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

Garofalo, Nicola; Lanconelli, Ermanno

doi:10.5802/aif.1215

Garofalo, Nicola ; Lanconelli, Ermanno

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

Résumé
Abstract

Bahouri a montré récemment qu’il n’y a généralement pas de résultat de prolongement à partir d’un ouvert, pour les solutions de $ℒ u = V u$ où $V \in C^{\infty}$ et $ℒ = \sum_{j = 1}^{N - 1} X_{j}^{2}$ est un opérateur (non elliptique) dans $R^{N}$ 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 $ℒ u = V u$ 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 $ℒ u = V u$ where $V \in C^{\infty}$ and $ℒ = \sum_{j = 1}^{N - 1} X_{j}^{2}$ is a (nonelliptic) operator in $R^{N}$ 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 $ℒ u = V u$ to have a finite order of vanishing at one point.

MR Zbl | 2 citations dans Numdam

DOI : 10.5802/aif.1215

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     author = {Garofalo, Nicola and Lanconelli, Ermanno},
     title = {Frequency functions on the {Heisenberg} group, the uncertainty principle and unique continuation},
     journal = {Annales de l'Institut Fourier},
     pages = {313--356},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {40},
     number = {2},
     year = {1990},
     doi = {10.5802/aif.1215},
     mrnumber = {91i:22014},
     zbl = {0694.22003},
     language = {en},
     url = {https://www.numdam.org/articles/10.5802/aif.1215/}
}

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Garofalo, Nicola; Lanconelli, Ermanno. Frequency functions on the Heisenberg group, the uncertainty principle and unique continuation. Annales de l'Institut Fourier, Tome 40 (1990) no. 2, pp. 313-356. doi : 10.5802/aif.1215. https://www.numdam.org/articles/10.5802/aif.1215/

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