Strong and Weak-Type Dispersive Estimates for the Energy-Critical Nonlinear Schrödinger Equation with an Inverse-Square Potential

We prove dispersive estimates for the three-dimensional defocusing energy-critical nonlinear Schrödinger equation associated with ${\mathcal{L}}_a=-\mathrm{\Delta }+a\mathrm{\mid }x{\mathrm{\mid }}^{-2}$. For nonnegative potentials, we extend the known finite-$p$ theory to the endpoint $L^1\to L^{\mathrm{\infty }}$. For negative potentials in the global well-posedness range, set $\sigma =\frac{1}{2}-\sqrt{\frac{1}{4}+a}$. We obtain the free strong decay rate for $2<p<3\mathrm{/}\sigma$ and the limiting Lorentz estimate from $L^{(3\mathrm{/}\sigma {)}^{\mathrm{\prime }},1}$ to $L^{3\mathrm{/}\sigma ,\mathrm{\infty }}$. The restriction $p<3\mathrm{/}\sigma$ is sharp for strong Lebesgue decay. The proof combines a finite-interval bootstrap, nonlinear real interpolation on bounded energy sets, and endpoint Sobolev–Lorentz estimates adapted to ${\mathcal{L}}_a$.