The Sobolev Wavefront Set of the Causal Propagator in Finite Regularity
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866909226987683840 |
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| author | Sanchez, Yafet Sanchez Schrohe, Elmar |
| author_facet | Sanchez, Yafet Sanchez Schrohe, Elmar |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2203_04362 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The Sobolev Wavefront Set of the Causal Propagator in Finite Regularity Sanchez, Yafet Sanchez Schrohe, Elmar Analysis of PDEs Mathematical Physics 58J47 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. |
| title | The Sobolev Wavefront Set of the Causal Propagator in Finite Regularity |
| topic | Analysis of PDEs Mathematical Physics 58J47 |
| url | https://arxiv.org/abs/2203.04362 |