Interpolation of Intersections Generated by a Linear Functional

Let $(X_0,X_1)$ be a Banach couple such that $X_0\cap X_1$ is dense in $X_0$ and $X_1$. By $(X_0,X_1)_{\theta,q}$, $0<\theta<1$, $1\le q<\infty$, we denote the spaces of the real interpolation method. Let $\psi$ be a nonzero linear functional defined on some linear space $M\subset X_0+X_1$ and such that $\psi\in(X_0\cap X_1)^*$, and let $N=\operatorname{Ker}\psi$. We examine conditions under which the natural formula
$$ (X_0\cap N,X_1\cap N)_{\theta,q}=(X_0,X_1)_{\theta,q}\cap N $$
is valid. In particular, the results obtained here imply those due to Ivanov and Kalton on the comparison of the interpolation spaces $(X_0,X_1)_{\theta,q}$ and $(N_0,X_1)_{\theta,q}$, where $\psi\in X_0^*$ and $N_0=\operatorname{Ker}\psi$. By way of application, we consider a problem, posed by Krugljak, Maligranda, and Persson, on the interpolation of intersections generated by an integral functional defined on weighted $L_p$-spaces.