``Potato kugel'' for sub-Laplacians
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 \(\Psi=\left| \nabla_{\mathcal{L}} d\right|^2\). The extension to the sub-Laplacian setting then reads as follows: Let \(D\) be a bounded open subset of \(\mathbb{G}\). Assume that \[ \int_D \frac{1}{d(y^{-1}\circ x)^{Q-2}}\Psi(y)dy= \frac{a}{d(x)^{Q-2}}+b\quad\forall x\in\mathbb{G}\setminus D. \] Then \(D\) is a \(d\)-ball centered at the origin, \(b=0\) and \(a=\Psi(D)\). This result is deduced as a corollary of an analogue of \textit{Ü. Kuran}'s theorem [Bull. Lond. Math. Soc. 4, 311--312 (1972; Zbl 0257.31006)] in the sub-Laplacian setting. Namely if \(D\) is a bounded open set in \(\mathbb{G}\) containing the origin and for all harmonic and integrable functions \(u\) in \(D\) \[ u(0)=\frac{1}{\Psi(D)}\int_D u(y)\Psi(y)dy, \] with \(\Psi(D)=\int_D \Psi dy\), then \(D\) is a \(d\)-ball.
scientific article; zbMATH DE number 6176546