The Hessian Sobolev inequality and its extensions. (English) Zbl 1335.35075
Summary: The Hessian Sobolev inequality of X.-J. Wang [Indiana Univ. Math. J. 43, No. 1, 25–54 (1994; Zbl 0805.35036)], and the Hessian Poincaré inequalities of N. S. Trudinger and X.-J. Wang [Calc. Var. Partial Differ. Equ. 6, No. 4, 315–328 (1998; Zbl 0927.58013)] 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.
MSC:
| 35J87 | Unilateral problems for nonlinear elliptic equations and variational inequalities with nonlinear elliptic operators |
| 31C45 | Other generalizations (nonlinear potential theory, etc.) |
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
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