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The inclusion problem for mixed norm spaces


Authors: Wayne Grey and Gord Sinnamon
Journal: Trans. Amer. Math. Soc. 368 (2016), 8715-8736
MSC (2010): Primary 46E30; Secondary 46A45, 26D15
DOI: https://doi.org/10.1090/tran6665
Published electronically: January 26, 2016
MathSciNet review: 3551586
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Abstract: Given two mixed norm Lebesgue spaces on an $ n$-fold product of arbitrary $ \sigma $-finite measure spaces, is one contained in the other? If so, what is the norm of the inclusion map? These questions are answered completely for a large range of Lebesgue indices and all measure spaces. When the measure spaces are atomless, both questions are settled for all indices. When the measure spaces are not purely atomic, the first question is settled for all indices. Some complete and some partial results are given in the remaining cases, but a wide variety of behaviour is observed. In particular, the norm problem for purely atomic measure spaces is seen to be intractable for certain ranges of the Lebesgue indices; it is equivalent to an optimization problem that includes a known NP-hard problem as a special case.


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Additional Information

Wayne Grey
Affiliation: Department of Mathematics, University of Western Ontario, London N6A 5B7, Canada
Email: wgrey@uwo.ca

Gord Sinnamon
Affiliation: Department of Mathematics, University of Western Ontario, London N6A 5B7, Canada
Email: sinnamon@uwo.ca

DOI: https://doi.org/10.1090/tran6665
Received by editor(s): October 30, 2014
Published electronically: January 26, 2016
Additional Notes: This work was supported by the Natural Sciences and Engineering Research Council of Canada
Article copyright: © Copyright 2016 American Mathematical Society