H\"older regularity and gradient estimates H\"older regularity and gradient estimates for SDEs driven by cylindrical $\alpha$-stable processes

We establish H"older regularity and gradient estimates for the transition semigroup of the solutions to the following SDE: $$\mathrm{d}{X}_t=\sigma (t,X_{t-})\mathrm{d}{Z}_t+b(t,X_t)\mathrm{d}t,X_0=x\in {\mathbb{R}}^d,$$ where $( Z_t){t\geq 0}$isa$d$-dimensionalcylindrical$\alpha$-stableprocesswith$\alpha \in (0, 2)$,$\sigma (t, x):{\mathbb R}+\times{\mathbb R}^d\to{\mathbb R}^d\otimes{\mathbb R}^d$ is bounded measurable, uniformly nondegenerate and Lipschitz continuous in $x$ uniformly in $t$, and $b (t, x):{\mathbb R}_+\times{\mathbb R}^d\to{\mathbb R}^d$isbounded$\beta$-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.