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).