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On the bilinear Hilbert transform along two polynomials


Author: Dong Dong
Journal: Proc. Amer. Math. Soc. 147 (2019), 4245-4258
MSC (2010): Primary 42B20; Secondary 47B38
DOI: https://doi.org/10.1090/proc/14518
Published electronically: June 27, 2019
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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

$\displaystyle \mathcal {M}_{P,Q}(f,g)(x)=\sup _{\epsilon >0}\frac {1}{2\epsilon }\int _{-\epsilon }^{\epsilon } \vert f(x-P(t))g(x-Q(t))\vert dt.$


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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

DOI: https://doi.org/10.1090/proc/14518
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
Article copyright: © Copyright 2019 American Mathematical Society