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

Nicola Garofalo; Ermanno Lanconelli

Annales de l'institut Fourier (1990)

  • Volume: 40, Issue: 2, page 313-356
  • ISSN: 0373-0956

Abstract

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

How to cite

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Garofalo, Nicola, and Lanconelli, Ermanno. "Frequency functions on the Heisenberg group, the uncertainty principle and unique continuation." Annales de l'institut Fourier 40.2 (1990): 313-356. <http://eudml.org/doc/74879>.

@article{Garofalo1990,
abstract = {A recent result of Bahouri shows that continuation from an open set fails in general for solutions of $\{\cal L\}u=Vu$ where $V\in C^\{\infty \}$ and $\{\cal L\}=\sum ^\{N-1\}_\{j=1\}X^ 2_ j$ is a (nonelliptic) operator in $\{\bf R\}^ N$ satisfying Hörmander’s condition for hypoellipticity. In this paper we study the model case when $\{\cal L\}$ 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 $\{\cal L\}u=Vu$ to have a finite order of vanishing at one point.},
author = {Garofalo, Nicola, Lanconelli, Ermanno},
journal = {Annales de l'institut Fourier},
keywords = {hypoellipticity; subelliptic Laplacian; Heisenberg group; unbounded; first order differential inequality; nontrivial solutions},
language = {eng},
number = {2},
pages = {313-356},
publisher = {Association des Annales de l'Institut Fourier},
title = {Frequency functions on the Heisenberg group, the uncertainty principle and unique continuation},
url = {http://eudml.org/doc/74879},
volume = {40},
year = {1990},
}

TY - JOUR
AU - Garofalo, Nicola
AU - Lanconelli, Ermanno
TI - Frequency functions on the Heisenberg group, the uncertainty principle and unique continuation
JO - Annales de l'institut Fourier
PY - 1990
PB - Association des Annales de l'Institut Fourier
VL - 40
IS - 2
SP - 313
EP - 356
AB - A recent result of Bahouri shows that continuation from an open set fails in general for solutions of ${\cal L}u=Vu$ where $V\in C^{\infty }$ and ${\cal L}=\sum ^{N-1}_{j=1}X^ 2_ j$ is a (nonelliptic) operator in ${\bf R}^ N$ satisfying Hörmander’s condition for hypoellipticity. In this paper we study the model case when ${\cal L}$ 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 ${\cal L}u=Vu$ to have a finite order of vanishing at one point.
LA - eng
KW - hypoellipticity; subelliptic Laplacian; Heisenberg group; unbounded; first order differential inequality; nontrivial solutions
UR - http://eudml.org/doc/74879
ER -

References

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  1. [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. Zbl0439.49028MR82g:49038
  2. [Ba] H. BAHOURI, Non prolongement unique des solutions d'opérateurs "Somme de Carrés", Ann. Inst. Fourier, Grenoble, 36-4 (1986), 137-155. Zbl0603.35008MR88c:35027
  3. [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. Zbl0176.09703MR41 #7486
  4. [Fe] H. FEDERER, Geometric measure theory (Die Grundlehren der mathematischen Wissenschaften, vol. 153), Berlin-Heidelberg-New York, Springer, 1969. Zbl0176.00801
  5. [F1] G. B. FOLLAND, A fundamental solution for a subelliptic operator, Bull. of the Amer. Math. Soc., 79 (2) (1973), 373-376. Zbl0256.35020MR47 #3816
  6. [F2] G. B. FOLLAND, Harmonic analysis in phase space, Annals of Math. Studies, Princeton Univ. Press, Princeton, N.J., 1989. Zbl0682.43001MR92k:22017
  7. [F3] G. B. FOLLAND, Applications of analysis on nilpotent groups to partial differential equations, Bull. Amer. Math. Soc., 83 (1977), 912-930. Zbl0371.35008MR56 #16132
  8. [FS] G. B. FOLLAND and E. M. STEIN, Hardy spaces on homogeneous groups, Mathematical Notes, Princeton Univ. Press, 1982. Zbl0508.42025MR84h:43027
  9. [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. Zbl0678.35015MR88b:35059
  10. [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. Zbl0674.35007MR88j:35046
  11. [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. Zbl0366.22010MR57 #1574
  12. [Gr] P. C. GREINER, Spherical harmonics on the Heisenberg group, Canad. Math. Bull., 23 (4) (1980), 383-396. Zbl0496.22012MR82e:43009
  13. [GrK] P. C. GREINER and T. H. KOORNWINDER, Variations on the Heisenberg spherical harmonics, preprint, Mathematical Centrum, Amsterdam, 1983. 
  14. [He] W. HEISENBERG, The physical principles of the quantum theory, Dover, 1949. 
  15. [Her] R. HERMANN, Lie Groups for Physicists, W. A. Benjamin, N. Y., 1966. Zbl0135.06901MR35 #4327
  16. [H1] L. HÖRMANDER, Hypoelliptic second order differential equations, Acta Math., 119 (1967), 147-171. Zbl0156.10701MR36 #5526
  17. [H2] L. HÖRMANDER, Uniqueness theorems for second-order elliptic differential equations, Comm. in PDE, 8 (1983), 21-64. Zbl0546.35023MR85c:35018
  18. [Ho] R. HOWE, On the role of the Heisenberg group in harmonic analysis, Bull. Amer. Math. Soc., 3 (1980), 821-843. Zbl0442.43002MR81h:22010
  19. [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. Zbl0311.47021
  20. [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. Zbl0252.43022

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