Characterizations of second-order differential operators
Włodzimierz Fechner, Eszter Gselmann, Aleksandra Świątczak
TL;DR
This work addresses the problem of characterizing second-order differential operators through a Leibniz-type identity for additive maps in an abstract ring/functional setting. It combines functional-equation preliminaries, operator-relations theory, and extension theorems with König–Milman results to show that the identity $T(fg)= fT(g)+T(f)g+2B(A(f),A(g))$ characterizes second-order differential operators on spaces like $\mathscr{C}^k(\Omega,\mathbb{R})$, and that weaker forms (Eq2 or bullet) are equivalent under additivity. A central contribution is an explicit decomposition: there exist continuous $b,c$ and $a$ such that for all $f$, $T(f)(x)= \langle f''(x) c(x), c(x) \rangle + \langle f'(x), b(x) \rangle + a(x) f(x) \ln|f(x)|$ and $A(f)(x)= \langle f'(x), c(x) \rangle$, with special simplifications $c\equiv 0$ for $k=1$ and $b\equiv c\equiv 0$ for $k=0$. These results unify first- and second-order operator characterizations and provide a principled algebraic route to identifying differential operators in spaces of smooth functions.
Abstract
{Let $N, k$ be positive integers with $k\geq 2$, and $Ω\subset \mathbb{R}^{N}$ be a domain.} By the well-known properties of the Laplacian and the gradient, we have \[ Δ(f\cdot g)(x)=g(x) Δf(x)+f(x) Δg(x)+2\langle \nabla f(x), \nabla g(x)\rangle \] for all $f, g\in \mathscr{C}^{k}(Ω, \mathbb{R})$. {Due to the results of H.~König and V.~Milman, Operator relations characterizing derivatives. Birkhäuser / Springer, Cham, 2018.,} the converse is also true, i.e. this operator equation characterizes the Laplacian and the gradient under some assumptions. Thus the main aim of this paper is to provide an extension of this result and to study the corresponding equation \[ T(f\cdot g)= fT(g)+T(f)g+2B(A(f), A(g)) \qquad \left(f, g\in P\right), \] where $Q$ and $R$ are commutative rings, $P$ is a subring of $Q$ and $T\colon P\to Q$ and $A\colon P\to R$ are additive, while $B\colon R\times R\to Q$ is a symmetric and bi-additive. Related identities with one function will also be considered.