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

Włodzimierz Fechner, Eszter Gselmann

TL;DR

This work addresses the problem of identifying second-order differential operators through a higher-order Leibniz-type identity involving three test functions. The authors develop a localization-to-pointwise framework, derive an Aichinger-type functional equation via Faà di Bruno, and prove that operators satisfying the identity must take the form $D(f)(x)= c_0(x) f(x)\ln|f(x)| + c_1(x) f'(x) + c_2(x) f''(x) + d_{00}(x) f(x) (\ln|f(x)|)^2$ with specific regularity-induced constraints, while isotropy yields constant coefficients. They relate these results to linear operators and to annihilation of low-degree polynomials, and provide nonlinear examples to illustrate the scope beyond linearity. The findings advance the understanding of second-order operator structure and offer a concrete characterization framework for differential operators from a cubic-argument identity.

Abstract

Let $Ω\subset \mathbb{R}$ be a nonempty and open set, then for all $f, g, h\in \mathscr{C}^{2}(Ω)$ we have \begin{multline*} \frac{\mathrm{d}^{2}}{\mathrm{d} x^{2}}(f\cdot g\cdot h) - f\frac{\mathrm{d}^{2}}{\mathrm{d} x^{2}}(g\cdot h)-g\frac{\mathrm{d}^{2}}{\mathrm{d} x^{2}}(f\cdot h)-h\frac{\mathrm{d}^{2}}{\mathrm{d} x^{2}}(f\cdot g) + f\cdot g\frac{\mathrm{d}^{2}}{\mathrm{d} x^{2}} h+f\cdot h\frac{\mathrm{d}^{2}}{\mathrm{d} x^{2}} g+g\cdot h \frac{\mathrm{d}^{2}}{\mathrm{d} x^{2}}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 $D\colon \mathscr{C}^{k}(Ω)\to \mathscr{C}(Ω)$, where $k$ is a given nonnegative integer and the above identity is supposed to hold for all $f, g, h \in \mathscr{C}^{k}(Ω)$. We show that besides the operators of first and second derivative, there are more solutions to this equation. Some special cases characterizing differential operators are also studied.

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

TL;DR

This work addresses the problem of identifying second-order differential operators through a higher-order Leibniz-type identity involving three test functions. The authors develop a localization-to-pointwise framework, derive an Aichinger-type functional equation via Faà di Bruno, and prove that operators satisfying the identity must take the form with specific regularity-induced constraints, while isotropy yields constant coefficients. They relate these results to linear operators and to annihilation of low-degree polynomials, and provide nonlinear examples to illustrate the scope beyond linearity. The findings advance the understanding of second-order operator structure and offer a concrete characterization framework for differential operators from a cubic-argument identity.

Abstract

Let be a nonempty and open set, then for all we have \begin{multline*} \frac{\mathrm{d}^{2}}{\mathrm{d} x^{2}}(f\cdot g\cdot h) - f\frac{\mathrm{d}^{2}}{\mathrm{d} x^{2}}(g\cdot h)-g\frac{\mathrm{d}^{2}}{\mathrm{d} x^{2}}(f\cdot h)-h\frac{\mathrm{d}^{2}}{\mathrm{d} x^{2}}(f\cdot g) + f\cdot g\frac{\mathrm{d}^{2}}{\mathrm{d} x^{2}} h+f\cdot h\frac{\mathrm{d}^{2}}{\mathrm{d} x^{2}} g+g\cdot h \frac{\mathrm{d}^{2}}{\mathrm{d} x^{2}}f=0 \end{multline*} The aim of this paper is to consider the corresponding operator equation for operators , where is a given nonnegative integer and the above identity is supposed to hold for all . We show that besides the operators of first and second derivative, there are more solutions to this equation. Some special cases characterizing differential operators are also studied.

Paper Structure

This paper contains 3 sections, 13 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. Operator relations and their connections
  3. Results

Key Result

Theorem 1

Let $\Omega\subset \mathbb{R}$ be a nonempty and open set and $k$ be a nonnegative integer. Suppose that the operator $T\colon \mathscr{C}^k(\Omega)\to \mathscr{C}(\Omega)$ satisfies the Leibniz rule, i.e., Then there exist functions $c, d\in \mathscr{C}^k(\Omega)$ such that for all $f\in \mathscr{C}(\Omega)$ and $x\in \Omega$ For $k=0$ we necessarily have $d=0$. Conversely, any such map $T$ sati

Theorems & Definitions (30)

  • Theorem 1: König-Milman
  • Definition 1
  • Definition 2
  • Theorem 2: König-Milman
  • Corollary 1
  • Remark 1: Leibniz rule $\Rightarrow$ identity \ref{['id_2']}
  • Remark 2: Second-order Leibniz rule $\Rightarrow$ \ref{['id_2']}
  • Remark 3
  • Lemma 1
  • proof
  • ...and 20 more