Hessian measures II


In our previous paper on this topic, we introduced the notion of $k$-Hessian measure associated with a continuous $k$-convex function in a domain $\Omega$ in Euclidean $n$-space, $k=1,\cdots,n$, and proved a weak continuity result with respect to local uniform convergence. In this paper, we consider $k$-convex functions, not necessaril continuous, and prove the weak continuity of the associated $k$-Hessian measure with respect to convergence in measure. The proof depends upon local integral estimates for the gradients of $k$-convex functions.


Neil S. Trudinger

Xu-Jia Wang