Open Access
2014 Marcinkiewicz interpolation theorems for Orlicz and Lorentz gamma spaces
Ron Kerman, Colin Phipps, Lubosš Pick
Publ. Mat. 58(1): 3-30 (2014).


Fix the indices $\alpha$ and $\beta$, $1<\alpha<\beta<\infty$, and suppose $\varrho$ is an Orlicz gauge or Lorentz gamma norm on the real-valued functions on a set $X$ which are measurable with respect to a~$\sigma$-finite measure $\mu$ on it. Set

$$M(\gamma,X):=\{f\colon X\to\mathbb R \text{ with } \sup_{\lambda>0}\lambda \mu(\{x\in X: |f(x)|>\lambda\})^{\frac1{\gamma}}<\infty\},$$

$\gamma=\alpha,\beta$. In this paper we obtain, as a special case, simple criteria to guarantee that a linear operator $T$ satisfies $T\colon L_{\varrho}(X)\to L_{\varrho}(X)$, whenever $T\colon M(\alpha,X)\to M(\alpha, X)$ and $T\colon M(\beta,X)\to M(\beta, X)$.


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Ron Kerman. Colin Phipps. Lubosš Pick. "Marcinkiewicz interpolation theorems for Orlicz and Lorentz gamma spaces." Publ. Mat. 58 (1) 3 - 30, 2014.


Published: 2014
First available in Project Euclid: 20 December 2013

zbMATH: 1305.46020
MathSciNet: MR3161506

Primary: 46E35
Secondary: 46E30

Keywords: optimality , rearrangement-invariant norms , Sobolev imbeddings , Supremum operators

Rights: Copyright © 2014 Universitat Autònoma de Barcelona, Departament de Matemàtiques

Vol.58 • No. 1 • 2014
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