Remarks on Aₚ-regular lattices of measurable functions

A Banach lattice X of measurable functions on a space of homogeneous type is said to be Aₚ-regular if every f ∈ X admits a majorant g ≥ |f| belonging to the Muckenhoupt class Aₚ with suitable control on the norm and the constant. It is well known that the Aₚ-regularity of the order dual $X’$ of X implies the boundedness of the Hardy–Littlewood maximal operator on X1 p for $p > 1$ (equivalently, the A₁-regularity of this lattice), provided that $X’$ is norming for X. This result admits a partial converse and an interesting characterization: the A₁-regularity of X1 p( ᵖ) implies the Aₚ-regularity of $X’$, and for lattices X with the Fatou property these conditions are equivalent to the A₁-regularity of both X1 p and (X1 p )’. As an application, an exact form of the self-duality of BMO-regularity is obtained, the Aq-regularity of the lattices L∞( ᵖ) for all 1 < p,q < ∞ is established, and in many cases it is shown that the A₁-regularity of both Y and $Y’$ yields the A₁-regularity of Y( ˢ) for all 1 < s < ∞, which implies the boundedness of the Calderón–Zygmund operators in Y( ˢ).