Generalized Ornstein–Uhlenbeck semigroups in weighted Lp-spaces on Riemannian manifolds

Abstract

Let E be a Hermitian vector bundle over a Riemannian manifold M with metric g, and let ∇ be a metric covariant derivative on E. We study the generalized Ornstein–Uhlenbeck differential expression P∇=∇†∇u+∇(dϕ)♯u−∇Xu+Vu, where ∇† is the formal adjoint of ∇, (dϕ)♯ is the vector field corresponding to dϕ via g, X is a smooth real vector field on M, and V is a self-adjoint locally integrable section of the endomorphsim bundle EndE. In the setting of a geodesically complete M, we establish a sufficient condition for the equality of the maximal and minimal realizations of P∇ in the (weighted) space ΓLμp(E) of Lμp-type sections of E, where 1<p<∞ and dμ=e−ϕdνg, with νg being the usual volume measure. Furthermore, we show that (the negative of) the maximal realization −Hp,max generates an analytic quasi-contractive semigroup in ΓLμp(E), 1<p<∞. Additionally, in the same context, we establish a coercivity estimate, which leads to the so-called separation property of the covariant Schrödinger operator (that is, P∇ with X≡0 and ϕ≡0) in the (unweighted) space ΓLp(E), 1<p<∞. With the generation result at our disposal, we describe a Feynman–Kac representation for the Lμp-semigroup generated by Hp,max, 1<p<∞. For the Ornstein–Uhlenbeck differential expression acting on functions, that is, Pd=Δu+(dϕ)♯u−Xu+Vu, where Δ is the (non-negative) scalar Laplacian on M and V is a locally integrable real-valued function, we consider another way of realizing Pd as an operator in Lμp(M) and, by imposing certain geometric conditions on M, we prove another semigroup generation result. The study of the mentioned realization of Pd depends, among other things, on the fulfillment of the so-called Lp-Calderón–Zygmund inequality on M.