Regularity of fundamental solutions for Lévy-type operators

For a class of non-symmetric non-local Lévy-type operators $\mathcal L^{\kappa}$, which include those of the form $$ \mathcal L^{\kappa}f(x):= \int _{\mathbb R^d}( f(x+z)-f(x)- \mathbf 1_{|z| \lt 1} \langle z,\nabla f(x)\rangle )\kappa (x,z)J(z)\, dz, $$