Fundamental Solutions of Nonlocal Hörmander’s Operators

Consider the following nonlocal integro-differential operator:forα∈(O,2),L_(σ,b)^(a)f(x)=p.v.∫_(R^(d))-{0}f(x+σ(x)z-f(x)/|z|^(d+α)dz+b(x).△f(x),whereσ:R^(d)×R^(d)and b:R^(d)→R^(d)are smooth and have bounded first-order derivatives,and p.v.stands for the Cauchy principal value.Let B1(x):=σ(x)and Bj+1(x):=b(x).△Bj(x)-△b(x)·Bj(x)for j∈N.Under the followingHormander's type condition:for any x∈R^(d)and some n=n(x)∈N,Rank[B1(x),B2(K),...,Bn(x)]=d,by using the Malliavin calculus,we prove the existence of the heat kernelρt(x,y)to the operator L_(σ,b)^(α)as well as the continuity of x→ρt(x,.)in L^(1)(R^(d))as a densityfunction for each t>0.Moreover,whenσ(x)=σis constant and Bj∈C_(b)^(∞)for eachj∈N,under the following uniform Hormander's type condition:for some j0∈N,we also show the smoothness of(t,x,y)→ρt(x,y)withρt(,.)∈b^(∞)(R^(d)×R^(d))for each t>0.