On the bilinear Hilbert transform along two polynomials

Dong Dong

doi:10.1090/proc/14518

On the bilinear Hilbert transform along two polynomials

Dong Dong^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

2 Scopus citations

Abstract

We prove that the bilinear Hilbert transform along two polynomials B_P,Q(f, g)(x) = ^∫ _R f(x − P(t))g(x − Q(t)) ^dt _t is bounded from L^p × Lq 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 M_P,Q(f, g)(x) = sup _ε> _{0 2} ¹ _ε ^∫ ₋ ^ε _ε |f(x − P(t))g(x − Q(t))|dt.

Original language	English (US)
Pages (from-to)	4245-4258
Number of pages	14
Journal	Proceedings of the American Mathematical Society
Volume	147
Issue number	10
DOIs	https://doi.org/10.1090/proc/14518
State	Published - 2019
Externally published	Yes

ASJC Scopus subject areas

Mathematics(all)
Applied Mathematics

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title = "On the bilinear Hilbert transform along two polynomials",

abstract = "We prove that the bilinear Hilbert transform along two polynomials BP,Q(f, g)(x) = ∫ R f(x − P(t))g(x − Q(t)) dt t is bounded from Lp × Lq to Lr 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 MP,Q(f, g)(x) = sup ε> 0 2 1 ε ∫ − ε ε |f(x − P(t))g(x − Q(t))|dt.",

author = "Dong Dong",

note = "Publisher Copyright: {\textcopyright}2019 American Mathematical Society",

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N2 - We prove that the bilinear Hilbert transform along two polynomials BP,Q(f, g)(x) = ∫ R f(x − P(t))g(x − Q(t)) dt t is bounded from Lp × Lq to Lr 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 MP,Q(f, g)(x) = sup ε> 0 2 1 ε ∫ − ε ε |f(x − P(t))g(x − Q(t))|dt.

AB - We prove that the bilinear Hilbert transform along two polynomials BP,Q(f, g)(x) = ∫ R f(x − P(t))g(x − Q(t)) dt t is bounded from Lp × Lq to Lr 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 MP,Q(f, g)(x) = sup ε> 0 2 1 ε ∫ − ε ε |f(x − P(t))g(x − Q(t))|dt.

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On the bilinear Hilbert transform along two polynomials

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