  We prove functional inequalities on vector fields on the Euclidean space when
it is equipped with a bounded measure that satisfies a Poincar\'e inequality,
and study associated self-adjoint operators. The weighted Korn inequality
compares the differential matrix, once projected orthogonally to certain
finite-dimensional spaces, with its symmetric part and, in an improved form of
the inequality, an additional term. We also consider Poincar\'e-Korn
inequalities for estimating a projection of the vector field by the symmetric
part of the differential matrix and zeroth-order versions of these inequalities
obtained using the Witten-Laplace operator. The constants depend on geometric
properties of the potential and the estimates are quantitative and
constructive. These inequalities are motivated by kinetic theory and related
with the Korn inequality (1906) in mechanics, on a bounded domain.
