A new interpolation approach to spaces of Triebel–Lizorkin type

Abstract

We introduce in this paper new interpolation methods for closed subspaces of Banach function spaces. For $q\in[1,\infty]$, the $l^{q}$-interpolation method allows to interpolate linear operators that have bounded $l^{q}$-valued extensions. For $q=2$ and if the Banach function spaces are $r$-concave for some $r<\infty$, the method coincides with the Rademacher interpolation method that has been used to characterize boundedness of the $H^{\infty}$-functional calculus. As a special case, we obtain Triebel–Lizorkin spaces $F^{2\theta}_{p,q}(\mathbb{R}^{d})$ by $l^{q}$-interpolation between $L^{p}(\mathbb{R}^{d})$ and $W^{2}_{p}(\mathbb{R}^{d})$ where $p\in(1,\infty)$. A similar result holds for the recently introduced generalized Triebel–Lizorkin spaces associated with $R_{q}$-sectorial operators in Banach function spaces. So, roughly speaking, for the scale of Triebel–Lizorkin spaces our method thus plays the role the real interpolation method plays in the theory of Besov spaces.