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 $\mathrm{\mid }1\mathrm{/}p-1\mathrm{/}2\mathrm{\mid }<\gamma \mathrm{/}(n+1)$. The key ingredients are the spectral measure kernel of the partial inverse-square operator ${\mathcal{H}}_a$ and the complex interpolation argument.