$L^p$ regularity for elliptic operators with unbounded coefficients

Under suitable conditions on the functions a, F and V we show that the operator Au = ∇(a∇u) + F · ∇u − V u with domain W^{2,p} intersected with the domai of the potential V generates a positive analytic semigroup on Lp, 1 < p < ∞. Analogous results are also established in the spaces L1 and C0. As an application we show that the generalized Ornstein–Uhlenbeck operator AΦ,Gu = Δu − ∇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.