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

Abstract

Recent results of A. Lerner concerning certain properties of the Fefferman-Stein maximal function are applied to show that $(\BMO, X)_θ= X^θ$, $0<\theta <1$, 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 $(X^{\alpha }{)}^{\prime }$ for some $0<\alpha ⩽1$.