The Richberg technique for subsolutions

This note adapts the sophisticated Richberg technique for approximation in pluripotential theory to the F-potential theory associated to a general nonlinear convex subequation F ⊂ J²(X) on a manifold X. The main theorem is the following “local to global” result. Suppose u is a continuous strictly F-subharmonic function such that each point x ∈ X has a fundamental neighborhood system consisting of domains for which a “quasi” form of C^∞ approximation holds. Then for any positive h ∈ C(X) there exists a strictly F-subharmonic function w ∈ C^∞(X) with u < w < u + h. Applications include all convex constant coefficient subequations on Rⁿ, various nonlinear subequations on complex and almost complex manifolds, and many more.