I have a problem, which is quite urgent, as I have only today discovered an error in a proof i had in a thesis which is to be handed in tomorrow.

The problem, if stated in as full generality as possible is: Given some extension property from $C_0^{\infty}$ to $L^2$ on $\mathbb{R}^n$, how can I utilize this to extend an operator from $C_0^{\infty}$ to $L^2$ on a manifold with a Riemannian metric.

The specific problem is that I am given a pseudodifferential operator $A$ in the space $L^0_{\rho,\delta}(M)$, where $M$ is a compact manifold. That $A$ lies in $L^0_{\rho,\delta}$ implies that the symbol of $A$ is locally bounded in every compact subset of $M$. I wish to extend this operator to an operator on $L^2(M)$. I thought I had done it, by considering $\phi_j A$, where the $\phi_j$ are a smooth partition of unity subordinate to some finite atlas of $M$, since I extend these. However, simply considering $\phi_j A \circ \kappa_i$, where $\kappa_i$ is the diffeomorphism corresponding to the chart $U_i$, doesn't work since supp$ Au$ might be entirely disjoint from supp$ u$.

Any help would be much appreciated!