Small local action of singular integrals on spaces of non-homogeneous type

Abstract

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 principal value integral to exist. 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 applications, we obtain a soft proof of a theorem of Tolsa and Ruiz-de-Villa on the non-existence of a measure with positive and finite upper density 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 $s\in (d-1,d)$.