``Potato kugel'' for sub-Laplacians
type: Item
instance of: scholarly article
title: ``Potato kugel'' for sub-Laplacians
zbMATH Open document ID: 1272.31011
author: Ermanno Lanconelli
published in: Israel Journal of Mathematics
publication date: 2013-06-18T00:00:00Z
review text: The ``Potato kugel'' theorem of \textit{D. Aharonov, M. M. Schiffer} and \textit{L. Zalcman} [Isr. J. Math. 40, 331--339 (1981; Zbl 0496.31006)] is the following: Suppose that \(D\) is a bounded open subset \(\mathbb{R}^N\) with \(N\geq 3\). Assume that \[ \int_D \frac{1}{\left| y-x\right|^{N-2}}dy= \frac{a}{\left| x\right|^{N-2}}+b\quad\text{for all} x\in\mathbb{R}^N\setminus D \] for suitable positive real constants \(a\) and \(b\). Then \(D\) is a Euclidean ball centered at the origin, \(b=0\) and \(a=\left| D\right|\). In the paper under review, the author is interested in extending this result to the sub-Laplacian setting. To state this extension requires a little notation. A sum of squares operator \[ \mathcal{L}=\sum_{j=1}^m X_j^2 \] is a sub-Laplacian in \(\mathbb{R}^N\) if (a) the \(X_j\)'s are smooth vector fields in \(\mathbb{R}^N\) and generate a Lie algebra \(\mathfrak{a}\) satisfying rank \(\mathfrak{a}(x)=\text{dim}\,\mathfrak{a}=N\) at any point \(x\in\mathbb{R}^N\) and (b) there exists a group of dilations \((\delta_\lambda)_{\lambda>0}\) in \(\mathbb{R}^N\) such that every vector field \(X_j\) is \(\delta_\lambda\) homogenous of degree one. The group of dilations are given by \(\delta_\lambda(x_1,\ldots,x_N)=(\lambda^{\sigma_1}x_1,\ldots \lambda^{\sigma_N}x_N)\) with the \(\sigma_j\) given by natural numbers. These two conditions imply a group law \(\circ\) so that \(\mathbb{G}=(\mathbb{R}^N, \circ, \delta_\lambda)\) is a stratified Lie group on which every vector field \(X_j\) is left translation invariant. The number \(Q=\sum_{j=1}^N \sigma_j\) is the homogeneous dimension of \(\mathbb{G}\). An \(\mathcal{L}\)-gauge is a continuous function \(d:\mathbb{G}\to[0,\infty)\), strictly positive, smooth outside of the origin, which is \(\delta_\lambda\)-homogeneous of degree one, and such that \[ \gamma(x)=\frac{1}{d(x)^{Q-2}} \] is \(\mathcal{L}\)-harmonic in \(\mathbb{G}\setminus\{0\}\). Let \(\nabla_{\mathcal{L}}=(X_1,\ldots, X_N)\) and \(\Psi=\left|…
Mathematics Subject Classification ID: 31C12
zbMATH DE Number: 6176546
zbMATH Keywords: potato kugel
zbMATH Keywords: sub-Laplacian
reviewed by: Brett Wick
MaRDI profile type: Publication
full work available at URL: s11856-012-0060-x
OpenAlex ID: W2078310346
cites work: Potato Kugel
cites work: Gauge functions, eikonal equations and Bôcher's theorem on stratified Lie groups
cites work: On left-invariant Hörmander operators in \(\mathbb R^N\). Applications to Kolmogorov-Fokker-Planck equations
cites work: Stratified Lie Groups and Potential Theory for their Sub-Laplacians
cites work: Hypoelliptic second order differential equations
cites work: On the Mean-Value Property of Harmonic Functions
cites work: A characterization of heat balls by a mean value property for temperatures
DOI: 10.1007/S11856-012-0060-X
description: scientific article; zbMATH DE number 6176546