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

Abstract

We consider the Calderón-Zygmund kernels $K_{\alpha ,n}(x)=(x_i^{2n-1}/|x{|}^{2n-1+\alpha }{)}_{i=1}^d$ in ${\mathbb{R}}^n$ 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 bi-Lipschitz invariance of this capacity. Furthermore we show that for any Borel set $E\subset {\mathbb{R}}^n$ 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.