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

Abstract

Let $Z=(Z^{1}, \ldots, Z^{d})$ be the $d$-dimensional L\'evy processes where $Z^{i}$'s are independent $1$-dimensional L\'evy processes with jump kernel $J^{\phi, 1}(u,w) =|u-w|^{-1}\phi(|u-w|)^{-1}$ for $u, w\in \mathbb R$. Here $\phi$ is an increasing function with weak scaling condition of order $\underline \alpha, \overline \alpha\in (0, 2)$. Let $J(x,y) \asymp J^\phi (x,y)$ be the symmetric measurable function where \begin{align*} J^\phi(x,y):=\begin{cases} J^{\phi, 1}(x^i, y^i)\qquad&\text{ if $x^i \ne y^i$ for some $i$ and $x^j = y^j$ for all $j \ne i$}\\ 0\qquad&\text{ if $x^i \ne y^i$ for more than one index $i$.} \end{cases} \end{align*} Corresponding to the jump kernel $J$, we show the existence of non-isotropic Markov processes $X:=(X^{1}, \ldots, X^{d})$ and obtain sharp two-sided heat kernel estimates for the transition density functions.