Interpolation of intersections by the real method

Let $(X_0,X_1)$ be a Banach couple with $X_0\cap X_1$ dense both in $X_0$ and in $X_1,$ and let $(X_0,X_1)_{\theta,q}$ $(0<\theta <1,$ $1\le q<\infty)$ denote the real interpolation spaces. Suppose $\psi$ is a linear functional defined on some linear subspace $M\subset X_0+X_1$ and satisfying $\psi\in (X_0\cap X_1)^*,$ $\psi\ne 0.$ Conditions are considered that ensure the natural identity

$(X_0\cap \operatorname{Ker}\psi,X_1\cap \operatorname{Ker}\psi)_{\theta,q} =(X_0,X_1)_{\theta,q}\cap \operatorname{Ker}\psi.$

The results obtained provide a solution for the problem posed by N. Krugljak, L. Maligranda, and L.-E. Persson and pertaining to interpolation of couples of intersections that are generated by an integral functional in a weighted $L_p$-scale. Furthermore, an expression is found for the $\mathcal{K}$-functional on a couple of intersections corresponding to a linear functional, and some other related questions are treated.