Transition density estimates for diagonal systems of SDEs driven by cylindrical $\alpha$-stable processes

We consider the system of stochastic differential equation $dX_t = A(X_{t-}) \, dZ_t$, $X_0 = x$, driven by cylindrical $\alpha$-stable process $Z_t$ in $\mathbb{R}^d$. We assume that $A(x) = (a_{ij}(x))$ is diagonal and $a_{ii}(x)$ are bounded away from zero, from infinity and H\"older continuous. We construct transition density $p^A(t,x,y)$ of the process $X_t$ and show sharp two-sided estimates of this density. We also prove H\"older and gradient estimates of $x \to p^A(t,x,y)$. Our approach is based on the method developed by Chen and Zhang.