Lᵖ-regularity for elliptic operators with unbounded coefficients

Under suitable conditions on the functions a∈ C¹( RN, RN²), F∈ C¹( RN, RN), and V: RN→ [0,∞), we show that the operator Au=∇ (a∇ u) +F· ∇ u-Vu with domain W2,pV( RN)= \ u∈ W2,p( RN):Vu∈ Lᵖ( RN) \ generates a positive analytic semigroup on Lᵖ( RN), 1 < p < ∞. Analogous results are also established in the spaces L¹( RN) and C₀( RN). As an application we show that the generalized Ornstein--Uhlenbeck operator AΦ,G u=Δ u -∇ Φ · ∇ u + G· ∇ u with domain W2,p( RN,μ) generates an analytic semigroup on the weighted space Lᵖ( RN,μ), where 1 < p < ∞ and μ(dx)=e-Φ(x)dx.