Global properties of generalized Ornstein--Uhlenbeck operators on \(L^p(\mathbb R^N, \mathbb R^N)\)with more than linearly growing coefficients (Q2517677)

The realization \(A_{p}\) of the elliptic operator \(Au= \text{div}(Q\nabla u)+F.\nabla u+Vu\) in \(L^{p}(\mathbb{R}^{N},\mathbb{R}^{N})\), \(p>1\), is studied in the present paper. It is shown that \(A_{p}\) generates a strongly continuous semigroup and the domain \(D(A_{p})\) is determined. The approach is related to the Monniaux--Prüss theorem on the sum of noncommuting operators. \(L^{p}-L^{q}\) estimates and pointwise gradient estimates are also proved.