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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Sharp Marchaud and converse inequalities in Orlicz spaces


Authors: Z. Ditzian and A. V. Prymak
Journal: Proc. Amer. Math. Soc. 135 (2007), 1115-1121
MSC (2000): Primary 26A15, 26B99, 41A27; Secondary 41A63, 46E30
Published electronically: October 27, 2006
MathSciNet review: 2262913
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Abstract: For $ L_p$ spaces on $ \mathbb{T}^d$, $ \mathbb{R}^d$ and $ S^{d-1}$, sharp versions of the classical Marchaud inequality are known. These results are extended here to Orlicz spaces (on $ \mathbb{T}^d$, $ \mathbb{R}^d$ and $ S^{d-1}$) for which $ M(u^{1/q})$ is convex for some $ q$, $ 1<q\le2$, where $ M(u)$ is the Orlicz function. Sharp converse inequalities for such spaces are deduced.


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Additional Information

Z. Ditzian
Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1
Email: zditzian@math.ualberta.ca

A. V. Prymak
Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1
Email: prymak@gmail.com

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08682-5
PII: S 0002-9939(06)08682-5
Keywords: Sharp Marchaud inequality, Orlicz spaces, power-type $q$.
Received by editor(s): November 9, 2005
Published electronically: October 27, 2006
Additional Notes: The first author was supported by NSERC grant of Canada A4816.
This research was done while the second author visited University of Alberta in 2005; the visit was supported by the first author’s NSERC grant of Canada A4816
Communicated by: Jonathan M. Borwein
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.