Fix d ≥ 2 and s ∈ (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 Ruizde-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 ∈ Z. Furthermore, we provide a considerable generalization of this theorem if s ∈ (d − 1, d). Date

In this paper, we study the relationship between two fundamental regularity properties of an s-dimensional Calderón-Zygmund operator (CZO) acting on a Borel measure μ in R d , with s ∈ (0, d).In the classical case when s = d and μ is equal to the Lebesgue measure, Calderón and Zygmund showed that if a CZO is bounded in L 2 , then the principal value integral exists almost everywhere. However, there are by now several examples showing that this implication may fail for lower dimensional kernels and measures, even when the CZO has a homogeneous kernel consisting of spherical harmonics.We introduce sharp geometric conditions on μ, in terms of certain scaled transportation distances, which ensure that an extension of the Calderón-Zygmund theorem holds. These conditions are necessary and sufficient in the cases of the Riesz transform and the Huovinen transform. Our techniques build upon prior work by Mattila and Verdera, and incorporate the machinery of symmetric measures, introduced to the area by Mattila.On the other hand, the CZO T exists in the sense of principal value if for every complex measure ν,( 1.2) For classical CZOs (s = d) acting in Euclidean space R d with μ = m d (the Lebesgue measure), a density argument ensures that the boundedness of a CZO in L 2 (m d ) implies the existence of the CZO in the sense of principal value m d -almost everywhere; see, for instance, [2,29].However, there are by now several examples which show that the Calderón-Zygmund theorem does not necessarily extend when the Lebesgue measure is changed to a different underlying measure, see, for example, [3,5]. It was shown in [13] that there is a measure μ satisfying μ(B(x, r)) r for every disc B(x, r) ⊂ C ∼ = R 2 such that the one-dimensional CZO associated to the Huovinen kernel K(z) = z k |z| k+1 , k 3 odd, is bounded in L 2 (μ) but the principal value integral fails to exist μ-almost everywhere. Huovinen [10] has previously studied the geometric consequences of the existence of the principal value integral associated to this kernel, which plays a significant role in the literature due to being the prototypical example of a onedimensional CZ kernel in the plane for which the Melnikov-Menger curvature formula (see, for example, [21]) fails to hold, see the survey papers [18,19].Notwithstanding these examples, it is expected that an analogue of the classical Calderón-Zygmund theorem should hold for the s-Riesz transform the CZO with kernel K(x) = x |x| s+1 (x ∈ R d ). Indeed, a long standing conjecture † states that if μ is a non-atomic measure, then whenever the s-Riesz transform operator is bounded in L 2 (μ), it also exists in principal value. This was proved for s = 1 by Tolsa ‡ (see [30]), and for s = (d − 1), where it can be proved by combining the deep results of Eiderman-Nazarov-Volberg [8], Nazarov-Tolsa-Volberg [24], and Mattila-Verdera (stated as Theorem 1.3 below) [23]. It is an open problem for s = 2, . . . , d − 2.The results described in the preceding paragraphs combine to show that the problem of when (1.1) implies (1...

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For a set E of positive and finite length, we prove that if the Huovinen transform (the convolution operator with kernel z k /|z| k+1 for an odd number k) associated to E exists in principal value, then E is rectifiable.1 building upon a number of important results including [Me, MMV, Dav3, DM, MM, NTV2]2 More precisely, these techniques play an important role in Theorem A below.3 We say that a Borel measure µ is s-rectifiable if there exist Lipschitz maps

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