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 ∈ [1,∞] the lq-interpolation method allows to interpolate linear operators that have bounded lq-valued extensions. For q = 2 and if the Banach function spaces are r-concave for some r <∞, the method coincides with the Rademacher interpolation method that has been used to characterize boundedness of the H∞-functional calculus. As a special case, we obtain Triebel-Lizorkin spaces F 2θ p,q(R) by lq-interpolation between Lp(Rd) and W 2 p (Rd) where p ∈ (1,∞). A similar result holds for the recently introduced generalized Triebel-Lizorkin spaces associated with Rq-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. AMS subject classification (MSC2010): 46 B 70, 47 A 60, 42 B 25 keywords: interpolation, sectorial operators, Triebel-Lizorkin spaces, real interpolation method, H∞-functional calculus, Rademacher interpolation method, γinterpolation method