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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



An operator equation characterizing the Laplacian

Authors: H. König and V. Milman
Original publication: Algebra i Analiz, tom 24 (2012), nomer 4.
Journal: St. Petersburg Math. J. 24 (2013), 631-644
MSC (2010): Primary 47A62
Published electronically: May 24, 2013
MathSciNet review: 3088010
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Abstract: The Laplace operator on $ \mathbb{R}^n$ satisfies the equation

$\displaystyle \Delta (f g)(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\,:\,C^2(\mathbb{R}^n,\mathbb{R})\to C(\mathbb{R}^n,\mathbb{R})$ and $ A\,:\,C^2(\mathbb{R}^n,\mathbb{R})\to C(\mathbb{R}^n,\mathbb{R}^n)$ are operators satisfying the equation

$\displaystyle T(f g)(x)=(Tf)(x)g(x)+f(x)(Tg)(x)+\langle (Af)(x), (Ag)(x)\rangle$ (1)

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,

$\displaystyle (Tf)(x)=\frac {d(\Vert x\Vert)^2}2 (\Delta f)(x),\quad (Af)(x)=d(\Vert x\Vert)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\,:\,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$.

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Additional Information

H. König
Affiliation: Mathematisches Seminar, Universität Kiel, 24098 Kiel, Germany

V. Milman
Affiliation: School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel

Keywords: Laplace operator, second order Leibniz rule, operator functional equations
Received by editor(s): November 1, 2011
Published electronically: May 24, 2013
Additional Notes: Supported in part by the Alexander von Humboldt Foundation, by ISF grant 387/09, and BSF grant 2006079.
Article copyright: © Copyright 2013 American Mathematical Society

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