Some Calderón–Zygmund kernels and their relations to Wolff capacities and rectifiability

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• 作者：Vasilis Chousionis ; Laura Prat
• 刊名：Mathematische Zeitschrift
• 出版年：2016
• 出版时间：February 2016
• 年：2016
• 卷：282
• 期：1-2
• 页码：435-460
• 全文大小：603 KB
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• 作者单位：Vasilis Chousionis (1)
  Laura Prat (2)
  
  1. Department of Mathematics, University of Connecticut, 196 Auditorium Road U-3009, Storrs, CT, 06269-3009, USA
  2. Departament de Matemàtiques, Universitat Autònoma de Barcelona, Catalonia, Spain
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-1823

文摘

We consider the Calderón–Zygmund kernels \(K_ {\alpha ,n}(x)=(x_i^{2n-1}/|x|^{2n-1+\alpha })_{i=1}^d\) in \({\mathbb R}^d\) for \(0<\alpha \le 1\) and \(n\in \mathbb {N}\). We show that, on the plane, for \(0<\alpha <1\), the capacity associated to the kernels \(K_{\alpha ,n}\) is comparable to the Riesz capacity \(C_{\frac{2}{3}(2-\alpha ),\frac{3}{2}}\) of non-linear potential theory. As consequences we deduce the semiadditivity and bilipschitz invariance of this capacity. Furthermore we show that for any Borel set \(E\subset {\mathbb R}^d\) with finite length the \(L^2(\mathcal {H}^1 \lfloor E)\)-boundedness of the singular integral associated to \(K_{1,n}\) implies the rectifiability of the set E. We thus extend to any ambient dimension, results previously known only in the plane. V. Chousionis is funded by the Academy of Finland Grant SA 267047. Also, partially supported by the ERC Advanced Grant 320501, while visiting Universitat Autònoma de Barcelona.L. Prat is supported by grants 2009SGR-000420 (Generalitat de Catalunya) and MTM2010-15657 (Spain).