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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Singular integral operators associated to curves with rational components
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by Magali Folch-Gabayet and James Wright PDF
Trans. Amer. Math. Soc. 360 (2008), 1661-1679 Request permission

Abstract:

We prove $L^p ({\mathbb R}^n), 1<p<\infty$, bounds for \[ Hf(x) = p.v. \int _{-\infty }^{\infty } f(x_1 - R_1 (t), \ldots , x_n - R_n (t) ) dt/t \] and \[ Mf(x) = \sup _{h>0} {1\over h} \int _{0}^{h} |f(x_1 - R_1 (t), \ldots , x_n - R_n (t) )| dt \] 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
  • MR Author ID: 325654
  • Email: j.r.wright@ed.ac.uk
  • 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.
  • © Copyright 2007 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 360 (2008), 1661-1679
  • MSC (2000): Primary 42B15
  • DOI: https://doi.org/10.1090/S0002-9947-07-04349-8
  • MathSciNet review: 2357709