An A_p --A_infty inequality for the Hilbert Transform

TL;DR

This paper proves a new weighted inequality for the Hilbert transform and related operators, linking the A_infty characteristic of weights to the L^p norm, extending previous results and conjecturing broader applicability.

Contribution

It establishes an A_p --A_infty inequality for Haar shift operators of bounded complexity, generalizing prior results and applying to all Calderon-Zygmund operators.

Findings

Proved an L^p(w) inequality involving A_infty characteristic.

Extended the inequality from p=2 to all p in (1, ∞).

Applied results to Calderon-Zygmund operators including the Hilbert transform.

Abstract

Continuing a theme of Lerner and Hytonen-Perez, we establish an L^p(w) inequality for a Haar shift operator of bounded complexity, that quantifies the contribution of the A_infty characteristic of the weight to the L^p norm. Here, 1<p<\infty. The Hytonen-Perez inequality is only for p=2, and we improve an inequality of the author and 6 other collaborators. As a corollary, the same inequality holds for all Calderon-Zygmund operators in the convex hull of Haar shifts of a bounded complexity, of which the canonical example is the Hilbert transform. We conjecture that the same inequality holds for all Calderon-Zygmund operators.