有限正则性下因果传播子的Sobolev波前集
The Sobolev Wavefront Set of the Causal Propagator in Finite Regularity
AI总结：
本文研究有限正则性整体双曲时空下Klein-Gordon算子因果传播子的Sobolev波前集，给出不同正则性条件下波前集的包含与等式关系，并验证传播子整体正则性与光滑情形一致。
摘要：
给定四维、正则性为$C^τ$的整体双曲时空$M=\n\bbr\times Σ$，我们估计Klein-Gordon算子的因果传播子$K_G$的Sobolev波前集。在光滑情形下，传播子满足$WF'(K_G)=C$，其中$C\ns\ubbcdot T^*(M\times M)$由满足以下条件的点$(\tilde{x},\tilde{ξ},\tilde{y},\tilde{η})$构成：$\tilde{ξ}$、$\tilde{η}$分别是零测地线$γ$在$\tilde{x}$、$\tilde{y}$处的余切向量，且沿$γ$互为平行移动。我们证明，当$τ>2$时，对任意$ε>0$，有$WF'^{-2+τ-ε}(K_G)\ns\ubbcdot C$。此外，在正则性为$C^{τ+2}$且$τ>2$的情形下，当$0<ε<τ+\frac{1}{2}$时，有$C\ns\ubbcdot WF'^{-\frac{1}{2}}(K_G)\ns\ubbcdot WF'^{τ-ε}(K_G)\ns\ubbcdot C$。在$Σ$紧致的超静情形下，我们证明当$ε>0$且$τ>2$时$WF'^{-\frac{3}{2}+τ-ε}(K_G)\ns\ubbcdot C$，当$τ>3$且$ε<τ-3$时$WF'^{-\frac{3}{2}+τ-ε}(K_G)= C$。另外，我们证明传播子$K_G$的整体正则性为$H^{-\frac{1}{2}-ε}_{loc}(M\times M)$，与光滑情形一致。
英文摘要：
Given a globally hyperbolic spacetime $M=\mathbb{R}\times Σ$ of dimension four and regularity $C^τ$, we estimate the Sobolev wavefront set of the causal propagator $K_G$ of the Klein-Gordon operator. In the smooth case, the propagator satisfies $WF'(K_G)=C$, where $C\subset T^*(M\times M)$ consists of those points $(\tilde{x},\tildeξ,\tilde{y},\tildeη)$ such that $\tildeξ,\tildeη$ are cotangent to a null geodesic $γ$ at $\tilde{x}$ resp. $\tilde{y}$ and parallel transports of each other along $γ$. We show that for $τ>2$, $WF'^{-2+τ-ε}(K_G)\subset C$ for every $ε>0$. Furthermore, in regularity $C^{τ+2}$ with $τ>2$, $C\subset WF'^{-\frac{1}{2}}(K_G)\subset WF'^{τ-ε}(K_G)\subset C$ holds for $0<ε<τ+\frac{1}{2}$. In the ultrastatic case with $Σ$ compact, we show $WF'^{-\frac{3}{2}+τ-ε}(K_G)\subset C$ for $ε>0$ and $τ>2$ and $WF'^{-\frac{3}{2}+τ-ε}(K_G)= C$ for $τ>3$ and $ε<τ-3$. Moreover, we show that the global regularity of the propagator $K_G$ is $H^{-\frac{1}{2}-ε}_{loc}(M\times M)$ as in the smooth case.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
