The inclusion problem for mixed norm spaces
The inclusion problem for mixed norm spaces
复制标题
混合范数空间的包含问题
DOI：
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发表时间：
2016
期刊：
影响因子：
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通讯作者：
G. Sinnamon
中科院分区：
文献类型：
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作者：
Wayne Grey;G. Sinnamon
给定任意σ-有限测度空间的n重积上的两个混合范数Lebesgue空间，当其中一个包含在另一个中时？如果是，那么包含映射的范数是什么？这些问题完全回答了大范围的勒贝格指数和所有的措施空间。当测度空间是无原子的时，这两个问题对于所有指数都得到了解决。当测度空间不是纯原子的时候，第一个问题对于所有的指数都得到了解决。在其余的情况下，给出了一些完整的和部分的结果，但观察到各种各样的行为。特别是，纯粹原子测度空间的范数问题在勒贝格指数的某些范围内被认为是难以处理的;它等价于一个包含已知NP难问题作为特例的优化问题。
Given two mixed norm Lebesgue spaces on an n-fold product of arbitrary σ-finite measure spaces, when 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.