Global properties of generalized Ornstein–Uhlenbeck operators on with more than linearly growing coefficients

We show that the realization A p of the elliptic operator A u = div ( Q ∇ u ) + F ⋅ ∇ u + V u in L p ( R N , R N ) , p ∈ [ 1 , + ∞ [ , generates a strongly continuous semigroup, and we determine its domain D ( A p ) = { u ∈ W 2 , p ( R N , R N ) : F ⋅ ∇ u + V u ∈ L p ( R N , R N ) } if 1 < p < + ∞ . The diffusion coefficients Q = ( q i j ) are uniformly elliptic and bounded together with their first-order derivatives, the drift coefficients F can grow as | x | log | x | , and V can grow logarithmically. Our approach relies on the Monniaux–Prüss theorem on the sum of noncommuting operators. We also prove L p – L q estimates and, under somewhat stronger assumptions, we establish pointwise gradient estimates and smoothing of the semigroup in the spaces W α , p ( R N , R N ) , α ∈ [ 0 , 1 ] , where 1 < p < + ∞ .