Multiplication in Sobolev spaces, revisited
Multiplication in Sobolev spaces, revisited
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DOI：
10.4310/arkiv.2021.v59.n2.a2
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发表时间：
2015-12
期刊：
影响因子：
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通讯作者：
A. Behzadan;Michael Holst
中科院分区：
文献类型：
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作者：
A. Behzadan;Michael Holst
在本文中，我们重新审视 Sobolev-Slobodeckij 空间中的一些经典点乘法定理，并在此过程中我们引用了一个简单的反例，说明当有界域被 Rn 替换时，某些乘法定理如何在 Sobolev-Slobodeckij 空间中失败。我们确定了故障的根源，并检查了为什么在贝塞尔势空间中没有遇到相同的故障。为了仔细分析这种情况，我们首先对 Zolesio 1977 年文章中陈述和证明的经典乘法结果进行调查，并仔细区分在所有 Rn 上定义的空间和在有界域（例如 Lipschitz 边界）上定义的空间的情况。然而，我们提供的调查有一些新的问题；我们提供的证明几乎完全基于插值理论，而不是 Littlewood-Paley 理论和 Besov 空间，并且我们给出的一些结果及其证明（包括负指数的结果）并未以此处介绍的方式出现在文献中。我们还包括乘法定理之一的一个特别重要的变体，该定理与广义相对论和其他领域中出现的非线性偏微分方程系统的研究相关。在 Sobolev-Slobodeckij 空间的情况下，乘法连续的条件有些微妙且相互交织，因此，Zolesio 在 1977 年的乘法定理在标准文献中被引用（不止一次），其概括性比 Zolesio 实际证明的要稍微普遍，并且在允许构建反例（例如此处包含的例子）的情况下。
In this article, we re-examine some of the classical pointwise multiplication theorems in Sobolev-Slobodeckij spaces, and along the way we cite a simple counter-example that illustrates how certain multiplication theorems fail in Sobolev-Slobodeckij spaces when a bounded domain is replaced by Rn. We identify the source of the failure, and examine why the same failure is not encountered in Bessel potential spaces. To analyze the situation carefully, we begin with a survey of the classical multiplication results stated and proved in the 1977 article of Zolesio, and we carefully distinguish between the case of spaces defined on the all of Rn and spaces defined on a bounded domain (with e.g. a Lipschitz boundary). However, the survey we give has a few new wrinkles; the proofs we include are based almost exclusively on interpolation theory rather than Littlewood-Paley theory and Besov spaces, and some of the results we give and their proofs, including the results for negative exponents, do not appear in the literature in the way presented here. We also include a particularly important variation of one of the multiplication theorems that is relevant to the study of nonlinear PDE systems arising in general relativity and other areas. The conditions for multiplication to be continuous in the case of Sobolev-Slobodeckij spaces are somewhat subtle and intertwined, and as a result, the multiplication theorems of Zolesio in 1977 have been cited (more than once) in the standard literature in slightly more generality than what is actually proved by Zolesio, and in cases that allow for the construction of counter-examples such as the one included here.