$\mathbf A_1$-regularity and boundedness of Calder\'on-Zygmund operators. II

TL;DR

This paper proves a necessary condition for the boundedness of Calderón-Zygmund operators on Banach lattices, linking it to the boundedness of the Hardy-Littlewood maximal operator, and improves related theoretical results.

Contribution

It establishes the 'only if' condition for Calderón-Zygmund operator boundedness in Banach lattices, removing fixed point theorem reliance and refining BMO-regularity divisibility results.

Findings

Boundedness of Calderón-Zygmund operators characterized by maximal operator boundedness.

Removed fixed point theorem from the proof of the main lemma.

Provided an improved divisibility result for BMO-regularity.

Abstract

A proof is given for the "only if" part of the result stated in the previous paper of the series that a suitably nondegenerate Calder\'on-Zygmund operator $T$ is bounded in a Banach lattice $X$ on $\mathbb R^n$ if and only if the Hardy-Littlewood maximal operator $M$ is bounded in both $X$ and $X'$, under the assumption that $X$ has the Fatou property and $X$ is $p$-convex and $q$-concave with some $1 < p, q < \infty$. We also get rid of a fixed point theorem in the proof of the main lemma and give an improved version of an earlier result concerning the divisibility of $\mathrm {BMO}$-regularity.