Fix $d\geq 2$ and $s\in (0,d)$. In this paper we introduce a notion called small local action associated to a singular integral operator, which is a necessary condition for the existence of the principal value integral. Our goal is to understand the geometric properties of a measure for which an associated singular integral has small local action. We revisit Mattila's theory of symmetric measures and relate, under the condition that the measure has finite upper density, the existence of small local action to the cost of transporting the measure to a collection of symmetric measures. As an application, we obtain a soft proof of a theorem of Ruiz de Villa and Tolsa on the measures for which the principal value integral associated with the $s$-Riesz transform exists if $s\not\in \mathbb{Z}$. Furthermore, we provide a considerable generalization of this theorem if $s\in (d-1,d)$.

中文翻译：

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非齐次空间上奇异积分的小局部作用

修复$ d \ geq 2 $和$ s \ in（0，d）$。在本文中，我们介绍了一个称为小局部行为的概念与奇异积分算子相关联，这是存在主值积分的必要条件。我们的目标是了解相关奇异积分具有较小局部作用的量度的几何特性。我们回顾了Mattila的对称测度理论，并在测度具有有限的上限密度的条件下，将小局部作用的存在与将测度传输到一组对称测度的成本相关联。作为一个应用程序，我们获得了Ruiz de Villa和Tolsa定理的软证明，其中关于如果$ s \ not \ in \ mathbb {Z} $中存在与$ s $ -Riesz变换相关的主值积分的度量。此外，如果$ s \ in（d-1，d）$，我们对该定理进行相当大的推广。