Microlocal analysis of Radon transforms over quadric surfaces
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arXiv
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| Auteurs principaux: | , , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866908831711232000 |
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| author | Ambartsoumian, Gaik Felea, Raluca Krishnan, Venkateswaran P. Nolan, Clifford J. Quinto, Eric Todd |
| author_facet | Ambartsoumian, Gaik Felea, Raluca Krishnan, Venkateswaran P. Nolan, Clifford J. Quinto, Eric Todd |
| contents | We study the microlocal properties of generalized Radon transforms over a family of quadric hypersurfaces whose centers lie on an orientable hypersurface $S$. The quadric surfaces we consider are level sets of the quadratic form associated to a symmetric, invertible matrix $A$, with real entries. We study the singularities of the right and left projections of the canonical relation associated with these operators and show that they are determined by the signature of the matrix $A$ and the hypersurface $S$. If the matrix is positive/negative definite (i.e., the surface of integration is an ellipsoid) and $S$ is strictly convex, we prove that the singularities are folds. If the matrix is indefinite (i.e., the surface of integration is a hyperboloid-type quadric) and $S$ is either strictly convex or a cylinder, then cusp, fold, or blowdown singularities are present. We also study the case when the surface of integration is a paraboloid and show that the Bolker condition is satisfied. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_12453 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Microlocal analysis of Radon transforms over quadric surfaces Ambartsoumian, Gaik Felea, Raluca Krishnan, Venkateswaran P. Nolan, Clifford J. Quinto, Eric Todd Classical Analysis and ODEs Analysis of PDEs 44A12, 58J40 We study the microlocal properties of generalized Radon transforms over a family of quadric hypersurfaces whose centers lie on an orientable hypersurface $S$. The quadric surfaces we consider are level sets of the quadratic form associated to a symmetric, invertible matrix $A$, with real entries. We study the singularities of the right and left projections of the canonical relation associated with these operators and show that they are determined by the signature of the matrix $A$ and the hypersurface $S$. If the matrix is positive/negative definite (i.e., the surface of integration is an ellipsoid) and $S$ is strictly convex, we prove that the singularities are folds. If the matrix is indefinite (i.e., the surface of integration is a hyperboloid-type quadric) and $S$ is either strictly convex or a cylinder, then cusp, fold, or blowdown singularities are present. We also study the case when the surface of integration is a paraboloid and show that the Bolker condition is satisfied. |
| title | Microlocal analysis of Radon transforms over quadric surfaces |
| topic | Classical Analysis and ODEs Analysis of PDEs 44A12, 58J40 |
| url | https://arxiv.org/abs/2602.12453 |