Grand and small Lebesgue spaces and their analogs
type: Item
instance of: scholarly article
title: Grand and small Lebesgue spaces and their analogs
zbMATH Open document ID: 1071.46023
DOI: 10.4171/ZAA/1215
author: A. Fiorenza
author: Georgi E. Karadzhov
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 2005-04-11T00:00:00Z
review text: Summary: We give the following, equivalent, explicit expressions for the norms of the small and grand Lebesgue spaces, which depend only on the non-decreasing rearrangement (we assume here that the underlying measure space has measure 1): \[ \|f\|_{L^(p}\approx\int^1_0(1-\ln t)^{-\frac 1p}\left(\int^t_0 \bigl[f^*(s) \bigr]^pds\right)^{\frac 1p}dt/t \quad (1<p<\infty) \] \[ \|f\|_{L^p)}\approx \sup_{0<t<1}(1-\ln t)^{-\frac 1p}\left(\int^1_t\bigl[f^*(s)\bigr]^pds \right)^{\frac 1p}\quad (1< p<\infty). \] Similar results are proven for the generalized small and grand spaces.
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 46B70
zbMATH DE Number: 2155001
zbMATH Keywords: extrapolation
zbMATH Keywords: interpolation
zbMATH Keywords: Lorentz-Karumata spaces
MaRDI profile type: Publication
cites work: Q3905932
cites work: Quelques problèmes de Dirichlet avec données dans de grands espaces de Sobolev
cites work: Q3994467
cites work: Extrapolation theory for the real interpolation method.
cites work: New extrapolation estimates
cites work: Formulae for the distance in some quasi-Banach spaces
cites work: Duality and reflexivity in grand Lebesgue spaces
cites work: Q3148784
cites work: New properties of small Lebesgue spaces and their applications
cites work: Q4376706
cites work: Sobolev imbedding theorems in borderline cases
cites work: A remark on the equality \(\text{det} Df=\text{Det} Df\)
cites work: Inverting the \(p\)-harmonic operator
cites work: Q5571194
cites work: Extrapolation theory with applications
cites work: Q4891389
cites work: Mappings of finite distortion: Monotonicity and continuity
cites work: On the integrability of the Jacobian under minimal hypotheses
cites work: Q4508977
cites work: Extrapolation and optimal decompositions: with applications to analysis
cites work: A note on extrapolation theory
cites work: Q5699008
cites work: Extrapolation results on general Besov-Hölder-Lipschitz spaces
cites work: Some new applications of the pointwise relations for the relative rearrangement
cites work: Q3838240
cites work: Q4523721
cites work: Notes on Fourier analysis. XXIX. An extrapolation theorem
full work available at URL: https://doi.org/10.4171/zaa/1215
OpenAlex ID: W2056039888
description: scientific article; zbMATH DE number 2155001