Range characterization of the ray transform on Sobolev spaces of symmetric tensor fields in two dimensions
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2025
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| _version_ | 1866908517260066816 |
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| author | Agrawal, Divyansh Krishnan, Venkateswaran P. Sharafutdinov, Vladimir A. |
| author_facet | Agrawal, Divyansh Krishnan, Venkateswaran P. Sharafutdinov, Vladimir A. |
| contents | The ray transform $I_m$ integrates a symmetric $m$ rank tensor field $f$ on $\mathbb{R}^n$ over lines. In the case of $n\ge3$, the range characterization of the operator $I_m$ on weighted Sobolev spaces $H^{s}_t({\mathbb R}^n;S^m{\mathbb R}^n)$ was obtained in [V. Krishnan and V. Sharafutdinov. Range characterization of ray transform on Sobolev spaces of symmetric tensor fields. Inverse Problems and Imaging, 18(6), 1272--1293, 2024]. Here we obtain a range characterization result in higher order weighted Sobolev spaces in two dimensions. Range characterization in the case of $n=2$ is very different from that for $n\ge3$, and this allows us to obtain such a result in higher order weighted Sobolev spaces $H^{r,s}_t(\mathbb{R}^2)$ for any real $r$. Nevertheless, our main tool is again the Reshetnyak formula stating that $\lVert I_mf\rVert_{H^{(r,s+1/2)}_{t+1/2}(T{\mathbb S}^{n-1})}=\lVert f\rVert_{H^{(r,s)}_t({\mathbb R}^n;S^m{\mathbb R}^n)}$ for a solenoidal tensor field $f$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_03046 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Range characterization of the ray transform on Sobolev spaces of symmetric tensor fields in two dimensions Agrawal, Divyansh Krishnan, Venkateswaran P. Sharafutdinov, Vladimir A. Analysis of PDEs Primary: 44A12, 65R32. Secondary: 46F12 The ray transform $I_m$ integrates a symmetric $m$ rank tensor field $f$ on $\mathbb{R}^n$ over lines. In the case of $n\ge3$, the range characterization of the operator $I_m$ on weighted Sobolev spaces $H^{s}_t({\mathbb R}^n;S^m{\mathbb R}^n)$ was obtained in [V. Krishnan and V. Sharafutdinov. Range characterization of ray transform on Sobolev spaces of symmetric tensor fields. Inverse Problems and Imaging, 18(6), 1272--1293, 2024]. Here we obtain a range characterization result in higher order weighted Sobolev spaces in two dimensions. Range characterization in the case of $n=2$ is very different from that for $n\ge3$, and this allows us to obtain such a result in higher order weighted Sobolev spaces $H^{r,s}_t(\mathbb{R}^2)$ for any real $r$. Nevertheless, our main tool is again the Reshetnyak formula stating that $\lVert I_mf\rVert_{H^{(r,s+1/2)}_{t+1/2}(T{\mathbb S}^{n-1})}=\lVert f\rVert_{H^{(r,s)}_t({\mathbb R}^n;S^m{\mathbb R}^n)}$ for a solenoidal tensor field $f$. |
| title | Range characterization of the ray transform on Sobolev spaces of symmetric tensor fields in two dimensions |
| topic | Analysis of PDEs Primary: 44A12, 65R32. Secondary: 46F12 |
| url | https://arxiv.org/abs/2509.03046 |