Let $f:R^n \to R$ be convex. Then there exist signed Radon measures $\mu^{ij}=\mu^{ji}$ such that $$ \int_{R^n} f \frac{\partial^2 \varphi}{\partial x_i \partial x_j} dx= \int_{R^n} \varphi d\mu^{ij} \quad (i,j=1,...,n) $$ for all $\varphi \in C^2_c(R^n)$. Let $[\frac{\partial^2 \varphi}{\partial x_i \partial x_j}]=[D^2 f]$ be the distributional Hessian metrix. The absolutely part w.r.t. n-dim Lebesgue measure $[D^2f]_{ac}$ is a non-negatively definite matrix.

For $C^2$ Riemannian manifolds, since we can take $C_c^2$ functions as test functions, we can define the distributional Hessian metrix just as the the case of $R^n$.

I wonder whether we can define the distributional Hessian metrix in spaces where we have not $C^2_c$ functions. For example, $C^0$ Riemannian manifold, Lipschitz manifold with $L^{\infty}$ Riemannian metric. Metric measure spaces such as $CD(K,N)$ spaces.