Bilinear Hilbert transforms along curves I. The monomial case

TL;DR

This paper proves a boundedness result for the bilinear Hilbert transform along monomial curves, connecting harmonic analysis with oscillatory integral techniques.

Contribution

It establishes the first L^2 x L^2 to L^1 estimate for bilinear Hilbert transforms along monomial curves, advancing understanding of multilinear singular integrals.

Findings

Proved L^2 x L^2 to L^1 boundedness for the bilinear Hilbert transform along monomials.

Connected the problem to multi-linear oscillatory integral estimates.

Provided techniques potentially applicable to more general curves.

Abstract

We establish an L^2 \times L^2 to L^1 estimate for the bilinear Hilbert transform along a curve defined by a monomial. Our proof is closely related to multi-linear oscillatory integrals.