Local Sobolev estimates of a function by means of its Radon transform

Abstract

In this article, we will define local and microlocalSobolev seminorms and prove local and microlocal inverse continuityestimates for the Radon hyperplane transform in these seminorms. Therelation between the Sobolev wavefront set of a function $f$ and ofits Radon transform is well-known [18]. However, Sobolevwavefront is qualitative and therefore the relation in[18] is qualitative. Our results will make the relationbetween singularities of a function and those of its Radon transformquantitative. This could be important for practical applications,such as tomography, in which the data are smooth but can have largederivatives.