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$L^p$ improving estimates for some classes of Radon transforms

Author: Chan Woo Yang
Journal: Trans. Amer. Math. Soc. 357 (2005), 3887-3903
MSC (2000): Primary 44A12; Secondary 35S30
Published electronically: May 4, 2005
MathSciNet review: 2159692
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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.

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  • [B] J-G. Bak, An $L^p-L^q$ estimate for Radon transforms associated to polynomials, Duke Math. Journal, 101 (2000), 259-269. MR 1738178 (2001b:42012)
  • [BOS] J-G. Bak, D. Oberlin and A. Seeger Two endpoint bounds for generalized Radon transforms in the plane, Rev. Mat. Iberoamericana, 18 (2002), 231-247. MR 1924693 (2003h:44002)
  • [Ch1] M. Christ, Hilbert transforms along curves, I. Nilpotent groups, Ann. Math., 122 (1985), 575-596. MR 0819558 (87f:42039a)
  • [Ch2] M. Christ, Failure of an endpoint estimate for integral along curves, Fourier analysis and partial differential equations, ed. by J. Garcia-Cuerva, E. Hernandez, F. Soria and J.L. Torrea, CRC Press, 1995. MR 1330238 (97e:44007)
  • [CSWW] A. Carbery, A. Seeger, S. Wainger and J. Wright, Classes of singular integral operators along variable lines, J. Geom. Anal., 9 (1999), 583-609. MR 1757580 (2001g:42026)
  • [GS] A. Greenleaf and A. Seeger, On oscillatory integral operators with folding canonical relations, Studia Math., 132(2)(1999), 125-139. MR 1669698 (2000g:58040)
  • [L] S. Lee, Endpoint $L^p-L^q$ estimates for degenerate Radon transforms in $\mathbb{R}^2$ associated with real analytic functions, Corrected reprint of Math. Z., 243 (2003), no. 2, 217-241 [MR 1961865 (2004g:47065a)]. Math. Z., 243 (2003), no. 4, 817-841. MR 1974584 (2004g:47065b)
  • [PSt1] D. H. Phong and E. M. Stein, Damped oscillatory integral operators with analytic phases, Adv. in Math., 134 (1998), 146-177. MR 1612395 (2000b:42009)
  • [PSt2] D. H. Phong and E. M. Stein, The Newton polyhedron and oscillatory integral operators, Acta Math., 179 (1997), 105-152. MR 1484770 (98j:42009)
  • [PSt3] D. H. Phong and E. M. Stein, Models of degenerate Fourier integral operators and Radon transforms, Ann. Math., 140 (1994), 703-722. MR 1307901 (96c:35206)
  • [R] V. S. Rychkov, Sharp $L^2$ bounds for oscillatory integral operators with $C^{\infty}$ phases, Math. Z., 236 (2001), 461-489. MR 1821301 (2002i:42016)
  • [S1] A. Seeger, Degenerate Fourier integral operators in the plane, Duke Math. J., 71 (1993), 685-745. MR 1240601 (94h:35292)
  • [S2] A. Seeger, Radon transforms and finite type conditons, J. Amer. Math. Soc., 11 (1998), 869-897. MR 1623430 (99f:58202)
  • [StW] E. M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, No. 32, Princeton University Press, 1971. MR 0304972 (46:4102)

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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

Keywords: Oscillatory integral operator, Radon transform
Received by editor(s): September 11, 2001
Received by editor(s) in revised form: October 29, 2002
Published electronically: May 4, 2005
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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