Heat kernels of non-symmetric L\'evy-type operators

We construct the fundamental solution (the heat kernel) $p^{\kappa }$ to the equation ${\mathrm{\partial }}_t={\mathcal{L}}^{\kappa }$, where under certain assumptions the operator ${\mathcal{L}}^{\kappa }$ takes one of the following forms, \begin{align*} $\mathcal{L}$^{\kappa}f(x)&:= \int_{$\mathbb{R}$^d}( f(x+z)-f(x)- 1_{|z|<1} \left<z,\nabla f(x)\right>)\kappa(x,z)J(z), dz ,, $\mathcal{L}$^{\kappa}f(x)&:= \int_{$\mathbb{R}$^d}( f(x+z)-f(x))\kappa(x,z)J(z), dz,, $\mathcal{L}$^{\kappa}f(x)&:= \frac1{2}\int_{$\mathbb{R}$^d}( f(x+z)+f(x-z)-2f(x))\kappa(x,z)J(z), dz,. \end{align*} In particular, $J\text{\hspace{0.17em}⁣}:{\mathbb{R}}^d\to [0,\mathrm{\infty }]$ is a L'evy density, i.e., ${\int }_{{\mathbb{R}}^d}(1\wedge \mathrm{\mid }x{\mathrm{\mid }}^2)J(x)dx<\mathrm{\infty }$. The function $\kappa (x,z)$ is assumed to be Borel measurable on ${\mathbb{R}}^d\times {\mathbb{R}}^d$ satisfying $0<{\kappa }_0\le \kappa (x,z)\le {\kappa }_1$, and $|\kappa(x,z)-\kappa(y,z)|\leq \kappa_2|x-y|^{\beta}$forsome$\beta\in (0, 1)$. We prove the uniqueness, estimates, regularity and other qualitative properties of $p^{\kappa }$.