Lelong–Jensen type formula, k-Hessian boundary measure and Lelong number for k-convex functions

Abstract

The k-convex functions are the viscosity subsolutions to the fully nonlinear elliptic equations Fk[u]=0, where Fk[u] is the elementary symmetric function of order k of the eigenvalues of Hessian matrix D2u, k=1,…,n. For example, F1[u] is the Laplacian Δu and Fn[u] is the real Monge–Ampère operator detD2u, while 1-convex functions and n-convex functions are subharmonic and convex in the classical sense, respectively. In this paper, we generalize the method of pluripotential theory for complex Monge–Ampère operator to that for the k-Hessian operator to establish the Lelong–Jensen type formula for real k-convex functions, show the comparison theorem for the k-Hessian boundary measure and introduce the generalized Lelong number for k-convex functions. We also show a relationship between the k-Hessian boundary measure and the generalized Lelong number.