  The Coifman-Fefferman inequality implies quite easily that a Calderon-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
discuss this phenomenon in some detail and establish a converse result under
the assumption that $X$ is $p$-convex and $q$-concave with some $1 < p, q <
\infty$ and satisfies the Fatou property: 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 $M$ has to be bounded in both $X$ and $X'$.
