一般奇异积分交换子的粗糙谱渐近
Rough spectral asymptotics for commutators of general singular integrals
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中文总结 AI 辅助
本文证明了度量测度空间上一般奇异积分交换子奇异值的谱渐近公式（用上下极限代替精确极限），并提出了Marcinkiewicz插值定理的新渐近版本及临界指标处交换子上界的新方法。
中文摘要 AI 辅助
最近，在Connes量子化微积分背景下产生的交换子的Schatten类成员资格已在一般框架中被刻画，该框架涵盖多个具体且具有重要性的情形。与此相反，关于这些交换子的奇异值渐近行为的相关结果仅局限于少数几个具体例子。在本文中，我们证明了一个谱渐近公式的版本，该版本以可比较的下极限和上极限代替精确极限，在度量测度空间上的奇异积分中具有广泛的普适性。一个具有独立意义的关键证明要素是Marcinkiewicz插值定理的一个新渐近版本。我们还提出了在临界指标处交换子上界估计的一种新方法；虽然该方法不如更精细的方法那样普遍，但它足以以更简单的方式重现Heisenberg群和Carnot群中已知的上界。
英文摘要
Recently, Schatten class membership of commutators arising from Connes' quantised calculus has been characterised in a general framework, which covers multiple concrete situations of interest. In contrast to this, related results dealing with the asymptotic behaviour of the singular values of these commutators have been restricted to a relatively short list of specific examples only. In this work, we show that a version of the spectral asymptotic formula, involving comparable lower and upper limits in place of an exact limit, remains valid in great generality of singular integrals over metric measure spaces. A key proof ingredient of independent interest is a new asymptotic version of the Marcinkiewicz interpolation theorem. We also present a new approach to commutator upper bounds at the critical index; while not quite as general as more elaborate methods, it is general enough to reproduce the known upper bounds in Heisenberg and Carnot groups in a much simpler way.
发表机构
- Aalto University（阿尔托大学）
机构由 AI 辅助整理，请以论文原文为准。
