论文标题
对称张量场的某些广义射线变换的唯一延续结果
Unique continuation results for certain generalized ray transforms of symmetric tensor fields
论文作者
论文摘要
令$ i_ {m} $表示欧几里得射线变换作用在紧凑的对称$ m $ -tensor field发行$ f $上，而$ i_ {m}^{*} $是其正式的$ l^2 $ axpaint。我们研究了普通运算符$ n_ {m} = i_ {m}^{*} i_ {m} $的唯一延续结果。更确切地说，我们表明，如果$ n_ {m} $以$ x_0 $的点消失，而不是圣人运营商$ w $在$ f $ natishes上代表$ f $ w $ nishes，则在包含$ x_0 $的开放套件上消失，那么$ f $是一个潜在的张量场。这概述了Ilmavirta和Mönkkönen最近的两项作品，他们证明了功能和矢量场/1形式的射线变换的独特延续结果。这项工作的主要贡献之一是将作用于高阶张量场作用的圣人操作员作为作用于1型的外部导数操作员的正确概括，从而使高阶张量球场的射线变换成为可能的独特连续性结果。在本文的后半部分，我们证明了动量射线和横向射线变换的类似独特的延续结果。
Let $I_{m}$ denote the Euclidean ray transform acting on compactly supported symmetric $m$-tensor field distributions $f$, and $I_{m}^{*}$ be its formal $L^2$ adjoint. We study a unique continuation result for the normal operator $N_{m}=I_{m}^{*}I_{m}$. More precisely, we show that if $N_{m}$ vanishes to infinite order at a point $x_0$ and if the Saint-Venant operator $W$ acting on $f$ vanishes on an open set containing $x_0$, then $f$ is a potential tensor field. This generalizes two recent works of Ilmavirta and Mönkkönen who proved such unique continuation results for the ray transform of functions and vector fields/1-forms. One of the main contributions of this work is identifying the Saint-Venant operator acting on higher order tensor fields as the right generalization of the exterior derivative operator acting on 1-forms, which makes unique continuation results for ray transforms of higher order tensor fields possible. In the second half of the paper, we prove analogous unique continuation results for momentum ray and transverse ray transforms.
