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

Abstract

We prove dispersive estimates for the three-dimensional defocusing energy-critical nonlinear Schrödinger equation associated with ${\mathcal{L}}_a=-\Delta +a|x{|}^{-2}$. For nonnegative potentials, we extend the known finite-$p$ theory to the endpoint $L^1\to L^{\infty }$. 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 $2<p<3/\sigma$ and the limiting Lorentz estimate from $L^{(3/\sigma {)}^{\prime },1}$ to $L^{3/\sigma ,\infty }$. The restriction $p<3/\sigma$ 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 ${\mathcal{L}}_a$.