Bilinear Hilbert transforms associated to plane curves

TL;DR

This paper establishes boundedness results for bilinear Hilbert transforms and maximal functions along specific plane curves, advancing understanding of their behavior in harmonic analysis.

Contribution

It proves boundedness of bilinear Hilbert transforms and maximal functions along certain plane curves from L^2 x L^2 to L^1, a novel extension in the field.

Findings

Boundedness of bilinear Hilbert transforms along plane curves.

Boundedness of maximal functions along plane curves.

Extension of classical results to more general curves.

Abstract

We prove that the bilinear Hilbert transforms and maximal functions along certain general plane curves are bounded from $L^2(\mathbb{R})\times L^2(\mathbb{R})$ to $L^1(\mathbb{R})$.