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Article Contents

# Ray transform on Sobolev spaces of symmetric tensor fields, I: Higher order Reshetnyak formulas

Dedicated to the memory of our teacher Yuri Grigor'evich Reshetnyak

The first author was supported by India SERB Matrics Grant MTR/2017/000837, and the second author was supported by RFBR, Grant 20-51-15004 (joint French – Russian grant)

• For an integer $r\ge0$, we prove the $r^{\mathrm{th}}$ order Reshetnyak formula for the ray transform of rank $m$ symmetric tensor fields on ${{\mathbb R}}^n$. Roughly speaking, for a tensor field $f$, the order $r$ refers to $L^2$-integrability of higher order derivatives of the Fourier transform $\widehat f$ over spheres centered at the origin. Certain differential operators $A^{(m,r,l)}\ (0\le l\le r)$ on the sphere ${{\mathbb S}}^{n-1}$ are main ingredients of the formula. The operators are defined by an algorithm that can be applied for any $r$ although the volume of calculations grows fast with $r$. The algorithm is realized for small values of $r$ and Reshetnyak formulas of orders $0,1,2$ are presented in an explicit form.

Mathematics Subject Classification: Primary: 44A12, 65R32; Secondary: 46F12.

 Citation:

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