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$,drivenbycylindrical$\alpha$-stableprocess$Z_t$in$$\mathbb{R}$^d$\mathrm{.}Weassumethat$A(x) = (a_{ij}(x))$isdiagonaland$a_{ii}(x)$ 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.