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

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