Hostname: page-component-848d4c4894-pftt2 Total loading time: 0 Render date: 2024-05-17T06:46:49.034Z Has data issue: false hasContentIssue false
  • English
  • Français

Sharp Sobolev regularity of restricted X-ray transforms

Part of: Harmonic analysis in several variables

Published online by Cambridge University Press:  30 August 2023

Hyerim Ko ,
Sanghyuk Lee  and
Sewook Oh
Hyerim Ko
Affiliation:
Department of Mathematical Sciences and RIM, Seoul National University, Seoul 08826, Republic of Korea (kohr@snu.ac.kr, shklee@snu.ac.kr)
Sanghyuk Lee
Affiliation:
Department of Mathematical Sciences and RIM, Seoul National University, Seoul 08826, Republic of Korea (kohr@snu.ac.kr, shklee@snu.ac.kr)
Sewook Oh
Affiliation:
June E Huh Center for Mathematical Challenge, Korea Institute for Advanced Study, 85 Hoegiro Dongdaemun-gu, Seoul 02455, Republic of Korea (sewookoh@kias.re.kr)
Get access Rights & Permissions [Opens in a new window]

Abstract

We study $L^p$-Sobolev regularity estimates for the restricted X-ray transforms generated by nondegenerate curves. Making use of the inductive strategy in the recent work by the authors, we establish the sharp $L^p$-regularity estimates for the restricted X-ray transforms in $\mathbb {R}^{d+1}$, $d\ge 3$. This extends the result due to Pramanik and Seeger in $\mathbb {R}^3$.

