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Spring 2015 A new interpolation approach to spaces of Triebel–Lizorkin type
Peer Christian Kunstmann
Illinois J. Math. 59(1): 1-19 (Spring 2015). DOI: 10.1215/ijm/1455203156

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.

Citation

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Peer Christian Kunstmann. "A new interpolation approach to spaces of Triebel–Lizorkin type." Illinois J. Math. 59 (1) 1 - 19, Spring 2015. https://doi.org/10.1215/ijm/1455203156

Information

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

zbMATH: 1346.46018
MathSciNet: MR3459625
Digital Object Identifier: 10.1215/ijm/1455203156

Subjects:
Primary: 42B25 , 46B70 , 47A60

Rights: Copyright © 2015 University of Illinois at Urbana-Champaign

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Vol.59 • No. 1 • Spring 2015
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