The Hardy spaces $H^p$ of holomorphic functions on the unit disk are Banach spaces.
Question: Are they also Banach lattices?
If yes, why is it less common to consider the Hardy spaces as Banach lattices than it is to consider $L^p$ as Banach lattices?
Motivation for question: the function spaces $L^p$ are often supplied with an order (pointwise order) to become Banach lattices. We can then derive properties like the Fatou property or order continuity of the norm. However, I cannot find any place where the Hardy space is given such an order to become a Banach lattice.