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Mathematische Zeitschrift

, Volume 255, Issue 4, pp 813–825 | Cite as

Extrapolation results for q < 1

  • María J. Carro
Article
  • 73 Downloads

Abstract

Given a sublinear operator T such that \({T:\ell\log\ell \rightarrow X}\) is bounded, it can be shown that \({T:\ell^q\rightarrow X}\) is bounded, with constant C/(1−q), for every 0 < q < 1. In this paper, we study the converse result, not only for sequence spaces, but for general measure spaces proving that, if T : L q (μ) → X is bounded, with constant C/(1−q), for every \({q_0\le q < 1}\) and X is Banach, then T : L log (1/L)(μ) → X is bounded. Moreover, this result is optimal. We also show that things are quite different if the Banach condition on X is dropped.

Keywords

Extrapolation theory Weak type estimates Operators acting on sequences 

Mathematics Subject Classification (2000)

47A30 47A63 

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Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Departament de Matemàtica Aplicada i AnàlisiUniversitat de BarcelonaBarcelonaSpain

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