Keywords

Lp Sobolev regularity

MSC classification

Primary: 42B25: Maximal functions, Littlewood-Paley theory
Secondary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.)
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 21
Check for updates
Copyright
Copyright © The Author(s), 2023. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Beltran, D., Guo, S., Hickman, J. and Seeger, A.. Sobolev improving for averages over curves in $\mathbb {R}^4$. Adv. Math. 393 (2021), 108089.CrossRefGoogle Scholar
Beltran, D., Guo, S., Hickman, J. and Seeger, A.. Sharp $L^p$ bounds for the helical maximal function. To appear in Am. J. Math. arXiv:2102.08272.Google Scholar
Bentsen, G.. $L^p$ regularity for a class of averaging operators on the Heisenberg group. Indiana Univ. Math. J. 71 (2022), 819–855.CrossRefGoogle Scholar
Bentsen, G.. Regularity estimates for a class of integral operators with fold blowdown singularities. J. Geom. Anal. 32 (2022), 89.CrossRefGoogle Scholar
Bourgain, J. and Demeter, C.. The proof of the $\ell ^2$ decoupling conjecture. Ann. Math. 182 (2015), 351–389.CrossRefGoogle Scholar
Bourgain, J., Demeter, C. and Guth, L.. Proof of the main conjecture in Vinogradov's mean value theorem for degrees higher than three. Ann. Math. 184 (2016), 633–682.CrossRefGoogle Scholar
Christ, M. and Erdoğan, B. M.. Mixed norm estimates for a restricted X-ray transform. J. Anal. Math. 87 (2002), 187–198.CrossRefGoogle Scholar
Christ, M. and Erdoğan, B. M.. Mixed norm estimates for certain generalized Radon transforms. Trans. Am. Math. Soc. 360 (2008), 5477–5488.CrossRefGoogle Scholar
Dendrinos, S. and Stovall, B.. Uniform estimates for the X-ray transform restricted to polynomial curves. J. Funct. Anal. 262 (2012), 4986–5020.CrossRefGoogle Scholar
Dendrinos, S. and Stovall, B.. Uniform bounds for convolution and restricted X-ray transforms along degenerate curves. J. Funct. Anal. 268 (2015), 585–633.CrossRefGoogle Scholar
Erdoğan, M. B.. Mixed-norm estimates for a restricted X-ray transform in $\mathbb {R}^4$ and $\mathbb {R}^5$. Int. Math. Res. Not. 2001 (2001), 575–600.CrossRefGoogle Scholar
Gelfand, I. M. and Graev, M. I.. Line complexes in the space $\mathbb {C}^n$. Funct. Ann. Appl. 2 (1968), 219–229.CrossRefGoogle Scholar
Greenleaf, A. and Seeger, A.. Fourier integral operators with fold singularities. J. Reine Angew. Math. 455 (1994), 35–56.Google Scholar
Greenleaf, A. and Seeger, A.. Fourier integral operators with cusp singularities. Am. J. Math. 120 (1998), 1077–1119.CrossRefGoogle Scholar
Greenleaf, A., Seeger, A. and Wainger, S.. On X-ray transforms for rigid line complexes and integrals over curves in $\mathbb {R}^4$. Proc. Am. Math. Soc. 127 (1999), 3533–3545.CrossRefGoogle Scholar
Greenleaf, A. and 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, pp. 93–141 (El Escorial, 2000).CrossRefGoogle Scholar
Greenleaf, A. and Uhlmann, G.. Nonlocal inversion formulas for the X-ray transform. Duke Math. J. 58 (1989), 205–240.CrossRefGoogle Scholar
Greenleaf, A. and Uhlmann, G.. Estimates for singular Radon transforms and pseudo-differential operators with singular symbols. J. Funct. Anal. 89 (1990), 202–232.CrossRefGoogle Scholar
Greenleaf, A. and Uhlmann, G.. Composition of some singular Fourier integral operators and estimates for restricted X-ray transforms I. Ann. Inst. Fourier (Grenoble) 40 (1990), 443–466.CrossRefGoogle Scholar
Gressman, P. T.. $L^p$-improving properties of averages on polynomial curves and related integral estimates. Math. Res. Lett. 16 (2009), 971–989.CrossRefGoogle Scholar
Ham, S. and Lee, S.. Restriction estimates for space curves with respect to general measure. Adv. Math. 254 (2014), 251–279.CrossRefGoogle Scholar
Ko, H., Lee, S. and Oh, S.. Maximal estimate for average over space curve. Invent. Math. 228 (2022), 991–1035.CrossRefGoogle Scholar
Ko, H., Lee, S. and Oh, S.. Sharp smoothing properties of averages over curves. Forum Math. Pi 11 (2023), e4.CrossRefGoogle Scholar
Laghi, N.. A note on restricted X-ray transforms. Math. Proc. Cambridge Philos. Soc. 46 (2009), 719–729.CrossRefGoogle Scholar
Oberlin, D. M.. An estimate for a restricted X-ray transform. Can. Math. Bull. 43 (2000), 472–476.CrossRefGoogle Scholar
Oberlin, D. and Smith, H.. A Bessel function multiplier. Proc. Am. Math. Soc. 127 (1999), 2911–2915.CrossRefGoogle Scholar
Oberlin, D., Smith, H. and Sogge, C. D.. Averages over curves with torsion. Math. Res. Lett. 5 (1998), 535–539.CrossRefGoogle Scholar
Pramanik, M., Rogers, K. M. and Seeger, A.. A Calderón–Zygmund estimate with applications to generalized Radon transforms and Fourier integral operators. Stud. Math. 202 (2011), 1–15.CrossRefGoogle Scholar
Pramanik, M. and Seeger, A.. $L^p$ regularity of averages over curves and bounds for associated maximal operators. Am. J. Math. 129 (2007), 61–103.CrossRefGoogle Scholar
Pramanik, M. and Seeger, A.. $L^p$ Sobolev estimates for a class of integral operators with folding canonical relations. J. Geom. Anal. 31 (2021), 6725–6765.CrossRefGoogle Scholar
Pramanik, M. and Seeger, A.. $L^p$ Sobolev regularity of a restricted X-ray transform in $\mathbb {R}^3$. Harmonic analysis and its applications, pp. 47–64 (Yokohama: Yokohama Publ., 2006).Google Scholar
Tao, T. and Vargas, A.. A bilinear approach to cone multipliers. II. Applications. Geom. Funct. Anal. 10 (2000), 216–258.CrossRefGoogle Scholar
Tao, T. and Wright, J.. $L^p$ improving bounds for averages along curves. J. Am. Math. Soc. 16 (2003), 605–638.CrossRefGoogle Scholar
Wolff, T.. A sharp bilinear cone restriction estimate. Ann. Math. (2) 153 (2001), 661–698.CrossRefGoogle Scholar