Divergence of combinatorial averages and the unboundedness of the trilinear Hilbert transform

TL;DR

This paper demonstrates the divergence of certain multilinear averages and the unboundedness of the trilinear Hilbert transform in specific $L^p$ spaces close to 1, revealing limitations in these harmonic analysis operators.

Contribution

It establishes the divergence of multilinear averages and unboundedness of the trilinear Hilbert transform in new $L^p$ ranges near 1, expanding understanding of their behavior.

Findings

Multilinear averages diverge in certain $L^p$ spaces near 1.

The trilinear Hilbert transform is unbounded in similar $L^p$ ranges.

Results highlight limitations of these operators in harmonic analysis.

Abstract

We consider multilinear averages in ergodic theory and harmonic analysis and prove their divergence in some range of $L^p$ spaces, with $p$ close enough to 1. We also prove that the trilinear Hilbert transform is unbounded in a similar range of $L^p$ spaces.