The weak convergence for a measure related to a class of conformally invariant fully nonlinear operator

Abstract

Trudinger-Wang introduced the notion of k-Hessian measure associated with k-convex functions, not necessarily continuous, and proved the weak continuity of the associated k-Hessian measure with respect to local $L^1$ convergence in 1999. In this paper we find a special divergence structure for the ${\sigma }_k$-Yamabe operator which is conformally invariant, and prove the weak continuity of the ${\sigma }_k$-Yamabe operator with respect to local $L^1$ convergence.