The geodesic X-ray transform with fold caustics
The geodesic X-ray transform with fold caustics
复制标题
具有折叠焦散的测地线 X 射线变换
DOI：
10.2140/apde.2012.5.219
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发表时间：
2010
期刊：
影响因子：
2.2
通讯作者：
G. Uhlmann
中科院分区：
文献类型：
--
作者：
Plamen Stefanov;G. Uhlmann
我们对具有折叠型共轭点的类测地线曲线族上的X射线变换进行了详细的微局部研究。我们表明正规算子是一个拟微分算子和一个傅里叶积分算子的和。我们计算了这两个算子的主象征以及与傅里叶积分算子相关的典范关系。在二维情况下，对于测地变换，我们表明奇点总是在某种阶数上相互抵消，并且我们给出了一个该阶数为无穷的例子；因此在这种情况下正规算子在微局部上不可逆。在三维或更高维的情况下，如果典范关系是一个局部典范图，我们表明正规算子的微局部可逆性。还研究了几个例子。
We give a detailed microlocal study of X-ray transforms over geodesics-like families of curves with conjugate points of fold type. We show that the normal operator is the sum of a pseudodifferential operator and a Fourier integral operator. We compute the principal symbol of both operators and the canonical relation associated to the Fourier integral operator. In two dimensions, for the geodesic transform, we show that there is always a cancellation of singularities to some order, and we give an example where that order is infinite; therefore the normal operator is not microlocally invertible in that case. In the case of three dimensions or higher if the canonical relation is a local canonical graph we show microlocal invertibility of the normal operator. Several examples are also studied.
Strichartz 范数的热流单调性
影响因子：
2.2
作者：
Bennett J
通讯作者：
Bennett J