Regularity of fundamental solutions for Lévy-type operators

Abstract

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