Fundamental solution for super-critical non-symmetric L\'evy-type operators

We prove the existence and give estimates of the fundamental solution (the heat kernel) for the equation ${\mathrm{\partial }}_t={\mathcal{L}}^{\kappa }$ for non-symmetric non-local operators $${\mathcal{L}}^{\kappa }f(x):={\int }_{{\mathbb{R}}^d}(f(x+z)-f(x)-1_{\mathrm{\mid }z\mathrm{\mid }<1}<z,\mathrm{\nabla }f(x)>)\kappa (x,z)J(z)dz,$$ under broad assumptions on $\kappa$ and $J$. Of special interest is the case when the order of the operator ${\mathcal{L}}^{\kappa }$ is smaller than or equal to 1. Our approach rests on imposing suitable cancellation conditions on the internal drift coefficient $${\int }_{r\le \mathrm{\mid }z\mathrm{\mid }<1}z\kappa (x,z)J(z)dz,0<r\le 1,$$ which allows us to handle the non-symmetry of $z\mapsto \kappa (x,z)J(z)$. The results are new even for the $1$-stable L'evy measure $J(z)=\mathrm{\mid }z{\mathrm{\mid }}^{-d-1}$.