Transactions of the American Mathematical Society

Author(s): Jonathan M. Bennett
Journal: Trans. Amer. Math. Soc. 354 (2002), 4871-4892.
MSC (2000): Primary 44A12, 42B20
Posted: July 16, 2002
Retrieve article in: PDF DVI PostScript
This article is available free of charge

Abstract: We study certain Hilbert transforms and maximal functions along variable flat curves in the plane. We obtain their $L^{2}(\mathbb{R} ^{2})$ boundedness by considering the oscillatory singular integrals which arise from an application of a partial Fourier transform.

1.: J. M. Bennett, Some oscillatory singular integrals with variable flat phases, and related operators, Ph.D. thesis, Edinburgh University (1998).
2.: A. Carbery, M. Christ, J. Vance, S. Wainger, and D. K. Watson, Operators associated to flat plane curves: $L^{p}$ estimates via dilation methods, Duke Mathematical Journal, Vol. 59, No. 3, 1989. MR 91m:42017
3.: A. Carbery and S. Pérez, Maximal functions and Hilbert transforms along variable flat curves, Math. Research Letters 6 (1999), 237-299. MR 2000f:42008
4.: A. Carbery, S. Wainger, and J. Wright, Hilbert transforms and maximal functions associated to flat curves on the Heisenberg group, J. Amer. Math. Soc., Volume 8, Number 1, (1995), 141-179. MR 95g:43010
5.: A. Carbery, S. Wainger, and J. Wright, Hilbert transforms and maximal functions along variable flat plane curves. J. Fourier Anal. Appl., Kahane Special Issue (1995), 119-139. MR 97b:42030
6.: H. Carlsson, M. Christ, A. Córdoba, J. Duoandikoetxea, J. L. Rubio de Francia, J. Vance, S. Wainger, and D. Weinberg, $L^{p}$ estimates for maximal functions and Hilbert transforms along flat convex curves in $\mathbb{R} ^{2}$ , Bull. Amer. Math. Soc. (N.S.) 14 (1986), 263-267. MR 87f:42044
7.: J. Kim, Hilbert transforms and maximal functions along curves in the Heisenberg group, Ph.D. thesis, University of Wisconsin-Madison (1998).
8.: D. H. Phong and E. M. Stein, Hilbert integrals, singular integrals and Radon transforms, I, Acta Math. 157 (1986), 99-157. MR 88i:42028a
9.: F. Ricci and E. M. Stein, Harmonic analysis on nilpotent groups and singular integrals I. Oscillatory integrals, J. Funct. Anal. 73 (1987), 179-194. MR 88g:42023
10.: A. Seeger, $L^{2}$ -estimates for a class of singular oscillatory integrals, Math. Research Letters 1 (1994), 65-73. MR 95e:42005

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 44A12, 42B20

Jonathan M. Bennett
Affiliation: Department of Mathematics and Statistics, JCMB, Kings Buildings, Mayfield Road, Edinburgh, EH9 3JZ, Scotland

DOI: 10.1090/S0002-9947-02-03087-8
PII: S 0002-9947(02)03087-8
Received by editor(s): May 4, 1999
Posted: July 16, 2002
Additional Notes: Partially supported by EPSRC Grant GR/L10024
Copyright of article: Copyright 2002, American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6850(e) ISSN 0002-9947(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2008, American Mathematical Society Privacy Statement		Search the AMS