Given a map $\Phi$ that sends functions $f(x_1,\dots,x_n)$ of $n$ variables to functions $\Phi f(y_1,\dots,y_m)$ of $m$ variables, there is the heuristic that solving the equation $\Phi f = g$ is only possible if the system is (over-)determined in the sense that $m\ge n$. I would like to know whether there is a setting in which this can be made precise.
Here is a first (failed) attempt: Suppose $X$ and $Y$ are compact manifolds of dimension $n$ and $m$ respectively and $\Phi$ is a map $\Phi:C(X)\rightarrow C(Y)$ between their spaces of continuous functions. I interpret the heuristics to say that $$ \Phi \text{ injective } \quad \Longrightarrow \quad m\ge n. $$ Now arguably the nicest form $\Phi$ can take is $\Phi=\varphi^*$, the pull-back by a continuous surjection $\varphi: Y\rightarrow X$. But alas, one can construct such a surjection from $S^1$ onto $S^2$ (starting with a space filling curve), and thus we do not necessarily have $m\ge n$.
Fair enough, continuous maps can be pretty wild. If we consider $\Phi:C^\infty(X)\rightarrow C^\infty(Y)$, then the counterexample from above does not carry over. Indeed, Sard's theorem implies that there can only be a smooth surjection $\varphi:Y\rightarrow X$ if $m\ge n$. This leads to:
Question. Let $X,Y$ be compact, smooth manifolds and $\Phi:C^\infty(X)\rightarrow C^\infty(Y)$ a continuous map. Which conditions on $\Phi$ ensure that $\Phi$ being injective implies that $\dim Y \ge \dim X$? For the start, it would be good to know whether it suffices to have $\Phi$ linear or whether additional structure on the Schwartz kernel is needed.