Illinois Journal of Mathematics

A new interpolation approach to spaces of Triebel–Lizorkin type

Peer Christian Kunstmann

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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.

Article information

Source
Illinois J. Math., Volume 59, Number 1 (2015), 1-19.

Dates
Received: 6 October 2014
Revised: 13 February 2015
First available in Project Euclid: 11 February 2016

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1455203156

Digital Object Identifier
doi:10.1215/ijm/1455203156

Mathematical Reviews number (MathSciNet)
MR3459625

Zentralblatt MATH identifier
1346.46018

Subjects
Primary: 46B70: Interpolation between normed linear spaces [See also 46M35] 47A60: Functional calculus 42B25: Maximal functions, Littlewood-Paley theory

Citation

Kunstmann, Peer Christian. A new interpolation approach to spaces of Triebel–Lizorkin type. Illinois J. Math. 59 (2015), no. 1, 1--19. doi:10.1215/ijm/1455203156. https://projecteuclid.org/euclid.ijm/1455203156


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