$\mathrm A_1$-regularity and boundedness of Riesz transforms in Banach lattices of measurable functions

Suppose that $X$ is a Banach lattice of measurable functions on $\mathbb R^n\times\Omega$ having the Fatou property. We show that the boundedness of all Riesz transforms $R_j$ in $X$ is equivalent to the boundedness of the Hardy–Littlewood maximal operator $M$ in both $X$ and $X'$, and thus to the boundedness of all Calderón–Zygmund operators in $X$. We also prove a result for the case of operators between lattices: if $Y\supset X$ is a Banach lattice with the Fatou property such that the maximal operator is bounded in $Y'$, then the boundedness of all Riesz transforms from $X$ to $Y$ is equivalent to the boundedness of the maximal operator from $X$ to $Y$, and thus to the boundedness of all Calderón–Zygmund operators from $X$ to $Y$.