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Let $(X, Y)$ be a suitable couple of quasi-Banach lattices of measurablefunctions on $\mathbb T \times \Omega$, and let $(X_A, Y_A)$ be the couple ofthe corresponding Hardy-type spaces. It has long been suspected that theBMO-regularity property of $(X, Y)$ is not only sufficient for the $\mathrmK$-closedness of $(X_A, Y_A)$ in $(X, Y)$ but also necessary. We establish theequivalence of these two properties for a general couple of Banach latticeshaving the Fatou property when $\Omega$ is a discrete measurable space, andalso for couples $(X, Y)$ where $X$ is allowed to be quasi-Banach but $Y$ isassumed to be $p$-convex with some $p > 1$ (here $\Omega$ is arbitrary). Weshow under certain mild restrictions that the "good interpolation" formula $$\left(X_A, \mathrm H_q\right)_{\theta, p} = \left[\left(X, \mathrmL_q\right)_{\theta, p}\right]_A $$ holds true if and only if $X$ isBMO-regular.

On K-closedness, BMO-regularity and real interpolation of Hardy-type spaces