property of Calderon-Zygmund kernel

Question

I have a kind of vague question about the property of Calderon-Zygmund kernel. If given a k dimensional Calderon-Zygmund kernel $K$, can we say immediately that it is Lipschitz continuous except at the origin, $K(rx)=r^{-k}K(x)$ for all $r>0$ and $x\neq0$, $\int_{S^{k-1}}Kd\sigma=0$ where $\sigma$ is the surface measure of the unit sphere? Is there any reference for this? Or we actually need some more assumptions to make this claim? Thanks!