Interpolation of Intersections Generated by a Linear Functional

S. V. Astashkin

Funkcionalʹnyj analiz i ego priloženiâ, Tome 39 (2005) no. 2, pp. 61-64

Résumé

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 $(X_0,X_1)_{\theta,q}$, $0\theta1$, $1\le q\infty$, 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.

Détail

Keywords: Banach space, subspace, Banach couple, subcouple, $\mathcal{K}$-functional, real interpolation method, weighted space.
Mots-clés : interpolation space