${\rm A}_{1}$-regularity and boundedness of Calderón–Zygmund operators
Volume 221 / 2014
Studia Mathematica 221 (2014), 231-247
MSC: Primary 46B42, 42B25, 42B20, 46E30, 47B38.
DOI: 10.4064/sm221-3-3
Abstract
The Coifman–Fefferman inequality implies quite easily that a Calderón–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 establish a converse result under the assumption that $X$ has the Fatou property and $X$ is $p$-convex and $q$-concave with some $1 < p, q < \infty $: 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 $M$ is bounded in both $X$ and $X'$.
