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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
DOI: https://doi.org/10.1090/S1061-0022-2013-01257-X
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$.


References [Enhancements On Off] (What's this?)

  • [A] J. Aczél, Lectures on functional equations and their applications, Math. Sci. Engrg., vol. 19, Acad. Press, New York-London, 1966. MR 0208210 (34:8020)
  • [AAM] S. Alesker, S. Artstein-Avidan, and V. Milman, A characterization of the Fourier transform and related topics, Linear and Complex Analysis, Amer. Math. Soc. Transl. Ser. 2, vol. 226, Amer. Math. Soc., Providence, RI, 2009, pp. 11-26. MR 2500506 (2010h:42010)
  • [AKM] S. Artstein-Avidan, H. König, and V. Milman, The chain rule as a functional equation, J. Funct. Anal. 259 (2010), 2999-3024. MR 2719283 (2012a:47078)
  • [AM1] S. Artstein-Avidan and V. Milman, The concept of duality in convex analysis, and the characterization of the Legendre transform, Ann. of Math. (2) 169 (2009), 661-674. MR 2480615 (2010a:26017)
  • [AM2] -, A characterization of the concept of duality, Electron. Res. Announc. Math. Sci. 14 (2007), 42-59. MR 2342714 (2008j:26018)
  • [AM3] -, The concept of duality for measure projections of convex bodies, J. Funct. Anal. 254 (2008), 2648-2666. MR 2406688 (2009d:52006)
  • [GS] H. Goldmann and P. Šemrl, Multiplicative derivations on $ C(X)$, Monatsh. Math. 121 (1996), 189-197. MR 1383530 (96m:46041)
  • [KM1] H. König and V. Milman, Characterizing the derivative and the entropy function by the Leibniz rule, with an appendix by D. Faifman, J. Funct. Anal. 261 (2011), 1325-1344. MR 2807102 (2012e:47102)
  • [KM2] -, An operator equation generalizing the Leibniz rule for the second derivative (submitted, 2011).

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Additional Information

H. König
Affiliation: Mathematisches Seminar, Universität Kiel, 24098 Kiel, Germany
Email: hkoenig@math.uni-kiel.de

V. Milman
Affiliation: School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel
Email: milman@post.tau.ac.il

DOI: https://doi.org/10.1090/S1061-0022-2013-01257-X
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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