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Families of Young functions and limits of Orlicz norms

Part of: Classical measure theory Linear function spaces and their duals

Published online by Cambridge University Press:  29 May 2023

Sullivan F. MacDonald  and
Scott Rodney
Sullivan F. MacDonald
Affiliation:
Department of Mathematics & Statistics, McMaster University, Hamilton, ON L8S4L8, Canada e-mail: macdos55@mcmaster.ca
Scott Rodney*
Affiliation:
Department of Mathematics, Physics and Geology, Cape Breton University, Sydney, NS B1Y3V3, Canada
*
e-mail: scott_rodney@cbu.ca
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Abstract

Given a $\sigma $-finite measure space $(X,\mu )$, a Young function $\Phi $, and a one-parameter family of Young functions $\{\Psi _q\}$, we find necessary and sufficient conditions for the associated Orlicz norms of any function $f\in L^\Phi (X,\mu )$ to satisfy

$$\begin{align*}\lim_{q\rightarrow \infty}\|f\|_{L^{\Psi_q}(X,\mu)}=C\|f\|_{L^\infty(X,\mu)}. \end{align*}$$

The constant C is independent of f and depends only on the family $\{\Psi _q\}$. Several examples of one-parameter families of Young functions satisfying our conditions are given, along with counterexamples when our conditions fail.

Keywords

Orlicz spacesfunctional analysisclassical analysis

MSC classification

Primary: 28A25: Integration with respect to measures and other set functions 46E30: Spaces of measurable functions ($L^p$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
Type
Article
Information
Canadian Mathematical Bulletin , Volume 67 , Issue 1 , March 2024 , pp. 26 - 39
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

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Footnotes

S. Rodney was supported by the NSERC Discovery Grant Program. S.F. MacDonald was supported by the NSERC USRA Program and the Department of Mathematics & Statistics at McMaster University.

References

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