Hölder regularity and gradient estimates for sdes driven by cylindrical α-stable processes^*

We establish Hölder regularity and gradient estimates for the transition semigroup of the solutions to the following SDE: dX_t = σ(t, X_t−)dZ_t + b(t, X_t)dt, X₀ = x ∈ R^d, where (Z_t)_t≥0 is a d-dimensional cylindrical α-stable process with α ∈ (0, 2), σ(t, x): R₊ × R^d → R^d ⊗ R^d is bounded measurable, uniformly nondegenerate and Lipschitz continuous in x uniformly in t, and b(t, x): R₊ × R^d → R^d is bounded β-Hölder continuous in x uniformly in t with β ∈ [0, 1] satisfying α + β > 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.