The Heisenberg ultrahyperbolic equation: the basic solutions as distributions
type: Item
instance of: scholarly article
title: The Heisenberg ultrahyperbolic equation: the basic solutions as distributions
zbMATH Open document ID: 1260.22009
author: Anthony C. Kable
published in: Pacific Journal of Mathematics
publication date: 2012-09-13T00:00:00Z
full work available at URL: p06.xhtml
reviewed by: Mladen Božičević
Mathematics Subject Classification ID: 22E30
Mathematics Subject Classification ID: 35R03
Mathematics Subject Classification ID: 22E25
zbMATH DE Number: 6082019
zbMATH Keywords: Kelvin transform
zbMATH Keywords: conformally invariant operator
zbMATH Keywords: Heisenberg group
MaRDI profile type: Publication
OpenAlex ID: W2095354329
DOI: 10.2140/PJM.2012.258.165
review text: Let \(N\) denote the Heisenberg group of dimension \(2d+1\). Recall that the elements of \(N\) are triples \((x,y,t)\), where \(x=(x_i),\; y=(y_i)\in \mathbb R^d\), and \(t\in \mathbb R\). For each \(z\in \mathbb C\), the Heisenberg ultrahyperbolic operator is defined by the formula \(\square _z=\triangle+(\mathbb E_x+(z+z_0))\frac{\partial}{\partial t}\). Here \(\triangle =\sum_{i=1}^{d}\frac{\partial^2} {\partial x_i\partial y_i}\) is the Euklidean ultrahyperbolic operator, \(\mathbb E_x=\sum_{i=1}^dx_i\frac{\partial} {\partial x_i}\) is the Euler operator with respect to \(x\), and \(z_0=d/2\). The group \(SL(d+2,\mathbb R)\) can be realized in four different ways as a group of conformal symmetries of \(\square_z\), with four realizations being indexed by two sign parameters \(\epsilon_1,\;\epsilon_2\in \{\pm 1\}\). Using this fact, the author introduces an analogue \(\mathbb K(z,\epsilon_1,\epsilon_2)\) of the classical Kelvin transform and defines basic solutions of the ultrahyperbolic equation \(\square _zf=0\) by the formula \(\varphi_0(z,\epsilon_1,\epsilon_2)= \mathbb K(z,\epsilon_1,\epsilon_2)^{-1}1\).NEWLINENEWLINEThe aim of this paper is to interpret the basic solutions as tempered distributions on \(N\) and to compute \(\square _z\varphi_0(z,\epsilon_1,\epsilon_2)\) in the distributional sense. The basic solutions are almost never locally integrable functions on \(N\) so they do not give rise to tempered distributions directly. To resolve this problem, the author associates a two parameter family of tempered distributions \(T(s_1,s_2,\epsilon_1,\epsilon_2)\) to the basic solutions and investigates their properties. The problem of computing \(\square _z\varphi_0(z,\epsilon_1,\epsilon_2)\) is reinterpreted as a problem of computing \(\square_zT(z-z_0,-(z+z_0),\epsilon_1,\epsilon_2)\).NEWLINENEWLINEThe main result of the paper shows that \(\square _zT(z-z_0,-(z+z_0),\epsilon,\epsilon)=0\), and \(\square _zT(z-z_0,-(z+z_0),\epsilon,-\epsilon)=a_\epsilon(z)\…
description: scientific article; zbMATH DE number 6082019