Heat kernels of non-symmetric jump processes: beyond the stable case

Let $J$ be the L\'evy density of a symmetric L\'evy process in $\mathbb{R}^d$ with its L\'evy exponent satisfying a weak lower scaling condition at infinity. Consider the non-symmetric and non-local operator $$ {\mathcal L}^{\kappa}f(x):= \lim_{\epsilon \downarrow 0} \int_{\{z \in \mathbb{R}^d: |z|>\epsilon\}}(f(x+z)-f(x))\kappa(x,z)J(z)\, dz\, , $$ where $\kappa(x,z)$ is a Borel measurable function on $\mathbb{R}^d\times \mathbb{R}^d$ satisfying $0<\kappa_0\le \kappa(x,z)\le \kappa_1$, $\kappa(x,z)=\kappa(x,-z)$ and $|\kappa(x,z)-\kappa(y,z)|\le \kappa_2|x-y|^{\beta}$ for some $\beta\in (0, 1)$. We construct the heat kernel $p^\kappa(t, x, y)$ of ${\mathcal L}^\kappa$, establish its upper bound as well as its fractional derivative and gradient estimates. Under an additional weak upper scaling condition at infinity, we also establish a lower bound for the heat kernel $p^\kappa$.