Interpolation for intersections of Hardy-type spaces

Let $(X,\mu)$ be a space with a finite measure $\mu$, let $A$ and $B$ be $w^*$-closed subalgebras of $L^{\infty}(\mu)$, and let $C$ and $D$ be closed subspaces of $L^p(\mu)$ ($1<p<\infty$) that are modules over $A$ and $B$, respectively. Under certain additional assumptions, the couple $(C\cap D, C\cap D\cap L^{\infty}(\mu))$ is $K$-closed in $(L^p(\mu), L^{\infty}(\mu))$. This statement covers, in particular, two cases analyzed previously: that of Hardy spaces on the two-dimensional torus and that of the coinvariant subspaces of the shift operator on the circle. Next, many situations when $A$ and $B$ are $w^*$-Dirichlet algebras also fit in this pattern.