## 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 \[ \mathcal {M}_{P,Q}(f,g)(x)=\sup _{\epsilon >0}\frac {1}{2\epsilon }\int _{-\epsilon }^{\epsilon } |f(x-P(t))g(x-Q(t))|dt.\]## References

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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