Operator relations characterizing higher-order differential operators

Abstract

Let $r$ be a positive integer, $N$ be a nonnegative integer and $\Omega \subset {\mathbb{R}}^r$ be a domain. Further, for all multi-indices $\alpha \in {\mathbb{N}}^r$, $|\alpha |\le N$, let us consider the partial differential operator $D^{\alpha }$ defined by \[ D^α= \frac{\partial^{|α|}}{\partial x_{1}^{α_{1}}\cdots \partial x_{r}^{α_{r}}}, \] where $\alpha =({\alpha }_1,\dots ,{\alpha }_r)$. Here by definition we mean $D^0\equiv \mathrm{id}$. An easy computation shows that if $f,g\in {\mathcal{C}}^N(\Omega )$ and $\alpha \in {\mathbb{N}}^r,|\alpha |\le N$, then we have \[ \tag{$\ast$} D^α(f\cdot g) = \sum_{β\leq α}\binomαβD^β(f)\cdot D^{α- β}(g). \] This paper is devoted to the study of identity $(\ast )$ in the space $\mathcal{C}(\Omega )$. 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 those mappings $T_{\alpha }:\mathcal{C}(\Omega )\to \mathcal{C}(\Omega )$, $\alpha \in {\mathbb{N}}^r,|\alpha |\le N$ that satisfy identity $(\ast )$ for all possible multi-indices $\alpha \in {\mathbb{N}}^r$, $|\alpha |\le N$. Our main result says that if the domain is $\mathcal{C}(\Omega )$, then the mappings $T_{\alpha }$ are of a rather special form. Related results in the space ${\mathcal{C}}^N(\Omega )$ are also presented.