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

Let $Z=(Z^{1}, \ldots, Z^{d})$ be the $d$-dimensional Lévy processes where $Z^{i}$'s are independent $1$-dimensional Lévy processes with jump kernel $J^{ϕ, 1}(u,w) =|u-w|^{-1}ϕ(|u-w|)^{-1}$ for $u, w\in \mathbb R$. Here $ϕ$ is an increasing function with weak scaling condition of order $\underline α, \overline α\in (0, 2)$. Let $J(x,y) \asymp J^ϕ(x,y)$ be the symmetric measurable function where \begin{align*} J^ϕ(x,y):=\begin{cases} J^{ϕ, 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.

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