Porosity and products of Orlicz spaces

Porosity and products of Orlicz spaces Let $$(\Omega , \Sigma , \mu )$$ be a measure space and let $$\varphi _1, \ldots , \varphi _n$$ and $$\varphi $$ be Young functions. In this paper, we, among other things, prove that the set $$E=\{(f_1, \ldots ,f_n)\in M^{\varphi _1}\times \cdots \times M^{\varphi _n}:\, N_\varphi (f_1\cdots f_n)<\infty \}$$ is a $$\sigma $$ - $$c$$ -lower porous set in $$M^{\varphi _1}\times \cdots \times M^{\varphi _n}$$ , under mild restrictions on the Young functions $$\varphi _1, \ldots , \varphi _n$$ and $$\varphi $$ . This generalizes a recent result due to Gł a̧ b and Strobin (J Math Anal Appl 368:382–390, 2010) to more general setting of Orlicz spaces. As an application of our results, we recover a sufficient and necessary condition for Orlicz spaces to be closed under the pointwise multiplication due to Hudzik (Arch Math 44:535–538, 1985) and Arens et al. (J Math Anal Appl 177:386–411, 1993). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

Porosity and products of Orlicz spaces

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Publisher
Springer Basel
Copyright
Copyright © 2012 by Springer Basel
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-012-0218-0
Publisher site
See Article on Publisher Site

Abstract

Let $$(\Omega , \Sigma , \mu )$$ be a measure space and let $$\varphi _1, \ldots , \varphi _n$$ and $$\varphi $$ be Young functions. In this paper, we, among other things, prove that the set $$E=\{(f_1, \ldots ,f_n)\in M^{\varphi _1}\times \cdots \times M^{\varphi _n}:\, N_\varphi (f_1\cdots f_n)<\infty \}$$ is a $$\sigma $$ - $$c$$ -lower porous set in $$M^{\varphi _1}\times \cdots \times M^{\varphi _n}$$ , under mild restrictions on the Young functions $$\varphi _1, \ldots , \varphi _n$$ and $$\varphi $$ . This generalizes a recent result due to Gł a̧ b and Strobin (J Math Anal Appl 368:382–390, 2010) to more general setting of Orlicz spaces. As an application of our results, we recover a sufficient and necessary condition for Orlicz spaces to be closed under the pointwise multiplication due to Hudzik (Arch Math 44:535–538, 1985) and Arens et al. (J Math Anal Appl 177:386–411, 1993).

Journal

PositivitySpringer Journals

Published: Dec 16, 2012

References

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