Abstract:
This is a part of a joint work by the author and K. Zlotnikov
Let $(X_0, X_1)$ be a compatible couple of Banach spaces, and let $Y_0, Y_1$ be closed subspaces of $X_0$ and $X_1$. The couple $(Y_0, Y_1)$ is said to be $K$-closed in $(X_0, X_1)$ if, whenever
$Y_0+Y_1\ni x=x_0+x_1$ with $x_i\in X_i$, $i=0,1$, we also have $x=y_0+y_1$ with $y_i\in Y_i$ and
$\|y_i\|\leq C\|x_i\|$, $i=1,2$.
We recall that $K$-closedness does occur in the scale of the Hardy spaces on the unit circle (viewed as subspaces of the corresponding Lebesgue spaces), but we are interested in the following two more complicated results (in them we assume that $1<p<\infty$).
1) The couple $(H^p(\mathbb{T}^2), H^{\infty}(\mathbb{T}^2))$ is $K$-closed in the couple
$(L^p(\mathbb{T}^2), L^{\infty}(\mathbb{T}^2))$ (Kislyakov and Xu, 1996).
2) For an inner function $\theta$ on the unit circle, the couple
$(H^p\cap\theta\overline{H^p}, H^{\infty}\cap\theta\overline{H^{\infty}})$ is $K$-closed in
$(L^p(\mathbb{T}), L^{\infty}(\mathbb{T}))$ (Kislyakov and Zlotnikov, 2018)
Surprisingly, the proofs of these facts are quite similar, signalizing that they may be particular cases of some general statement. Such a statement exists indeed and looks roughly like this. Again, here $1<p<\infty$.
Theorem.
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 $L^p(\mu)$ that are moduli 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 $(L^p(\mu), L^{\infty}(\mu))$.
The additional assumptions say, in particular, that some analogs of the harmonic conjugation operator relative to the algebras $A$ and $B$ have the usual properties, as, for instance, is in the case of $w^*$-Dirichlet algebras. However, the condition for $A$ and $B$ to be $w^*$-Dirichlet is too restrictive (in particular, we do not insist that a multiple of $\mu$ represent some multiplicative linear functional on either $A$ or $B$). Also, note that the proofs of statements 1) and 2) known previously relied upon the fact that, in those settings, the corresponding harmonic conjugations (or Riesz projections) were classical singular integral operators. In particular, the two proofs started with employing Calderón–Zygmund decomposition, which is not available in the generality adopted in the theorem. Also, some assumptions on the mutual position of the annihilators of $C$ and $D$ are required (in the context of statements 1 and 2, these assumptions are satisfied trivially).
