Complex interpolation of couple (X, BMO) for $A_1$-regular lattices

TL;DR

This paper investigates the complex interpolation between BMO and a Banach lattice X, showing that under certain conditions, the interpolation space equals a power of X, extending understanding of function space relationships.

Contribution

It establishes a new interpolation result for couples involving BMO and Banach lattices with specific properties, utilizing recent advances in maximal function analysis.

Findings

$( ext{BMO}, X)_ heta = X^ heta$ for $0< heta<1$ under given conditions

Interpolation holds for Banach lattices with the Fatou property and order continuous norm

Boundedness of Hardy-Littlewood maximal operator in dual spaces is crucial

Abstract

Recent results of A. Lerner concerning certain properties of the Fefferman-Stein maximal function are applied to show that $(\BMO, X)_\theta = X^\theta$, $0 < \theta < 1$, for a Banach lattice $X$ of measurable functions on $\mathbb R^n$ satisfying the Fatou property such that $X$ has order continuous norm and the Hardy-Littlewood maximal operator $M$ is bounded in $(X^\alpha)'$ for some $0 < \alpha \leqslant 1$.