The Coifman-Fefferman inequality implies quite easily that a Calderón-Zygmund operator T acts boundedly in a Banach lattice X on ℝⁿ 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 < ∞: 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'.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm221-3-3,
author = {Dmitry V. Rutsky},
title = {A1-regularity and boundedness of Calderon-Zygmund operators},
journal = {Studia Mathematica},
volume = {223},
year = {2014},
pages = {231-247},
zbl = {1316.46020},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm221-3-3}
}
Dmitry V. Rutsky. A₁-regularity and boundedness of Calderón-Zygmund operators. Studia Mathematica, Tome 223 (2014) pp. 231-247. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm221-3-3/