On bilinear Hilbert transform along two polynomials

TL;DR

This paper establishes boundedness results for the bilinear Hilbert transform along two polynomials and the associated maximal function, extending the understanding of such operators in harmonic analysis.

Contribution

It proves boundedness of the bilinear Hilbert transform along two polynomials with distinct degrees and the related maximal function, a significant extension in polynomial-based harmonic analysis.

Findings

Boundedness from L^p × L^q to L^r for the bilinear Hilbert transform along polynomials.

Boundedness of the corresponding bilinear maximal function.

Applicable for a large range of (p,q,r) with polynomials having distinct degrees.

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 $\mathcal{M}_{P,Q}(f,g)(x)=\sup_{\epsilon>0}\frac{1}{2\epsilon}\int_{-\epsilon}^{\epsilon} |f(x-P(t))g(x-Q(t))|dt$.