Abstract
We prove sharp \(L^p\) regularity results for a class of generalized Radon transforms for families of curves in a three-dimensional manifold associated with a canonical relation with fold and blowdown singularities. The proof relies on decoupling inequalities by Wolff and Bourgain–Demeter for plate decompositions of thin neighborhoods of cones and \(L^2\) estimates for related oscillatory integrals.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Gelfand, I.M., Graev, M.I.: Line complexes in the space \(C^{n}\). Funkcional. Anal. i Priložen. 2(3), 39–52 (1968)
Greenleaf, A., Uhlmann, G.: Nonlocal inversion formulas for the X-ray transform. Duke Math. J. 58(1), 205–240 (1989). https://doi.org/10.1215/S0012-7094-89-05811-0
Greenleaf, A., Uhlmann, G.: Composition of some singular Fourier integral operators and estimates for restricted X-ray transforms. Ann. Inst. Fourier (Grenoble) 40(2), 443–466 (1990)
Greenleaf, A., Seeger, A.: Fourier integral operators with fold singularities. J. Reine Angew. Math. 455, 35–56 (1994). https://doi.org/10.1515/crll.1994.455.35
Pramanik, M., Seeger, A.: \(L^p\) Sobolev regularity of a restricted X-ray transform in \(mathbb{R}^3\). In: Harmonic Analysis and Its Applications, pp. 47–64. Yokohama Publisher, Yokohama (2006)
Wolff, T.: Local smoothing type estimates on \(L^p\) for large \(p\). Geom. Funct. Anal. 10(5), 1237–1288 (2000). https://doi.org/10.1007/PL00001652
Bourgain, J., Demeter, C.: The proof of the \(l^2\) decoupling conjecture. Ann. Math. 182(1), 351–389 (2015). https://doi.org/10.4007/annals.2015.182.1.9
Pramanik, M., Seeger, A.: \(L^p\) Sobolev estimates for a class of integral operators with folding canonical relations. arXiv preprint arXiv: 1909.04173, To appear in J. Geom. Anal. (2019)
Pramanik, M., Seeger, A.: \(L^p\) regularity of averages over curves and bounds for associated maximal operators. Am. J. Math. 129(1), 61–103 (2007). https://doi.org/10.1353/ajm.2007.0003
Bentsen, G.: \({L}^p\) regularity for a class of averaging operators on the Heisenberg group. arXiv preprint arXiv:2002.01917, To appear in Indiana Univ. Math. J. (2020)
Helgason, S.: The Radon Transform. Progress in Mathematics, vol. 5, 2nd edn. Birkhäuser, Boston (1999)
Hörmander, L.: Fourier integral operators. I. Acta Math. 127(1–2), 79–183 (1971). https://doi.org/10.1007/BF02392052
Phong, D.H.: Singular integrals and Fourier integral operators. In: Essays on Fourier analysis in honor of Elias M. Stein (Princeton, NJ, 1991), Princeton Mathematical Series, vol. 42, pp. 286–320. Princeton University Press, Princeton (1995)
Guillemin, V., Sternberg, S.: Geometric Asymptotics. Mathematical Surveys, No. 14, American Mathematical Society, Providence (1977)
Comech, A.: Integral operators with singular canonical relations. In: Spectral Theory, Microlocal Analysis, Singular Manifolds. Mathematical Topic, vol. 14, pp. 200–248. Akademie, Berlin (1997)
Greenleaf, A., Seeger, A.: Oscillatory and Fourier integral operators with degenerate canonical relations. In: Proceedings of the 6th International Conference on Harmonic Analysis and Partial Differential Equations (El Escorial, 2000), pp. 93–141 (2002). https://doi.org/10.5565/PUBLMAT_Esco02_05
Comech, A.: Optimal regularity of Fourier integral operators with one-sided folds. Commun. Partial Differ. Equ. 24(7–8), 1263–1281 (1999). https://doi.org/10.1080/03605309908821465
Phong, D.H., Stein, E.M.: The Newton polyhedron and oscillatory integral operators. Acta Math. 179(1), 105–152 (1997). https://doi.org/10.1007/BF02392721
Pramanik, M., Rogers, K.M., Seeger, A.: A Calderón–Zygmund estimate with applications to generalized Radon transforms and Fourier integral operators. Stud. Math. 202(1), 1–15 (2011). https://doi.org/10.4064/sm202-1-1
Secco, S.: \(L^p\)-improving properties of measures supported on curves on the Heisenberg group. Stud Math. 132(2), 179–201 (1999). https://doi.org/10.4064/sm-132-2-179-201
do Carmo, M.P.a.: Riemannian Geometry. Mathematics: Theory & Applications. Birkhäuser, Boston (1992). https://doi.org/10.1007/978-1-4757-2201-7. https://doi-org.ezproxy.library.wisc.edu/10.1007/978-1-4757-2201-7. Translated from the second Portuguese edition by Francis Flaherty
Klingenberg, W.: Riemannian Geometry. de Gruyter Studies in Mathematics, vol. 1. Walter de Gruyter & Co., Berlin (1982)
Phong, D.H., Stein, E.M.: Radon transforms and torsion. Int. Math. Res. Notices 4, 49–60 (1991). https://doi.org/10.1155/S1073792891000077
Muscalu, C., Schlag, W.: Classical and multilinear harmonic analysis. In: Vol, I. (ed.) Cambridge Studies in Advanced Mathematics, vol. 137. Cambridge University Press, Cambridge (2013)
Cuccagna, S.: \(L^2\) estimates for averaging operators along curves with two-sided \(k\)-fold singularities. Duke Math. J. 89(2), 203–216 (1997). https://doi.org/10.1215/S0012-7094-97-08910-9
Anderson, T.C., Cladek, L., Pramanik, M., Seeger, A.: Spherical means on the Heisenberg group: stability of a maximal function estimate. arXiv preprint arXiv:1801.06981, To appear in J. Anal. Math. (2018)
Seeger, A.: Degenerate Fourier integral operators in the plane. Duke Math. J. 71(3), 685–745 (1993). https://doi.org/10.1215/S0012-7094-93-07127-X
Triebel, H.: Theory of Function Spaces. Monographs in Mathematics, vol. 78. Birkhäuser, Basel (1983). https://doi.org/10.1007/978-3-0346-0416-1
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Research supported in part by NSF Grant DMS 1764295.
Rights and permissions
About this article
Cite this article
Bentsen, G. \(L^p\) Regularity Estimates for a Class of Integral Operators with Fold Blowdown Singularities. J Geom Anal 32, 89 (2022). https://doi.org/10.1007/s12220-021-00791-1
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s12220-021-00791-1