Gevrey Type Regularity of the Riesz–Feller Operator Perturbed by Gradient in Lp(R)

Abstract

In the paper, we consider the perturbation of the Riesz–Feller operator Dθβ by the gradient operator b∇ . First, it is shown that for any p∈ [1 , ∞) , the perturbation Dθβ+b∇ generates a C semigroup in Lp(R) if β∈ (0 , 2] and | θ| < 1 . Next, we prove that the semigroup generated by Dθβ+b∇ is analytic if β∈ [1 , 2] and | θ| < 1 , and is of Gevrey class γ for any γ>1β if β∈ (0 , 1) . In the last, it is verified that the semigroup is not of Gevrey class γ with γ≤1β if β∈ (0 , 1) .