On the two dimensional Bilinear Hilbert Transform

TL;DR

This paper studies the two-dimensional Bilinear Hilbert Transform and the convergence of bilinear averages in ergodic theory, introducing new phase-space analysis techniques that blend one and a half dimensional and one-dimensional methods.

Contribution

It presents novel analytical techniques for the bilinear Hilbert transform in the plane and ergodic averages, advancing understanding in harmonic analysis and ergodic theory.

Findings

New phase-space analysis methods developed

Results on pointwise convergence of bilinear averages

Enhanced understanding of bilinear Hilbert transform in 2D

Abstract

We investigate the Bilinear Hilbert Transform in the plane and the pointwise convergence of bilinear averages in Ergodic theory, arising from $\Z^2$ actions. Our techniques combine novel one and a half dimensional phase-space analysis with more standard one dimensional theory.