Operator relations characterizing higher-order differential operators

Let \(r\) be a positive integer, \(N\) a nonnegative integer and \(\Omega \subset \mathbb{R}^{r}\) be a domain. Further, for all multi-indices \(\alpha \in \mathbb{N}^{r}\), \(|\alpha|\leq N\), let us consider the partial differential operator \(D^{\alpha}\) defined by \(D^{\alpha}= \frac{\partial^{|\alpha|}}{\partial x_{1}^{\alpha_{1}}\cdots \partial x_{r}^{\alpha_{r}}},\)where \(\alpha= (\alpha_{1}, \ldots, \alpha_{r})\). Here, by definition, we mean \(D^{0}\equiv \mathrm{id}\). A straightforward computation shows that if \(f, g\in \mathscr{C}^{N}(\Omega)\) and \(\alpha \in \mathbb{N}^{r}\) with \(|\alpha|\leq N\), then we have

$$D^{\alpha}(f\cdot g) = \sum_{\beta\leq \alpha}\binom{\alpha}{\beta}D^{\beta}(f)\cdot D^{\alpha - \beta}(g).$$

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This paper is devoted to the study of the identity \((\ast)\) in the space \(\mathscr{C}(\Omega)\). More precisely, if \(r\) is a positive integer, \(N\) is a nonnegative integer and \(\Omega \subset \mathbb{R}^{r}\) is a domain, then we describe all mappings (not necessarily linear) that satisfy the identity \((\ast)\) for all possible multi-indices \(\alpha\in \mathbb{N}^{r}\), \(|\alpha|\leq N\). Our main result states that if the domain is \(\mathscr{C}(\Omega)\), then the mappings in question take a particularly specific form. Related results for the space \(\mathscr{C}^{N}(\Omega)\) are also presented.