This note adapts the sophisticated Richberg technique for approximation in pluripotential theory to the $F$-potential theory associated to a general nonlinear convex subequation $F \subset J^2 (X)$ 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 $x \in X$ has a fundamental neighborhood system consisting of domains for which a “quasi” form of $C^\infty$ approximation holds. Then for any positive $ \in C(X)$ there exists a strictly $F$-subharmonic function $w \in C^\infty (X)$ with $u \lt w \lt u + h$. Applications include all convex constant coefficient subequations on $\mathbb{R}^n$, various nonlinear subequations on complex and almost complex manifolds, and many more.

中文翻译：

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解决方案的Richberg技术

本说明使复杂的Richberg技术在多能理论中的近似适应于与流形$ X $上的一般非线性凸子方程$ F \ subset J ^ 2（X）$相关的$ F $势能理论。主要定理是以下“局部到全局”结果。假设$ u $是连续的严格$ F $-次谐波函数，因此X $中的每个点$ x \ x具有一个基本的邻域系统，该域由包含$ C ^ \ infty $近似值的“拟”形式的域组成。那么对于C（X）$中的任何正$，存在一个严格的$ F $次谐波函数$ w \ in C ^ \ infty（X）$，其中$ u \ lt w \ lt u + h $。应用包括$ \ mathbb {R} ^ n $上的所有凸常数系数子方程，复杂和几乎复杂的流形上的各种非线性子方程，等等。