Iterated Weak and Weak Mixed-Norm Spaces with Applications to Geometric Inequalities
Iterated Weak and Weak Mixed-Norm Spaces with Applications to Geometric Inequalities
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DOI：
10.1007/s12220-019-00243-x
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发表时间：
2020-12
期刊：
影响因子：
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通讯作者：
Ting Chen;Wenchang Sun
中科院分区：
文献类型：
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作者：
Ting Chen;Wenchang Sun
本文考虑了具有混合范数的勒贝格空间中的两类弱范数：弱混合范数和迭代弱范数。我们研究了两个弱模的性质，并给出了它们之间的关系。即使对于普通的勒贝格空间，这两个弱范数也不是等价的，它们中的任何一个都不能控制另一个。我们分别给出了这两个弱范数的一些收敛和完备性结果。我们研究了截断范数的收敛问题，它取代了混合范数的勒贝格空间的测度收敛。并给出了截断范数收敛的一个刻画。我们证明了在弱混合范数的勒贝格空间上Hölder不等式并不总是成立的，并且我们给出了允许Hölder不等式的指数的完全刻画。作为应用，我们分别在弱混合范数空间和迭代弱空间中建立了与分数次积分有关的几何不等式，本质上推广了Hardy-Littlewood-Soblev不等式。
In this paper, we consider two types of weak norms, the weak mixed-norm and the iterated weak norm, in Lebesgue spaces with mixed norms. We study properties of two weak norms and present their relationship. Even for the ordinary Lebesgue spaces, the two weak norms are not equivalent and any one of them can not control the other one. We give some convergence and completeness results for the two weak norms, respectively. We study the convergence in the truncated norm, which is a substitution of the convergence in measure for mixed-norm Lebesgue spaces. And we give a characterization of the convergence in the truncated norm. We show that Hölder’s inequality is not always true on weak mixed-norm Lebesgue spaces and we give a complete characterization of indices which admit Hölder’s inequality. As applications, we establish some geometric inequalities related to fractional integrals in weak mixed-norm spaces and in iterated weak spaces, respectively, which essentially generalize the Hardy–Littlewood–Sobolev inequality.