Operator relations characterizing higher-order differential operators

Let $r$ be a positive integer, $N$ be a nonnegative integer and $\mathrm{\Omega }\subset {\mathbb{R}}^r$ be a domain. Further, for all multi-indices $\alpha \in {\mathbb{N}}^r$, $\mathrm{\mid }\alpha \mathrm{\mid }\le N$, let us consider the partial differential operator $D^{\alpha }$ defined by $$D^{\alpha }=\frac{{\mathrm{\partial }}^{\mathrm{\mid }\alpha \mathrm{\mid }}}{\mathrm{\partial }{x}_1^{{\alpha }_1}\cdots \mathrm{\partial }{x}_r^{{\alpha }_r}},$$ where $\alpha =({\alpha }_1,\dots ,{\alpha }_r)$. Here by definition we mean $D^0\equiv \mathrm{i}\mathrm{d}$. An easy computation shows that if $f,g\in {\mathcal{C}}^N(\mathrm{\Omega })$ and $\alpha \in {\mathbb{N}}^r,\mathrm{\mid }\alpha \mathrm{\mid }\le N$, then we have $$\begin{array}{cccc}& D^{\alpha }(f\cdot g)=\sum _{\beta \le \alpha }\left(\genfrac{}{}{0px}{}{\alpha }{\beta }\right)D^{\beta }(f)\cdot D^{\alpha -\beta }(g)\mathrm{.}& & \text{(}\ast \text{)}\end{array}$$ This paper is devoted to the study of identity $(\ast )$ in the space $\mathcal{C}(\mathrm{\Omega })$. More precisely, if $r$ is a positive integer, $N$ is a nonnegative integer and $\mathrm{\Omega }\subset {\mathbb{R}}^r$ is a domain, then we describe those mappings $T_{\alpha }\text{\hspace{0.17em}⁣}:\mathcal{C}(\mathrm{\Omega })\to \mathcal{C}(\mathrm{\Omega })$, $\alpha \in {\mathbb{N}}^r,\mathrm{\mid }\alpha \mathrm{\mid }\le N$ that satisfy identity $(\ast )$ for all possible multi-indices $\alpha \in {\mathbb{N}}^r$, $\mathrm{\mid }\alpha \mathrm{\mid }\le N$. Our main result says that if the domain is $\mathcal{C}(\mathrm{\Omega })$, then the mappings $T_{\alpha }$ are of a rather special form. Related results in the space ${\mathcal{C}}^N(\mathrm{\Omega })$ are also presented.