AI 中文总结
本文证明带反平方势的三维能量临界非线性薛定谔方程的色散估计，涵盖非负势端点情形和负势强衰减及Lorentz估计，限制尖锐，方法结合自举、实插值和端点Sobolev-Lorentz估计。
AI 中文摘要
我们证明了与$\mathcal L_a=-\Delta+a|x|^{-2}$相关的三维聚焦能量临界非线性薛定谔方程的色散估计。对于非负势，我们将已知的有限$p$理论推广到端点$L^1\to L^\infty$。对于全局适定性范围内的负势，设$\sigma=\frac12-\sqrt{\frac14+a}$。我们获得了$2<p<3/\sigma$的自由强衰减率，以及从$L^{(3/\sigma)',1}$到$L^{3/\sigma,\infty}$的极限Lorentz估计。限制$p<3/\sigma$对于强Lebesgue衰减是尖锐的。证明结合了有限区间自举、有界能量集上的非线性实插值，以及适应于$\mathcal L_a$的端点Sobolev-Lorentz估计。
英文摘要
We prove dispersive estimates for the three-dimensional defocusing energy-critical nonlinear Schrödinger equation associated with $\mathcal L_a=-Δ+a|x|^{-2}$. 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 $σ=\frac12-\sqrt{\frac14+a}$. We obtain the free strong decay rate for $2<p<3/σ$ and the limiting Lorentz estimate from $L^{(3/σ)',1}$ to $L^{3/σ,\infty}$. The restriction $p<3/σ$ 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 $\mathcal L_a$.
