Weighted Calderón–Zygmund decomposition with some applications to interpolation

Let $X$ be an $\mathrm A_1$-regular lattice of measurable functions and let $Q$ be a projection which is also a Calderón–Zygmund operator. Then it is possible to define a space $X^Q$ consisting of the functions $f\in X$ that satisfy $Qf=f$ in a certain sense. By using the Bourgain approach to interpolation, we establish that the couple $(\mathrm L_1^Q,X^Q)$ is $\mathrm K$-closed in $(\mathrm L_1,X)$. This result is sharp in the sense that, in general, $\mathrm A_1$-regularity cannot be replaced by weaker conditions such as $\mathrm A_p$-regularity for $p>1$.