一类具有部分逆平方势的色散方程的衰减估计
Decay estimates for a class of dispersive equations with partial inverse-square potentials
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中文总结 AI 辅助
研究一类含部分逆平方势的色散方程，利用谱测度显式表示及频率局部化、平稳相位法等，推导其色散半群衰减估计及分数薛定谔算子边界Strichartz估计，统一简化现有估计并扩展理论到更一般情况。
中文摘要 AI 辅助
设\(\mathcal{L}_a = -\Delta_x - \Delta_y + \frac{a}{2}|x|^{-2}\)（\(a > 0\)）为\(L^2(\mathbb{R}^2_x\times \mathbb{R}^n_y)\)上的薛定谔算子，涉及奇异部分逆平方势。本文有两个目的。首先，基于Zhang-Zhang建立的与算子\(\mathcal{L}_a\)相关的谱测度的显式表示，研究一类形如\(e^{it\phi(\sqrt{\mathcal{L}_a})}\)的色散半群的衰减估计，其中\(\phi:\mathbb{R}^+ \to \mathbb{R}\)是光滑函数，采用频率局部化和平稳相位法处理相位函数\(\phi\)的非齐次性带来的技术困难。其次，推导分数薛定谔算子\(e^{it\mathcal{L}_a^\nu}\)（\(0 < \nu \neq \frac{1}{2}\)）的边界Strichartz估计。作为衰减估计的应用，进一步得到与算子\(\mathcal{L}_a\)相关的一些具体波动方程的Strichartz估计，对应\(\phi(r)=r, r^2, r^2+r^4, \sqrt{1+r^2}, \sqrt{1+r^4}\)和\(r^\mu,0<\mu\neq 1\)。最显著的是，结果统一并简化了算子\(\mathcal{L}_a\)现有的色散估计，同时将相关理论扩展到更一般的情形。
英文摘要
Let $\mathcal{L}_a=-Δ_x-Δ_y+\frac{a}{2}|x|^{-2}$ with $a>0$ denote the Schrödinger operator on $L^2(\mathbb{R}^2_x\times \mathbb{R}^n_y)$, 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 $e^{itϕ(\sqrt{\mathcal{L}_a})}$, where $ϕ: \mathbb{R}^+ \to \mathbb{R}$ is a smooth function. To handle the technical difficulty arising from the inhomogeneity of the phase function $ϕ$, 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 $e^{it\mathcal{L}_a^ν}$, $0<ν\neq\frac{1}{2}$. 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 $ϕ(r)=r, r^2, r^2+r^4, \sqrt{1+r^2}, \sqrt{1+r^4}$, and $r^μ,0<μ\neq 1$. 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.
