Second-order derivations of function spaces -- a characterization of second-order differential operators

TL;DR

This paper characterizes all operators satisfying a specific third-order product rule, extending beyond derivatives to include other solutions, and explores their relation to differential operators.

Contribution

It provides a complete characterization of solutions to a third-order product rule operator equation, revealing new operators beyond derivatives.

Findings

Identifies operators satisfying the third-order product rule.

Classifies all solutions, including derivatives and others.

Analyzes special cases related to differential operators.

Abstract

Let $\Omega \subset \mathbb{R}$ be a nonempty and open set, then for all $f, g, h\in \mathscr{C}^{2}(\Omega)$ we have \begin{multline*} \diff{2}{x}(f\cdot g\cdot h) -f\diff{2}{x}(g\cdot h)-g\diff{2}{x}(f\cdot h)-h\diff{2}{x}(f\cdot g) + f\cdot g\diff{2}{x}h+f\cdot h\diff{2}{x}g+g\cdot h\diff{2}{x}f=0 \end{multline*} The aim of this paper is to consider the corresponding operator equation \[ D(f\cdot g \cdot h) - fD(g\cdot h) - gD(f\cdot h) - hD(f \cdot g) + f\cdot g D(h) + f\cdot h D(g) +g\cdot h D(f) =0 \] for operators $D\colon \mathscr{C}^{k}(\Omega)\to \mathscr{C}(\Omega)$, where $k$ is a given nonnegative integer and the above identity is supposed to hold for all $f, g, h \in \mathscr{C}^{k}(\Omega)$. We show that besides the operators of first and second derivative, there are more solutions to this equation, and we characterize all solutions. Some special cases characterizing differential operators are also studied.