论文标题
具有粗糙衰减电位的高阶Schrödinger操作员的全球Kato平滑和Strichartz的估计
Global Kato smoothing and Strichartz estimates for higher-order Schrödinger operators with rough decay potentials
论文作者
论文摘要
令\（h =（-Δ）^m + v \）是\（l^2（\ athbb {r}^n）\）上的高阶椭圆算子，其中\（v \）是一般有限的衰减电位。本文重点介绍了与\（H \）相关的Schrödinger-type方程解决方案的解决方案的估计。特别是，我们首先基于\（h \）绝对连续部分的Kato-yajima类型的均匀分辨率估计值，建立了\（e^{ith} \）的敏锐全局平滑估计。结果，我们还获得了最佳的局部衰减估计。然后，使用这些局部衰减估计值，我们证明了整个Strichartz估计值，包括端点情况。值得注意的是，我们得出了对具有粗糙电位的高阶病例的急剧平滑作用的估计值，这些效果适用于非线性高阶Schrödinger方程的研究。最后，我们引入了Kenig-Ruiz-Sogge类型的新的均匀Sobolev估计值，其中包含了额外的衍生项，这对于建立尖锐的Kato平滑估计值至关重要。
Let \( H = (-Δ)^m + V \) be a higher-order elliptic operator on \( L^2(\mathbb{R}^n) \), where \( V \) is a general bounded decaying potential. This paper focuses on the global Kato smoothing and Strichartz estimates for solutions to Schrödinger-type equation associated with \( H \). In particular, we first establish sharp global Kato smoothing estimates for \( e^{itH} \), based on uniform resolvent estimates of Kato-Yajima type for the absolutely continuous part of \( H \). As a consequence, we also obtain optimal local decay estimates. Using these local decay estimates, we then prove the full set of Strichartz estimates, including the endpoint case. Notably, we derive Strichartz estimates with sharp smoothing effects for higher-order cases with rough potentials, which are applicable to the study of nonlinear higher-order Schrödinger equations. Finally, we introduce new uniform Sobolev estimates of the Kenig-Ruiz-Sogge type, incorporating an additional derivative term, which are crucial for establishing the sharp Kato smoothing estimates.
