The Hessian Sobolev inequality and its extensions

TL;DR

This paper provides direct elliptic proofs for the Hessian Sobolev and Poincaré inequalities, extends them to trace inequalities with general measures, and introduces new techniques based on global solution estimates and duality arguments.

Contribution

It offers novel elliptic proof methods and extends fundamental inequalities to more general measure settings in differential geometry and PDEs.

Findings

Elliptic proofs for Hessian Sobolev and Poincaré inequalities.

Extensions to trace inequalities with arbitrary measures.

Development of new techniques using Wolff's potentials and duality.

Abstract

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.