The x-ray transform I integrates a function f on over lines: . The range characterization of the x-ray transform on the Schwartz space is well known, the main ingredient of the characterization is some system of second order differential equations that are called John’s equations. The Reshetnyak formula equates the norm to some special norm of If, it was also known before. We prove a new version of the Reshetnyak formula that involves first order derivatives of If with respect to the ξ-variable. On using the latter formula, we obtain the range characterization of the x-ray transform on Sobolev spaces.

中文翻译：

---

Sobolev空间上的X射线变换

x射线变换我整合了功能˚F上过线：。Schwartz空间上X射线变换的范围表征是众所周知的，该表征的主要成分是一些称为John方程的二阶微分方程组。Reshetnyak公式将规范等同于If的某些特殊规范，这在以前也是已知的。我们证明了Reshetnyak公式的新版本，该公式涉及If相对于ξ变量的一阶导数。使用后一个公式，我们获得了Sobolev空间上X射线变换的范围表征。