The Real Interpolation Method on Couples of Intersections

Suppose that $(X_0,X_1)$ is a Banach couple, $X_0\cap X_1$ is dense in $X_0$ and $X_1$, $(X_0,X_1)_{\theta,q}$ ($0<\theta<1$, $1\le q<\infty$) are the spaces of the real interpolation method, $\psi\in(X_0\cap X_1)^*$, $\psi\ne 0$, is a linear functional, $N=\operatorname{Ker}\psi$, and $N_i$ stands for $N$ with the norm inherited from $X_i$ ($i=0,1$). The following theorem is proved: the norms of the spaces $(N_0,N_1)_{\theta,q}$ and $(X_0,X_1)_{\theta,q}$ are equivalent on $N$ if and only if $\theta\in(0,\alpha)\cup(\beta_\infty,\alpha_0)\cup(\beta_0,\alpha_\infty)\cup(\beta,1)$, where $\alpha$, $\beta$, $\alpha_0$, $\beta_0$, $\alpha_\infty$, and $\beta_\infty$ are the dilation indices of the function $k(t)=\mathcal{K}(t,\psi;X_0^*,X_1^*)$.