$L^p$ improving estimates for some classes of Radon transforms
HTML articles powered by AMS MathViewer
- by Chan Woo Yang PDF
- Trans. Amer. Math. Soc. 357 (2005), 3887-3903 Request permission
Abstract:
In this paper, we give $L^p-L^q$ estimates and the $L^p$ regularizing estimate of Radon transforms associated to real analytic functions, and we also give estimates of the decay rate of the $L^p$ operator norm of corresponding oscillatory integral operators. For $L^p-L^q$ estimates and estimates of the decay rate of the $L^p$ operator norm we obtain sharp results except for extreme points; however, for $L^p$ regularity we allow some restrictions on the phase function.References
- Jong-Guk Bak, An $L^p\text {-}L^q$ estimate for Radon transforms associated to polynomials, Duke Math. J. 101 (2000), no. 2, 259–269. MR 1738178, DOI 10.1215/S0012-7094-00-10125-1
- Jong-Guk Bak, Daniel M. Oberlin, and Andreas Seeger, Two endpoint bounds for generalized Radon transforms in the plane, Rev. Mat. Iberoamericana 18 (2002), no. 1, 231–247. MR 1924693, DOI 10.4171/RMI/317
- Michael Christ, Hilbert transforms along curves. I. Nilpotent groups, Ann. of Math. (2) 122 (1985), no. 3, 575–596. MR 819558, DOI 10.2307/1971330
- Michael Christ, Failure of an endpoint estimate for integrals along curves, Fourier analysis and partial differential equations (Miraflores de la Sierra, 1992) Stud. Adv. Math., CRC, Boca Raton, FL, 1995, pp. 163–168. MR 1330238
- Anthony Carbery, Andreas Seeger, Stephen Wainger, and James Wright, Classes of singular integral operators along variable lines, J. Geom. Anal. 9 (1999), no. 4, 583–605. MR 1757580, DOI 10.1007/BF02921974
- Allan Greenleaf and Andreas Seeger, On oscillatory integral operators with folding canonical relations, Studia Math. 132 (1999), no. 2, 125–139. MR 1669698
- Sanghyuk Lee, Endpoint $L^p-L^q$ estimates for degenerate Radon transforms in $\Bbb R^2$ associated with real-analytic functions, Math. Z. 243 (2003), no. 2, 217–241. MR 1961865, DOI 10.1007/s00209-002-0454-2
- D. H. Phong and E. M. Stein, Damped oscillatory integral operators with analytic phases, Adv. Math. 134 (1998), no. 1, 146–177. MR 1612395, DOI 10.1006/aima.1997.1704
- D. H. Phong and E. M. Stein, The Newton polyhedron and oscillatory integral operators, Acta Math. 179 (1997), no. 1, 105–152. MR 1484770, DOI 10.1007/BF02392721
- D. H. Phong and E. M. Stein, Models of degenerate Fourier integral operators and Radon transforms, Ann. of Math. (2) 140 (1994), no. 3, 703–722. MR 1307901, DOI 10.2307/2118622
- Vyacheslav S. Rychkov, Sharp $L^2$ bounds for oscillatory integral operators with $C^\infty$ phases, Math. Z. 236 (2001), no. 3, 461–489. MR 1821301, DOI 10.1007/PL00004838
- Andreas Seeger, Degenerate Fourier integral operators in the plane, Duke Math. J. 71 (1993), no. 3, 685–745. MR 1240601, DOI 10.1215/S0012-7094-93-07127-X
- Andreas Seeger, Radon transforms and finite type conditions, J. Amer. Math. Soc. 11 (1998), no. 4, 869–897. MR 1623430, DOI 10.1090/S0894-0347-98-00280-X
- Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, N.J., 1971. MR 0304972
Additional Information
- Chan Woo Yang
- Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
- Address at time of publication: Department of Mathematics, Korea University, 1 Anam-dong, Sungbuk-ku, Seoul, Korea 136-701
- Received by editor(s): September 11, 2001
- Received by editor(s) in revised form: October 29, 2002
- Published electronically: May 4, 2005
- © Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 357 (2005), 3887-3903
- MSC (2000): Primary 44A12; Secondary 35S30
- DOI: https://doi.org/10.1090/S0002-9947-05-03807-9
- MathSciNet review: 2159692