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Extrapolation of Lp Data from a Modular Inequality

Published online by Cambridge University Press:  20 November 2018

Steven Bloom  and
Ron Kerman
Steven Bloom
Affiliation:
Siena College, Loudonville, NY 12211, USA
Ron Kerman
Affiliation:
Brock University, St. Catharines, Ontario, L2S 3A1
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Abstract

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If an operator $T$ satisfies a modular inequality on a rearrangement invariant space ${{L}^{\rho}}\left( \Omega ,\mu \right)$, and if $p$ is strictly between the indices of the space, then the Lebesgue inequality $\int{|Tf{{|}^{P}}\le C\int{|f{{|}^{P}}}}$ holds. This extrapolation result is a partial converse to the usual interpolation results. A modular inequality for Orlicz spaces takes the form $\int{\Phi \left( |Tf| \right)\le }\int{\Phi \left( C|f| \right)}$, and here, one can extrapolate to the (finite) indices $i\left( \Phi \right)$ and $I\left( \Phi \right)$ as well.

Keywords

42B25
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 45 , Issue 1 , 01 March 2002 , pp. 25 - 35
Copyright
Copyright © Canadian Mathematical Society 2002

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