Singular distributions, dimension of support, and symmetry of Fourier transform

Kozma, Gady; Olevskii, Alexander

doi:10.5802/aif.2801

Cited by 3 publications

(1 citation statement)

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“…If order of smoothness −s may be taken arbitrarily close to zero, then the lower bound of Hausdorff dimension remains the same (though this trade-off formally costs us the result about projections). This answer is obtained by the following lemma which employs a technique used in [15]. Lemma 3.9.…”

Section: Our Efforts Can Be Summerized As Followsmentioning

confidence: 99%

On dimension and regularity of bundle measures

Ayoush,

Wojciechowski

2017

Preprint

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In this paper we quantify the notion of antisymmetry of the Fourier transform of certain vector valued measures. The introduced scale is related to the condition appearing in Uchiyama's theorem and is used to give a lower bound for the rectifiable dimension of those measures. Moreover, we obtain an estimate of the lower Hausdorff dimension assuming certain more restrictive version of the 2-wave cone condition. Results of our considerations can be viewed as an uncertainty-type principle in the following way: it is impossible to simultaneously localize a (bundle) measure and a direction of its Fourier transform on small sets. The investigated class is modeled on the example of gradients of BV functions. The article contains also a theorem concerning regularity: we prove that elements of considered class vanish on 1-purely unrectifiable sets. Our results can be applied to studying the properties of PDE-constrainted measures.

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Section: Our Efforts Can Be Summerized As Followsmentioning

confidence: 99%