Regularity of fundamental solutions for L\'evy-type operators

For a class of non-symmetric non-local L'evy-type operators ${\mathcal{L}}^{\kappa }$, which include those of the form $${\mathcal{L}}^{\kappa }f(x):={\int }_{{\mathbb{R}}^d}(f(x+z)-f(x)-1_{\mathrm{\mid }z\mathrm{\mid }<1}<z,\mathrm{\nabla }f(x)>)\kappa (x,z)J(z)dz,$$ we prove regularity of the fundamental solution $p^{\kappa }$ to the equation $\partial_t =$\mathcal{L}$^{\kappa}$.