${\rm A}_{1}$-regularity and boundedness of Calderón–Zygmund operators

Dmitry V. Rutsky ¹

¹ Steklov Mathematical Institute St. Petersburg Branch Fontanka 27 191023 St. Petersburg, Russia

Studia Mathematica, Tome 221 (2014) no. 3, pp. 231-247

Résumé

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 $X'$.

Détail

DOI : 10.4064/sm221-3-3

Keywords: coifman fefferman inequality implies quite easily calder zygmund operator acts boundedly banach lattice mathbb hardy littlewood maximal operator bounded establish converse result under assumption has fatou property p convex q concave infty linear operator bounded nondegenerate certain sense example riesz transform bounded nbsp