The Heisenberg ultrahyperbolic equation: the basic solutions as distributions
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 \(\square _zT(z-z_0,-(z+z_0),\epsilon,-\epsilon)=a_\epsilon(z)\delta_0\), where \(\delta_0\) denotes the Dirac delta at the identity and \(a_\epsilon(z)\) is an elementary function. An analogous property for the Heisenberg Laplacian was established by \textit{G. B. Folland} and \textit{E. M. Stein} [Commun. Pure Appl. Math. 27, 429--522 (1974; Zbl 0293.35012)].
scientific article; zbMATH DE number 6082019