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arXiv · 1505.00518

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

Dmitry V. Rutsky
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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.

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Dmitry V. Rutsky. 2015-05-04. $\mathbf A_1$-regularity and boundedness of Calder\'on-Zygmund operators. II. https://arxiv.org/abs/1505.00518

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