The inclusion problem for mixed norm spaces

Grey, Wayne; Sinnamon, Gord

doi:10.1090/tran6665

The inclusion problem for mixed norm spaces
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by Wayne Grey and Gord Sinnamon PDF

Trans. Amer. Math. Soc. 368 (2016), 8715-8736 Request permission

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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