An operator equation characterizing the Laplacian

H. König ; V. Milman

Algebra i analiz, Tome 24 (2012) no. 4, pp. 137-155.

Résumé

The Laplace operator on $\mathbb R^n$ satisfies the equation $$ \Delta(fg)(x)=(\Delta f)(x)g(x)+f(x)(\Delta g)(x)+2\langle f'(x),g'(x)\rangle $$ for all $f,g\in C^2(\mathbb R^n,\mathbb R)$ and $x\in\mathbb R^n$. In the paper, an operator equation generalizing this product formula is considered. Suppose $T\colon C^2(\mathbb R^n,\mathbb R)\to C(\mathbb R^n,\mathbb R)$ and $A\colon C^2(\mathbb R^n,\mathbb R)\to C(\mathbb R^n,\mathbb R^n)$ are operators satisfying the equation \begin{equation} T(fg)(x)=(Tf)(x)g(x)+f(x)(Tg)(x)+\langle(Af)(x),(Ag)(x)\rangle \tag{1} \end{equation} for all $f,g\in C^2(\mathbb R^n,\mathbb R)$ and $x\in\mathbb R^n$. Assume, in addition, that $T$ is $O(n)$-invariant and annihilates the affine functions, and that $A$ is nondegenerate. Then $T$ is a multiple of the Laplacian on $\mathbb R^n$, and $A$ a multiple of the derivative, $$ (Tf)(x)=\frac{d(\|x\|)^2}2(\Delta f)(x),\quad (Af)(x)=d(\|x\|)f'(x), $$ where $d\in C(\mathbb R_+,\mathbb R)$ is a continuous function. The solutions are also described if $T$ is not $O(n)$-invariant or does not annihilate the affine functions. For this, all operators $(T,A)$ satisfying (1) for scalar operators $A\colon C^2(\mathbb R^n,\mathbb R)\to C(\mathbb R^n,\mathbb R)$ are determined. The map $A$, both in the vector and the scalar case, is closely related to $T$ and there are precisely three different types of solution operators $(T,A)$. No continuity or linearity requirement is imposed on $T$ or $A$.

Détail

Keywords: Laplace operator, second order Leibniz rule, operator functional equations.

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