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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  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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