Variations of the Bourgain method for $\mathrm{K}$-closedness of certain subcouples

In the early nineties J. Bourgain proved that the couple $(\mathrm{L}_ {1}^P, \mathrm{L}_ {p}^P)$ is $\mathrm{K}$-closed in $(\mathrm{L}_ {1}, \mathrm{L}_ {p})$, $1 < p < \infty$, where the subspaces $\mathrm{L}_ {q}^P$ of $\mathrm{L}_ {q}$ are defined by $\{P f = f\}$ with a projection $P$ that is a Calderón–Zygmund operator. $\mathrm{K}$-closedness means that arbitrary measurable decompositions in $\mathrm{L}_ {1} + \mathrm{L}_ {p}$ of functions from $\mathrm{L}_ {1}^P + \mathrm{L}_ {p}^P$ can be replaced by decompositions in $\mathrm{L}_ {1}^P + \mathrm{L}_ {p}^P$ with suitable norm estimates. In the present work we consider some variations of J. Bourgain's argument that natually lead to many of its known generalizations. To illustrate this, we prove the following generalization of a result by S. V. Kislyakov and Q. Xu about $\mathrm{K}$-closedness of Hardy spaces on the bidisk: spaces of functions on $\mathbb R^2$ with Fourier transform supported on an arbitrary finite union of polygons are $\mathrm{K}$-closed in $(\mathrm{L}_ {1}, \mathrm{L}_ {\infty})$. On the other hand, some counterexamples reveal certain hard limitations of such methods if one tries to apply them in higher dimensions and to more complicated spaces of functions on the line and on the plane. Among other things, we show how a recent result by S. V. Kislyakov and I. K. Zlotnikov about $\mathrm{K}$-closedness of the coinvariant subspaces of the shift operator ${\mathcal K}_\theta^p$ can be derived directly from J. Bourgain's original result to achieve $\mathrm{K}$-closedness of the entire scale $(\mathcal K^{1}_\theta, \mathcal K_\theta^{\infty})$.