Fefferman's SAK principle in one dimension

Frédéric Hérau

Annales de l'institut Fourier (2000)

  • Volume: 50, Issue: 4, page 1229-1264
  • ISSN: 0373-0956

Abstract

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In this article we give a complete proof in one dimension of an a priori inequality involving pseudo-differential operators: if a and b are symbols in S 1 , 0 2 such that | a | ≤ b , then for all ϵ > 0 we have the estimate ∥ a w u ∥ s 2 ≤ C ϵ ( ∥ b w u ