论文标题
有限规律性的因果传播器的Sobolev波前集
The Sobolev Wavefront Set of the Causal Propagator in Finite Regularity
论文作者
论文摘要
给定全球双曲线$ M = \ MATHBB {r} \ timesσ$ dimension四和规则性$ c^τ$，我们估计klein-gordon操作员的因果传播剂$ k_g $的Sobolev Wavefront集。在平稳的情况下，传播器满足$ wf'（k_g）= c $，其中$ c \ subset t^*（m \ times m）$由这些点$（\ tilde {x}，\ tilde {\ tildeT，\ tilde了，\ tilde {y} {y}，\tildeη） $γ$ at $ \ tilde {x} $ resp。 $ \ tilde {y} $和沿$γ$的彼此的并行运输。 我们表明，对于$τ> 2 $，$ wf'^{ - 2+τ-ε}（k_g）\ subset c $对于每个$ε> 0 $。此外，定期$ c^{τ+2} $，带有$τ> 2 $，$ c \ c \ subset wf'^{ - \ frac {1} {2}}}}}（k_g）\ subset wf'^{° 在具有$σ$ compact的超级情况下，我们显示$ wf'^{ - \ frac {3} {2} {2}+τ-ε}（k_g）\ subset c $ for $ε> 0 $ and $ to $ and $τ> 2 $ and $ wf'^{ - \ frac { - \ frac {3} {3} {2}+th $ t $ t $ g__ $ε<τ-3 $。此外，我们表明，传播器$ k_g $的全局规则性是$ h^{ - \ frac {1} {2}-ε} _ {loc}（m \ times m）$，如平滑的情况下。
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.
