${\mathbf{A}}_1$-regularity and boundedness of Calderón-Zygmund operators. II

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 Calderón-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 $X^{\prime }$, 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.