Remarks on $\mathrm {\mathbf A}_p$-regular lattices of measurable functions

Abstract:

A Banach lattice $X$ of measurable functions on a space of homogeneous type is said to be $\mathrm {A}_p$-regular if every $f \in X$ admits a majorant $g \geq |f|$ belonging to the Muckenhoupt class $\mathrm {A}_p$ with suitable control on the norm and the constant. It is well known that the $\mathrm {A}_p$-regularity of the order dual $X’$ of $X$ implies the boundedness of the Hardy–Littlewood maximal operator on $X^{\frac 1 p}$ for $p > 1$ (equivalently, the $\mathrm {A}_1$-regularity of this lattice), provided that $X’$ is norming for $X$. This result admits a partial converse and an interesting characterization: the $\mathrm {A}_1$-regularity of $X^{\frac 1 p}(\ell ^{p})$ implies the $\mathrm {A}_p$-regularity of $X’$, and for lattices $X$ with the Fatou property these conditions are equivalent to the $\mathrm {A}_1$-regularity of both $X^{\frac 1 p}$ and $\big (X^{\frac 1 p}\big )’$. As an application, an exact form of the self-duality of $\mathrm {BMO}$-regularity is obtained, the $\mathrm {A}_q$-regularity of the lattices $\mathrm {L}_{\infty }(\ell ^p)$ for all $1 < p,q < \infty$ is established, and in many cases it is shown that the $\mathrm {A}_1$-regularity of both $Y$ and $Y’$ yields the $\mathrm {A}_1$-regularity of $Y(\ell ^s)$ for all $1 < s < \infty$, which implies the boundedness of the Calderón–Zygmund operators in $Y(\ell ^s)$.