A_1-regularity and boundedness of Calderon-Zygmund operators

TL;DR

This paper explores the relationship between Calderon-Zygmund operators and the Hardy-Littlewood maximal operator in Banach lattices, establishing conditions under which boundedness of one implies boundedness of the other.

Contribution

It provides a converse result showing that boundedness of certain Calderon-Zygmund operators implies the boundedness of the maximal operator in specific Banach lattice settings.

Findings

Boundedness of Calderon-Zygmund operators implies boundedness of the maximal operator in Banach lattices.

The converse holds under p-convexity, q-concavity, and Fatou property assumptions.

Nondegenerate Calderon-Zygmund operators like Riesz transforms enforce maximal operator boundedness.

Abstract

The Coifman-Fefferman inequality implies quite easily that a Calderon-Zygmund operator $T$ acts boundedly in a Banach lattice $X$ on $\mathbb R^n$ if the Hardy-Littlewood maximal operator $M$ is bounded in both $X$ and $X'$. We discuss this phenomenon in some detail and establish a converse result under the assumption that $X$ is $p$-convex and $q$-concave with some $1 < p, q < \infty$ and satisfies the Fatou property: if a linear operator $T$ is bounded in $X$ and $T$ is nondegenerate in a certain sense (for example, if $T$ is a Riesz transform) then $M$ has to be bounded in both $X$ and $X'$.