  We consider the Calder\'on-Zygmund kernels $K_
{\alpha,n}(x)=(x_i^{2n-1}/|x|^{2n-1+\alpha})_{i=1}^d$ in $\mathbb{R}^n$ for
$0<\alpha\leq 1$ 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 $C_{\frac23(2-\alpha),\frac 3 2}$ 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.
