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Remarks on the degenerate Radon transform in $\mathbf{R}^{2}$

Author(s): Sang Hyuk Lee
Journal: Proc. Amer. Math. Soc. 129 (2001), 3373-3378.
MSC (2000): Primary 44A12; Secondary 47G10
Posted: April 25, 2001
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Abstract: The aim of this note is to prove endpoint boundedness of the generalized Radon transform which was introduced by Phong and Stein. M. Christ's combinatorial method is used to obtain restricted weak type at the endpoints. Also we show that the results of this note are essentially optimal.

References:

1.: J.-G. Bak, An $L^{p}-L^{q}$ estimate for Radon transforms associated to polynomials, Duke Math. J. 101 (2000), 259-269. CMP 2000:07
2.: A. Carbery, M. Christ, and J. Wright, Multidimensional van der Corput and sublevel set estimates, J. Amer. Math. Soc. 12 (1999), 981-1015. MR 2000h:42010
3.: M. Christ, Convolution, curvature and combinatorics a case study, Internat. Res. notices 19 (1998), 1033-1048. MR 2000a:42026
4.: D.H. Phong and E.M. Stein, Models of degenerate Fourier integral operators and Radon transforms, Ann. of Math. 140 (1994), 703-722. MR 93c:35206
5.: E.M.Stein, Harmonic analysis: Real-variable methods, Orthogonality, and Oscillatory integrals, Princeton University Press, Princeton, 1993. MR 95c:42002
6.: E.M. Stein, G. Weiss, Introduction to Harmonic Analysis on Euclidean spaces, Princeton University Press, Princeton, 1971.

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Additional Information:

Sang Hyuk Lee
Affiliation: Department of Mathematics, Pohang University of Science and Technology, Pohang 790-784, Korea
Email: huk@euclid.postech.ac.kr

DOI: 10.1090/S0002-9939-01-05956-1
PII: S 0002-9939(01)05956-1
Keywords: Radon transform, degenerate
Received by editor(s): July 2, 1999
Received by editor(s) in revised form: April 3, 2000
Posted: April 25, 2001
Additional Notes: The author was supported in part by KOSEF grant no. 1999-2-102-003-5 and BK21 Project.
Communicated by: Christopher D. Sogge
Copyright of article: Copyright 2001, American Mathematical Society


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