Remarks on the degenerate Radon transform in 𝐑²

Lee, Sang

doi:10.1090/S0002-9939-01-05956-1

Remarks on the degenerate Radon transform in $\mathbf {R}^{2}$
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by Sang Hyuk Lee PDF

Proc. Amer. Math. Soc. 129 (2001), 3373-3378 Request permission

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

J.-G. Bak, An $L^{p}-L^{q}$ estimate for Radon transforms associated to polynomials, Duke Math. J. 101 (2000), 259–269.
Anthony Carbery, Michael Christ, and James Wright, Multidimensional van der Corput and sublevel set estimates, J. Amer. Math. Soc. 12 (1999), no. 4, 981–1015. MR 1683156, DOI 10.1090/S0894-0347-99-00309-4
Michael Christ, Convolution, curvature, and combinatorics: a case study, Internat. Math. Res. Notices 19 (1998), 1033–1048. MR 1654767, DOI 10.1155/S1073792898000610
Erich Rothe, Topological proofs of uniqueness theorems in the theory of differential and integral equations, Bull. Amer. Math. Soc. 45 (1939), 606–613. MR 93, DOI 10.1090/S0002-9904-1939-07048-1
Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192
E.M. Stein, G. Weiss, Introduction to Harmonic Analysis on Euclidean spaces, Princeton University Press, Princeton, 1971.