Non existence of principal values of signed Riesz transforms of non integer dimension

Villa, Aleix Ruiz de; Tolsa, Xavier

doi:10.1512/iumj.2010.59.3884

Cited by 10 publications

(6 citation statements)

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“…The Mayboroda-Volberg Theorem. Building on the tools developed in [Tol1,RdVT], Mayboroda and Volberg [MV1,MV2] proved that if µ is a non-trivial finite measure with H s (supp(µ)) < ∞, and S µ (1) < ∞ µ-almost everywhere, then s ∈ Z and supp(µ) is s-rectifiable (see Section 2.6 below for the definition). When combined with Theorem 1.1 of Azzam-Tolsa [AT], Theorems 1.1 and 1.2 above provide another demonstration of this result.…”

Section: Introductionmentioning

confidence: 99%

The measures with an associated square function operator bounded in L2

Jaye

Nazarov

Tolsa

2018

Advances in Mathematics

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Section: Introductionmentioning

confidence: 99%

The measures with an associated square function operator bounded in L2

Jaye

Nazarov

Tolsa

2018

Advances in Mathematics

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“…Theorems 1.3 and 1.4 are both sharp for the s-Riesz kernel K(x) = x |x| s+1 . Indeed, we recall the following theorem, which is a consequence of results by Mattila-Preiss [22], Tolsa [32] and Ruiz de Villa-Tolsa [28]. For a relatively simple direct proof, see [11,Theorem 1.2].…”

Section: )mentioning

confidence: 93%

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

Jaye

Merchán

2019

Proc. London Math. Soc.

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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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“…Let us mention that, for the Cauchy transform, the same results were obtained in [19] with some density assumptions, and in [29] by using the notion of curvature of measures. For other results dealing with principal values, Hausdorff measures, rectifiability, and related questions, see also [8], [23], [6], [33], [27], and [28], for example.…”

Section: Vii-4mentioning

confidence: 99%