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

Garofalo, Nicola; Lanconelli, Ermanno

Annales de l'Institut Fourier, Tome 40 (1990), p. 313-356 / Harvested from Numdam

Access to full text
Full (PDF)
Full (DJVU)

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.

Publié le : 1990-01-01

MR 91i:22014 | Zbl 0694.22003

DOI : https://doi.org/10.5802/aif.1215

@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},
     volume = {40},
     year = {1990},
     pages = {313-356},
     doi = {10.5802/aif.1215},
     mrnumber = {91i:22014},
     zbl = {0694.22003},
     language = {en},
     url = {http://dml.mathdoc.fr/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, Tome 40 (1990) pp. 313-356. doi : 10.5802/aif.1215. http://gdmltest.u-ga.fr/item/AIF_1990__40_2_313_0/

Bibliographie

[A] F. T. Almgren, Jr., Dirichlet's problem for multiple valued functions and the regularity of mass minimizing integral currents, in Minimal Submanifolds and Geodesics (M. Obata, Ed.), North-Holland, Amsterdam, 1979, pp. 1-6. | MR 82g:49038 | Zbl 0439.49028

[Ba] H. Bahouri, Non prolongement unique des solutions d'opérateurs "Somme de Carrés", Ann. Inst. Fourier, Grenoble, 36-4 (1986), 137-155. | Numdam | MR 88c:35027 | Zbl 0603.35008

[B] J. M. Bony, Principe du maximum, inégalité de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés, Ann. Inst. Fourier, Grenoble, 19-1 (1969), 277-304. | Numdam | MR 41 #7486 | Zbl 0176.09703

[Fe] H. Federer, Geometric measure theory (Die Grundlehren der mathematischen Wissenschaften, vol. 153), Berlin-Heidelberg-New York, Springer, 1969. | Zbl 0176.00801

[F1] G. B. Folland, A fundamental solution for a subelliptic operator, Bull. of the Amer. Math. Soc., 79 (2) (1973), 373-376. | MR 47 #3816 | Zbl 0256.35020

[F2] G. B. Folland, Harmonic analysis in phase space, Annals of Math. Studies, Princeton Univ. Press, Princeton, N.J., 1989. | MR 92k:22017 | Zbl 0682.43001

[F3] G. B. Folland, Applications of analysis on nilpotent groups to partial differential equations, Bull. Amer. Math. Soc., 83 (1977), 912-930. | MR 56 #16132 | Zbl 0371.35008

[FS] G. B. Folland and E. M. Stein, Hardy spaces on homogeneous groups, Mathematical Notes, Princeton Univ. Press, 1982. | MR 84h:43027 | Zbl 0508.42025

[GL1] N. Garofalo and F. H. Lin, Monotonicity properties of variational integrals, Ap weights and unique continuation, Indiana Univ. Math. J., 35 (2) (1986), 245-268. | MR 88b:35059 | Zbl 0678.35015

[GL2] N. Garofalo and F. H. Lin, Unique continuation for elliptic operators : A geometric-variational approach, Comm. in Pure and Appl. Math. XL (1987), 347-366. | MR 88j:35046 | Zbl 0674.35007

[G] B. Gaveau, Principe de moindre action, propagation de la chaleur et estimées sous elliptiques sur certains groups nilpotents, Acta Math., 139 (1977), 95-153. | MR 57 #1574 | Zbl 0366.22010

[Gr] P. C. Greiner, Spherical harmonics on the Heisenberg group, Canad. Math. Bull., 23 (4) (1980), 383-396. | MR 82e:43009 | Zbl 0496.22012

[GrK] P. C. Greiner and T. H. Koornwinder, Variations on the Heisenberg spherical harmonics, preprint, Mathematical Centrum, Amsterdam, 1983.

[He] W. Heisenberg, The physical principles of the quantum theory, Dover, 1949.

[Her] R. Hermann, Lie Groups for Physicists, W. A. Benjamin, N. Y., 1966. | MR 35 #4327 | Zbl 0135.06901

[H1] L. Hörmander, Hypoelliptic second order differential equations, Acta Math., 119 (1967), 147-171. | MR 36 #5526 | Zbl 0156.10701

[H2] L. Hörmander, Uniqueness theorems for second-order elliptic differential equations, Comm. in PDE, 8 (1983), 21-64. | MR 85c:35018 | Zbl 0546.35023

[Ho] R. Howe, On the role of the Heisenberg group in harmonic analysis, Bull. Amer. Math. Soc., 3 (1980), 821-843. | MR 81h:22010 | Zbl 0442.43002

[KSWW] H. Kalf, U. W. Schmincke, J. Walter and R. Wüst, On the spectral theory of Schrödinger and Dirac operators with strongly singular potentials, in "Spectral theory and differential equations" (W. N. Everitt, Ed.), Lecture Notes in Math. 448, Springer-Verlag, 1975. | Zbl 0311.47021

[S] E. M. Stein, Some problems in harmonic analysis suggested by symmetric spaces and semi-simple groups, Actes, Congrès Intern. Math., Nice, 1 (1970), 179-189. | Zbl 0252.43022