  The Hessian Sobolev inequality of X.-J. Wang, and the Hessian Poincar\'e
inequalities of Trudinger and Wang are fundamental to differential and
conformal geometry, and geometric PDE. These remarkable inequalities were
originally established via gradient flow methods. In this paper, direct
elliptic proofs are given, and extensions to trace inequalities with general
measures in place of Lebesgue measure are obtained. The new techniques rely on
global estimates of solutions to Hessian equations in terms of Wolff's
potentials, and duality arguments making use of a non-commutative inner product
on the cone of k-convex functions.
