Real interpolation of Hardy-type spaces and BMO-regularity

Abstract

Let $(X,Y)$ be a couple of quasi-Banach lattices of measurable functions on $\mathbb{T}\times \Omega$ satisfying some additional assumptions. The K-closedness of a couple of Hardy-type spaces $(X_A,Y_A)$ in $(X,Y)$ and the stability of the real interpolation $(X_A,Y_A{)}_{\theta ,p}=(X_A+Y_A)\cap (X,Y{)}_{\theta ,p}$ are shown to be equivalent to each other and to the BMO-regularity of the associated lattices ${\left(L_1,{\left(X^r\right)}^{\prime }{Y}^r\right)}_{\delta ,q}$. The inclusion ${\left(X^{1-\theta }{Y}^{\theta }\right)}_A\subset {\left(X_A,Y_A\right)}_{\theta ,\infty }$ is also characterized in these therms. New examples of couples $(X_A,Y_A)$ with this stability are given, proving that this property is strictly weaker than the BMO-regularity of $(X,Y)$.