A direct approach to generation of analytic semigroups by generalized Ornstein–Uhlenbeck operators in weighted LpLp spaces

Generation of analytic semigroups by generalized Ornstein–Uhlenbeck operators A=−Δ+∇Φ⋅∇−G⋅∇+VA=−Δ+∇Φ⋅∇−G⋅∇+V in the weighted space Lp(RN,e−Φ(x)dx)Lp(RN,e−Φ(x)dx) is shown. The results improve the recent results established by Metafune–Prüss–Rhandi–Schnaubelt (2005) [8] (V≡0V≡0) and Kojima–Yokota (2010) [6] (V≢0V≢0). The proofs of those are based on a generation result acting on the unweighted space Lp(RN)Lp(RN)[8, Theorem 3.4], while ours is carried out by an application of LpLp-theory for second-order elliptic operators [13, Theorem 1.3] obtained by Sobajima (2012). Consequently, the additional condition |divG|≤ε(|∇Φ|2+V)+Cε assumed in both [8] and [6] is completely removed.