Second-order derivations of function spaces -- a characterization of second-order differential operators
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910007946117120 |
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| author | Fechner, Włodzimierz Gselmann, Eszter |
| author_facet | Fechner, Włodzimierz Gselmann, Eszter |
| contents | 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*}
\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 $D\colon \mathscr{C}^{k}(Ω)\to \mathscr{C}(Ω)$, 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, and we characterize all solutions. Some special cases characterizing differential operators are also studied. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_09028 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Second-order derivations of function spaces -- a characterization of second-order differential operators Fechner, Włodzimierz Gselmann, Eszter Classical Analysis and ODEs 47J05, 47A62, 47B38, 39B22 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*} \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 $D\colon \mathscr{C}^{k}(Ω)\to \mathscr{C}(Ω)$, 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, and we characterize all solutions. Some special cases characterizing differential operators are also studied. |
| title | Second-order derivations of function spaces -- a characterization of second-order differential operators |
| topic | Classical Analysis and ODEs 47J05, 47A62, 47B38, 39B22 |
| url | https://arxiv.org/abs/2502.09028 |