Frequency functions on the Heisenberg group, the uncertainty principle and unique continuation

Garofalo, Nicola; Lanconelli, Ermanno

Annales de l'Institut Fourier, Tome 40 (1990) no. 2, pp. 313-356.

Bahouri a montré récemment qu’il n’y a généralement pas de résultat de prolongement à partir d’un ouvert, pour les solutions de $ℒu=Vu$ où $V\in C^{\infty }$ et $ℒ={\sum }_{j=1}^{N-1}{X}_j^2$ est un opérateur (non elliptique) dans ${\mathbf{R}}^N$ vérifiant la condition d’hypoellipticité de Hörmander. Dans cet article, nous étudions le cas où $ℒ$ est le laplacien sous-elliptique sur le groupe d’Heisenberg et $V$ est un terme d’ordre zéro non nécessairement borné. On détermine une condition suffisante, qui est une inégalité différentielle du premier ordre, pour que les solutions non triviales de $ℒu=Vu$ aient des zéros d’ordre fini en un point.

A recent result of Bahouri shows that continuation from an open set fails in general for solutions of $ℒu=Vu$ where $V\in C^{\infty }$ and $ℒ={\sum }_{j=1}^{N-1}{X}_j^2$ is a (nonelliptic) operator in ${\mathbf{R}}^N$ satisfying Hörmander’s condition for hypoellipticity. In this paper we study the model case when $ℒ$ is the subelliptic Laplacian on the Heisenberg group and $V$ is a zero order term which is allowed to be unbounded. We provide a sufficient condition, involving a first order differential inequality, for nontrivial solutions of $ℒu=Vu$ to have a finite order of vanishing at one point.

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