On the problem of existence in principal value of a Calderón–Zygmund operator on a space of non‐homogeneous type

Abstract

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...

Rectifiability; a survey

“…The needed L 2 boundedness again follows by T (b) theorems. The proof is also based on the earlier work [JM20a] and [JM20b] of these authors.…”

“…In [MV09a] we proved with Verdera under very general conditions that the L 2 -boundedness together with zero lower density implies the existence of principal values. Jaye and Merchán [JM20a] strengthened this by replacing zero density by the condition that modifications of Tolsa's αs (recall Section 7.4) tend to zero. See also [JM20b] for related results and recall the discussion in 10.5 on [JM21].…”

The Huovinen transform and rectifiability of measures

“…We have thus far recorded two necessary conditions for the existence of principal value in Theorems A and B. Taken individually, neither condition needs to imply the existence of principal value, but by building on prior work of Mattila-Verdera [MV], we showed in [JM2] that when combined, these two necessary conditions are indeed sufficient:…”

“…It appears that all variants of the Legér scheme (for instance in [L,CMPT,To5,JTV]) have relied on this local flattening property in one way or another. In this paper we circumvent these difficulties with a novel decomposition of a measure involving a modified density, relying significantly on our previous papers [JM1,JM2], which we recall in the next sections.…”

“…Theorem C. [JM2,Theorem 1.5] Suppose that µ is a finite non-atomic Borel measure satisfying the transportation coefficient condition (1.4), and the L 2 -boundedness condition (1.5) holds. Then the principal value limit (1.2) exists.…”