Rectifiability of harmonic measure

Azzam, Jonas; Hofmann, Steve; Martell, José María; Mayboroda, Svitlana; Mourgoglou, Mihalis; Tolsa, Xavier; Volberg, Alexander

doi:10.1007/s00039-016-0371-x

Cited by 65 publications

(45 citation statements)

References 36 publications

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“…Now the main the result from [4] asserts that ω + | G pd m (x 0 ,r 0 ) is n-rectifiable and proves the lemma.…”

Section: Note Now Thatmentioning

confidence: 61%

“…The above lemma was originally stated in [4] for bounded domains, but it holds for unbounded domains with the same proof using the fact that, for n ≥ 2, any domain Ω ⊂ R n+1 is Greenian and, if it is unbounded, ∞ is a Wiener regular point (see [3,Theorem 6.7…”

Section: Background On Harmonic Measurementioning

confidence: 99%

“…The connection between the Riesz transform and harmonic measure is due to the fact that the Riesz kernel is the gradient of the Newtonian potential, and the relationship between the Riesz transform and rectifiability is a subject that has been in constant development for the last twenty years and has culminated in the solution of the David-Semmes conjecture by Nazarov, Tolsa and Volberg [29,30]. For recent examples of Riesz transform techniques used to study harmonic measure, see for instance [10,4,28]. The arguments in the current paper use a new recent result by Girela-Sarrión and Tolsa [18] on the connection between Riesz transforms and quantititative rectifiability for general Radon measures (see Theorem 3.1 below for more details).…”

Section: Theorem Bmentioning

confidence: 99%

“…The notation "´d " stands for "zero density," and "pd " stands for "positive density." Now the main result from [4] asserts that ! C j G pd m .x 0 ;r 0 / is n-rectifiable and proves the lemma.…”

Section: Proof Formentioning

confidence: 99%

See 3 more Smart Citations

Mutual Absolute Continuity of Interior and Exterior Harmonic Measure Implies Rectifiability

Azzam

Mourgoglou

Tolsa

2017

Comm Pure Appl Math

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“…Now the main the result from [4] asserts that ω + | G pd m (x 0 ,r 0 ) is n-rectifiable and proves the lemma.…”

Section: Note Now Thatmentioning

confidence: 61%

Section: Background On Harmonic Measurementioning

confidence: 99%

Section: Theorem Bmentioning

confidence: 99%

Section: Proof Formentioning

confidence: 99%

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Mutual Absolute Continuity of Interior and Exterior Harmonic Measure Implies Rectifiability

Azzam

Mourgoglou

Tolsa

2017

Comm Pure Appl Math

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“…We set Spike k to be the collection of all such k-spike measures over L ∈ G (1,2), ω ∈ C, and m | k.…”

Section: On the Huovinen Kernelsmentioning

confidence: 99%

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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Uniform Rectifiability and Elliptic Operators Satisfying a Carleson Measure Condition

et al. 2021

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Rectifiability of harmonic measure

Cited by 65 publications

References 36 publications

Mutual Absolute Continuity of Interior and Exterior Harmonic Measure Implies Rectifiability

Mutual Absolute Continuity of Interior and Exterior Harmonic Measure Implies Rectifiability

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

Uniform Rectifiability and Elliptic Operators Satisfying a Carleson Measure Condition

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