In this paper we consider nonnegative functions f on \(\mathbb {R}^n\) which are defined either by \(f(x)=\min \,(f_1(x_1),\ldots ,f_n(x_n))\) or by \(f(x)=\min \,(f_1(\hat{x}_1),\ldots ,f_n(\hat{x}_n)).\) Such minimum-functions are useful, in particular, in embedding theorems. We prove sharp estimates of rearrangements and Lorentz type norms for these functions, and we find the link between their Lorentz norms and geometric properties of their level sets.

中文翻译：

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关于某些Fournier–Gagliardo型不等式

在本文中，我们考虑\（\ mathbb {R} ^ n \）上的非负函数f，它们由\（f（x）= \ min \，（f_1（x_1），\ ldots，f_n（x_n））定义\）或通过\（F（X）= \分钟\，（F_1（\帽子{X} _1），\ ldots，F_N（\帽子{X} _n））。\）这样的最小功能是有用的，在特别是在嵌入定理时。我们证明了对这些函数的重排和Lorentz类型规范的清晰估计，并且我们发现了它们的Lorentz规范与它们的级别集的几何特性之间的联系。