no. 2
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.

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∈C ∞ et ℒ=∑ 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 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∈C ∞ and ℒ=∑ 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 ℒu=Vu to have a finite order of vanishing at one point.

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     title = {Frequency functions on the {Heisenberg} group, the uncertainty principle and unique continuation},
     journal = {Annales de l'Institut Fourier},
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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://aif.centre-mersenne.org/articles/10.5802/aif.1215/

[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 | Zbl

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

[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 | Zbl

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

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

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

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

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

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

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

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

[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

[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

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