Real interpolation of Hardy-type spaces: an announcement with some remarks

We consider the couples $(X_A, Y_A)$ of Hardy-type spaces defined for quasi-Banach lattices of measurable functions on $\mathbb T \times \Omega$. Under certain fairly general assumptions, the following conditions are shown to be equivalent: $(X_A, Y_A)$ is $K$-closed in $(X, Y)$, this couple is stable with respect to the real interpolation in the sense that $(X_A, Y_A)_{\theta, p} = (X_A + Y_A) \cap (X, Y)_{\theta, p}$, the inclusion $\left(X^{1 - \theta} Y^\theta\right)_A \subset \left(X_A, Y_A\right)_{\theta, \infty}$ holds true, and the lattices $\left(\mathrm{L}_1, \left(X^r\right)' Y^r\right)_{\delta, q}$ are $\mathrm{BMO}$-regular for some values of the parameters. The last property is weaker than the $\mathrm{BMO}$-regularity of $(X, Y)$, and it requires further study. Some new (compared to the main article) results are given concerning the characterization of this property in terms of the boundedness of the standard harmonic analysis operators such as the Hilbert transform and the Hardy-Littlewood maximal operator.