Gradient potential estimates for k-Hessian equations

Abstract

For $2\le k<n$, we prove pointwise gradient bounds for proper weak $k$-convex functions in terms of a local average and a Wolff potential of the full Hessian measure $F_k[u]$. The estimate yields local Lipschitz regularity for $F_k[u]=f$ with $f\in L^{n/k,1/k}_{\mathrm{loc}}$, and for $F_k[u]=|Du|^p$ when $0<p<nk/(n-k)$. Both the Lorentz index $1/k$ and the upper bound on $p$ are sharp.