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Proceedings of the American Mathematical Society

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

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On the bilinear Hilbert transform along two polynomials
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by Dong Dong PDF
Proc. Amer. Math. Soc. 147 (2019), 4245-4258 Request permission

Abstract:

We prove that the bilinear Hilbert transform along two polynomials $B_{P,Q}(f,g)(x)=\int _{\mathbb {R}}f(x-P(t))g(x-Q(t))\frac {dt}{t}$ is bounded from $L^p \times L^q$ to $L^r$ for a large range of $(p,q,r)$, as long as the polynomials $P$ and $Q$ have distinct leading and trailing degrees. The same boundedness property holds for the corresponding bilinear maximal function \[ \mathcal {M}_{P,Q}(f,g)(x)=\sup _{\epsilon >0}\frac {1}{2\epsilon }\int _{-\epsilon }^{\epsilon } |f(x-P(t))g(x-Q(t))|dt.\]
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Additional Information
  • Dong Dong
  • Affiliation: Center for Scientific Computation and Mathematical Modeling, University of Maryland, College Park, Maryland 20742
  • Email: ddong12@cscamm.umd.edu
  • Received by editor(s): January 25, 2017
  • Published electronically: June 27, 2019
  • Additional Notes: The author acknowledges the support from the Gene H. Golub Fund of the Mathematics Department at the University of Illinois.
  • Communicated by: Alexander Iosevich
  • © Copyright 2019 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 147 (2019), 4245-4258
  • MSC (2010): Primary 42B20; Secondary 47B38
  • DOI: https://doi.org/10.1090/proc/14518
  • MathSciNet review: 4002539