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

Abstract: 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.