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We study the stochastic differential equation $dX_t = A(X_{t-}) \, dZ_t$, $X_0 = x$, where $Z_t = (Z_t^{(1)},\ldots,Z_t^{(d)})^T$ and $Z_t^{(1)}, \ldots,Z_t^{(d)}$ are independent one-dimensional L{\'e}vy processes withcharacteristic exponents $\psi_1, \ldots, \psi_d$. We assume that each $\psi_i$satisfies a weak lower scaling condition WLSC($\alpha,0,\underline{C}$), a weakupper scaling condition WUSC($\beta,1,\overline{C}$) (where $0< \alpha \le\beta < 2$) and some additional regularity properties. We consider two mutuallyexclusive assumptions: either (i) all $\psi_1, \ldots, \psi_d$ are the same and$\alpha, \beta$ are arbitrary, or (ii) not all $\psi_1, \ldots, \psi_d$ are thesame and $\alpha > (2/3)\beta$. We also assume that the determinant of $A(x) =(a_{ij}(x))$ is bounded away from zero, and $a_{ij}(x)$ are bounded andLipschitz continuous. In both cases (i) and (ii) we prove that for any fixed$\gamma \in (0,\alpha) \cap (0,1]$ the semigroup $P_t$ of the process $X$satisfies $|P_t f(x) - P_t f(y)| \le c t^{-\gamma/\alpha} |x - y|^{\gamma}||f||_\infty$ for arbitrary bounded Borel function $f$. We also show theexistence of a transition density of the process $X$.

Semigroup properties of solutions of SDEs driven by L{\'e}vy processes with independent coordinates