$\mathbf A_1$-regularity and boundedness of Calderón-Zygmund operators. II

Dmitry V. Rutsky

Published 2015-05-04Version 1

A proof is given for the "only if" part of the result stated in the previous article 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.