Decay estimates for Nonlinear Schrödinger equation with the inverse-square potential

In this paper, we study the dispersive decay estimates for solution to the $3\mathrm{D}$ energy-critical nonlinear Schrödinger equation with an inverse-square operator ${\mathcal{L}}_a$ where the operator is denoted by ${\mathcal{L}}_a:=-\mathrm{\Delta }+\frac{a}{\mathrm{\mid }x{\mathrm{\mid }}^2}$ with the constant $a\ge 0$. Inspired by the work of \cite{KMVZZ1,K}, we first establish that the solutions exhibit ${\dot{H}}^1({\mathbb{R}}^3)$ uniform regularity, derive the Lorentz-Strichartz estimates, and then obtain the desired decay estimates using the bootstrap argument. The key ingredients of our approach include the equivalence of Sobolev norms and the fractional product rule.