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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Singular integral operators associated to curves with rational components


Authors: Magali Folch-Gabayet and James Wright
Journal: Trans. Amer. Math. Soc. 360 (2008), 1661-1679
MSC (2000): Primary 42B15
Published electronically: August 22, 2007
MathSciNet review: 2357709
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Abstract: We prove $ L^p ({\mathbb{R}}^n), 1<p<\infty$, bounds for

$\displaystyle Hf(x) = p.v. \int_{-\infty}^{\infty} f(x_1 - R_1 (t), \ldots , x_n - R_n (t) ) \, dt/t $

and

$\displaystyle Mf(x) = \sup_{h>0} {1\over h} \int_{0}^{h} \vert f(x_1 - R_1 (t), \ldots , x_n - R_n (t) )\vert \, dt $

where $ R_j (t) = P_j(t)/Q_j(t), j=1,2,\ldots, n$, are rational functions. Our bounds depend only on the degrees of the polynomials $ P_j, Q_j$ and, in particular, they do not depend on the coefficients of these polynomials.


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Additional Information

Magali Folch-Gabayet
Affiliation: Instituto de Matemáticas, UNAM, Area de la Investigación Científica, Circuito Exterior, Ciudad Universitaria, México, D.F. 04510
Email: folchgab@matem.unam.mx

James Wright
Affiliation: School of Mathematics, University of Edinburgh, JCMB, King’s Buildings, Mayfield Road, Edinburgh EH9 3JZ, Scotland
Email: j.r.wright@ed.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9947-07-04349-8
PII: S 0002-9947(07)04349-8
Received by editor(s): October 19, 2004
Received by editor(s) in revised form: May 30, 2006
Published electronically: August 22, 2007
Additional Notes: The first author acknowledges financial support from CONACyT (37046-E) and DGAPA-UNAM (PAPIIT IN101303).
The second author would like to thank the warm hospitality of the Instituto de Matemáticas, Universidad Nacional Autónoma de México where most of the research for this paper was conducted.
Article copyright: © Copyright 2007 American Mathematical Society