  It is known that by dualizing the Bochner-Lichnerowicz-Weitzenb\"{o}ck
formula, one obtains Poincar\'e-type inequalities on Riemannian manifolds
equipped with a density, which satisfy the Bakry-\'Emery Curvature-Dimension
condition (combining a lower bound on its generalized Ricci curvature and an
upper bound on its generalized dimension). When the manifold has a boundary, an
appropriate generalization of the Reilly formula may be used instead. By
systematically dualizing this formula for various combinations of boundary
conditions of the domain (convex, mean-convex) and the function (Neumann,
Dirichlet), we obtain new Brascamp-Lieb type inequalities on the manifold. All
previously known inequalities of Lichnerowicz, Brascamp-Lieb, Bobkov-Ledoux and
Veysseire are recovered, extended to the Riemannian setting and generalized
into a single unified formulation, and their appropriate versions in the
presence of a boundary are obtained. Our framework allows to encompass the
entire class of Borell's convex measures, including heavy-tailed measures, and
extends the latter class to weighted-manifolds having negative generalized
dimension.
