Vector-valued boundedness of harmonic analysis operators

Let $S$ be a space of homogeneous type, $X$ a Banach lattice of measurable functions on $S \times \Omega$ with the Fatou property and nontrivial convexity, and $Y$ some Banach lattice of measurable functions with the Fatou property. Under the assumption that the Hardy–Littlewood maximal operator $M$ is bounded both in $X$ and $X’$, it is proved that the boundedness of $M$ in $X (Y)$ is equivalent to its boundedness in $\mathrm L_{s}(Y)$ for some (equivalently, for all) $1 < s < \infty$. With $S = \mathbb R^n$, the last condition is known as the Hardy–Littlewood property of $Y$ and is related to the $\mathrm {UMD}$ property. For lattices $X$ with nontrivial convexity and concavity, the UMD property implies the boundedness of all Calderón–Zygmund operators in $X (Y)$ and is equivalent to the boundedness of a single nondegenerate Calderón–Zygmund operator. The $\mathrm {UMD}$ property of $Y$ is characterized in terms of the $\mathrm A_{p}$-regularity of both $\mathrm L_{\infty } (Y)$ and $\mathrm L_{\infty } (Y’)$. The arguments are based on an improved version of the divisibility property for $\mathrm A_{p}$-regularity.