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

DOI:
https://doi.org/10.1090/S0002-9947-05-03807-9

Published electronically:
May 4, 2005

MathSciNet review:
2159692

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we give estimates and the regularizing estimate of Radon transforms associated to real analytic functions, and we also give estimates of the decay rate of the operator norm of corresponding oscillatory integral operators. For estimates and estimates of the decay rate of the operator norm we obtain sharp results except for extreme points; however, for regularity we allow some restrictions on the phase function.

**[B]**J-G. Bak,*An**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**estimates for degenerate Radon transforms in**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**bounds for oscillatory integral operators with**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)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
44A12,
35S30

Retrieve articles in all journals with MSC (2000): 44A12, 35S30

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

DOI:
https://doi.org/10.1090/S0002-9947-05-03807-9

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.