Frequency functions on the Heisenberg group, the uncertainty principle and unique continuation
Annales de l'Institut Fourier, Volume 40 (1990) no. 2, p. 313-356

A recent result of Bahouri shows that continuation from an open set fails in general for solutions of u=Vu where VC and = 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 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 u=Vu to have a finite order of vanishing at one point.

Bahouri a montré récemment qu’il n’y a généralement pas de résultat de prolongement à partir d’un ouvert, pour les solutions de u=VuVC et = 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 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 u=Vu aient des zéros d’ordre fini en un point.

@article{AIF_1990__40_2_313_0,
     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},
     publisher = {Imprimerie Durand},
     address = {28 - Luisant},
     volume = {40},
     number = {2},
     year = {1990},
     pages = {313-356},
     doi = {10.5802/aif.1215},
     zbl = {0694.22003},
     mrnumber = {91i:22014},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1990__40_2_313_0}
}
Garofalo, Nicola; Lanconelli, Ermanno. Frequency functions on the Heisenberg group, the uncertainty principle and unique continuation. Annales de l'Institut Fourier, Volume 40 (1990) no. 2, pp. 313-356. doi : 10.5802/aif.1215. http://www.numdam.org/item/AIF_1990__40_2_313_0/

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