On dimension and regularity of bundle measures

Abstract

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.

Dimensional estimates and rectifiability for measures satisfying linear PDE constraints

“…We note that the same conjecture has also been advanced by Raita in [28, Question 5.11]; also see [10,Conjecture 1.5].…”

Hardy-Littlewood-Sobolev inequality for $p=1$

“…The question for general Ω seems to be open. Partial results were obtained in [3], [5], [38] and [54]. The author supposes that the methods in the present text can also help in this related problem; see the preprint [51].…”

“…One can prove that the Schwartz functions are dense in W Ω 1 (see the preprint [52]). A good definition of the distributional space W related to Ω was introduced by Ayoush and Wojciechowski in [5]. For any L ∈ G(l, k), we denote by π L the orthogonal projection of C l onto L. Definition 1.…”