Decay estimates for a class of dispersive equations with partial inverse-square potentials

Abstract

Let ${\mathcal{L}}_a=-\hat{I}{”}_x-\hat{I}{”}_y+\frac{a}{2}|x{|}^{-2}$ with $a>0$ denote the Schrödinger operator on $L^2({\mathbb{R}}_x^2\times {\mathbb{R}}_y^n)$, 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 $e^{it\ddot{I}•(\sqrt{{\mathcal{L}}_a})}$, where $\ddot{I}•:{\mathbb{R}}^{+}\to \mathbb{R}$ is a smooth function. To handle the technical difficulty arising from the inhomogeneity of the phase function $\ddot{I}•$, 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 Schrödinger operator $e^{it{\mathcal{L}}_a^{\hat{I}}½}$, $0<\hat{I}½\ne \frac{1}{2}$. 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 $\ddot{I}•(r)=r,r^2,r^2+r^4,\sqrt{1+r^2},\sqrt{1+r^4}$, and $r^{\hat{I}}¼,0<\hat{I}¼\ne 1$. 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.