  We establish H\"older regularity and gradient estimates for the transition
semigroup of the solutions to the following SDE: $$ {\rm d} X_t=\sigma (t,
X_{t-}){\rm d} Z_t+b (t, X_t){\rm d} t,\ \ X_0=x\in{\mathbb R}^d, $$ where $(
Z_t)_{t\geq 0}$ is a $d$-dimensional cylindrical $\alpha$-stable process with
$\alpha \in (0, 2)$, $\sigma (t, x):{\mathbb R}_+\times{\mathbb R}^d\to{\mathbb
R}^d\otimes{\mathbb R}^d$ is bounded measurable, uniformly nondegenerate and
Lipschitz continuous in $x$ uniformly in $t$, and $b (t, x):{\mathbb
R}_+\times{\mathbb R}^d\to{\mathbb R}^d$ is bounded $\beta$-H\"older continuous
in $x$ uniformly in $t$ with $\beta\in[0,1]$ satisfying $\alpha+\beta>1$.
Moreover, we also show the existence and regularity of the distributional
density of $X (t, x)$. Our proof is based on Littlewood-Paley's theory.
