New uniform bounds for a Walsh model of the bilinear Hilbert transform

TL;DR

This paper establishes new uniform $L^p$ bounds for a Walsh model of the bilinear Hilbert transform, covering a full range of exponents and employing a multi-frequency Calderon-Zygmund decomposition for near-1 exponents.

Contribution

It provides the first uniform bounds across degenerate and non-degenerate cases for the Walsh model of the bilinear Hilbert transform, extending known bounds.

Findings

Established full range of $L^p$ bounds for the quartile operator.

Introduced a multi-frequency Calderon-Zygmund decomposition for near-1 exponents.

Unified bounds for degenerate and non-degenerate cases.

Abstract

We prove old and new $L^p$ bounds for the quartile operator, a Walsh model of the bilinear Hilbert transform, uniformly in the parameter that models degeneration of the bilinear Hilbert transform. We obtain the full range of exponents that can be expected from known bounds in the degenerate and non-degenerate cases. For the new estimates with exponents p close to 1 the argument relies on a multi-frequency Calderon-Zygmund decomposition.