Heat kernel bounds for a large class of Markov process with singular jump

Let Z=(Z1,…,Zd) be the d-dimensional Lévy processes where Zi’s are independent 1-dimensional with jump kernel Jϕ,1(u,w)=|u−w|−1ϕ(|u−w|)−1 for u,w∈R. Here ϕ is an increasing function weak scaling condition of order α̲,α¯∈(0,2). J(x,y)≍Jϕ(x,y) symmetric measurable Jϕ(x,y)≔Jϕ,1(xi,yi)if xi≠yi some i and xj=yj all j≠i0if more than one index iCorresponding to J, we show existence non-isotropic Markov X≔(X1,…,Xd) obtain sharp two-sided heat estimates transition density functions.