The Sobolev Wavefront Set of the Causal Propagator in Finite Regularity

期刊论文
出版模式：

开放获取
发表日期：

2024年7月27日

查看全文

Yafet E. Sanchez Sanchez,
Elmar Schrohe

摘要

Given a globally hyperbolic spacetime $M={\mathbb {R}}\times \Sigma $ $M = R \times Σ$ of dimension four and regularity $C^\tau $ $C^{τ}$ , we estimate the Sobolev wavefront set of the causal propagator $K_G$ $K_{G}$ of the Klein–Gordon operator. In the smooth case, the propagator satisfies $WF'(K_G)=C$ $W F^{'} (K_{G}) = C$ , where $C\subset T^*(M\times M)$ $C \subset T^{*} (M \times M)$ consists of those points $(\tilde{x},\tilde{\xi },\tilde{y},\tilde{\eta })$ $(\tilde{x}, \tilde{ξ}, \tilde{y}, \tilde{η})$ such that $\tilde{\xi },\tilde{\eta }$ $\tilde{ξ}, \tilde{η}$ are cotangent to a null geodesic $\gamma $ $γ$ at $\tilde{x}$ $\tilde{x}$ resp. $\tilde{y}$ $\tilde{y}$ and parallel transports of each other along $\gamma $ $γ$ . We show that for $\tau >2$ $τ > 2$ , $\begin{aligned} WF'^{-2+\tau -{\epsilon }}(K_G)\subset C \end{aligned}$ $\begin{matrix} W F^{' - 2 + τ - ϵ} (K_{G}) \subset C \end{matrix}$ for every ${\epsilon }>0$ $ϵ > 0$ . Furthermore, in regularity $C^{\tau +2}$ $C^{τ + 2}$ with $\tau >2$ $τ > 2$ , $\begin{aligned} C\subset WF'^{-\frac{1}{2}}(K_G)\subset WF'^{\tau -\epsilon }(K_G)\subset C \end{aligned}$ $\begin{matrix} C \subset W F^{' - \frac{1}{2}} (K_{G}) \subset W F^{' τ - ϵ} (K_{G}) \subset C \end{matrix}$ holds for $0<\epsilon <\tau +\frac{1}{2}$ $0 < ϵ < τ + \frac{1}{2}$ . In the ultrastatic case with $\Sigma $ $Σ$ compact, we show $WF'^{-\frac{3}{2}+\tau -\epsilon }(K_G)\subset C$ $W F^{' - \frac{3}{2} + τ - ϵ} (K_{G}) \subset C$ for $\epsilon >0$ $ϵ > 0$ and $\tau >2$ $τ > 2$ and $WF'^{-\frac{3}{2}+\tau -\epsilon }(K_G)= C$ $W F^{' - \frac{3}{2} + τ - ϵ} (K_{G}) = C$ for $\tau >3$ $τ > 3$ and $\epsilon <\tau -3$ $ϵ < τ - 3$ . Moreover, we show that the global regularity of the propagator $K_G$ $K_{G}$ is $H^{-\frac{1}{2}-\epsilon }_{loc}(M\times M)$ $H_{loc}^{- \frac{1}{2} - ϵ} (M \times M)$ as in the smooth case.