Definition:Partial Differential Operator

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Let $U \subseteq \R^n$ be a open set.

Let $\mathcal C \subseteq \mathcal C^k \left(U, \R\right)$ be a set of $k$-times continuously differentiable functions.

Let $\displaystyle \partial_i = \frac {\partial} {\partial x_i}$ denote the partial derivative, $i = 1, \ldots, n$.

For a multiindex $\alpha = \left({\alpha_1, \ldots, \alpha_n}\right)$ indexed by $\left\{{1, \ldots, n}\right\}$ let $\partial^\alpha = \partial_1^{\alpha_1} \cdots \partial_n^{\alpha_n}$.

A mapping $T : \mathcal C \to \mathcal C^k \left(U, \R\right)$ is a partial differential operator if there exist $r \in \N$ and functions $f_\alpha : \R^n \to \C$ for each multiindex $\alpha$ with $\left\vert{\alpha}\right\vert \le r$ such that for all $g \in \mathcal C$:

$\displaystyle T \left({g}\right) = \sum_{\left\vert{\alpha}\right\vert \le r} f_\alpha \partial^\alpha g$