Decay estimates for a class of dispersive equations with partial inverse-square potentials
Let $\mathcal{L}_a=-\Delta_x-\Delta_y+\frac{a}{2}|x|^{-2}$ with $a>0$ denote the Schr\"odinger operator on $L^2(\mathbb{R}^2_x\times \mathbb{R}^n_y)$, which involves a singular partial inverse-square potential. The purpose of this manuscript is twofold. First, relying on the explicit representation for the spectral measure associated with the operator $\mathcal{L}_a$ established by Zhang-Zhang [J. Geom. Anal. \textbf{35}(3), Paper No. 71, 27pp (2025)], we investigate the decay estimate for a class of dispersive semigroups of the form $e^{it\phi(\sqrt{\mathcal{L}_a})}$, where $\phi: \mathbb{R}^+ \to \mathbb{R}$ is a smooth function. To handle the technical difficulty arising from the inhomogeneity of the phase function $\phi$, we adopt the frequency localization and the stationary phase method. In the second part of the paper, we first derive boundary Strichartz estimates for the fractional Schr\"odinger operator $e^{it\mathcal{L}_a^\nu}$, $0<\nu\neq\frac{1}{2}$. As applications of the established decay estimates, we further obtain Strichartz estimates for some concrete wave equations associated with the operator $\mathcal{L}_a$, which corresponds to $\phi(r)=r, r^2, r^2+r^4, \sqrt{1+r^2}, \sqrt{1+r^4}$, and $r^\mu,0<\mu\neq 1$. Most notably, our results unify and simplify existing dispersive estimates for the operator $\mathcal{L}_a$, while extending the relevant theory to more general scenarios.