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 $\partial_t =\mathcal{L}^{\kappa}$ for non-symmetric non-local operators $$\mathcal{L}^{\kappa}f(x):= \int_{\mathbb{R}^d}( f(x+z)-f(x)- 1_{|z|<1} \left<z,\nabla f(x)\right>)\kappa(x,z)J(z)\, dz\,,$$ 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 $$\int_{r\leq |z|<1} z \kappa(x,z)J(z)dz\,,\qquad 0<r\leq 1\,,$$ 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 $J(z)=|z|^{-d-1}$.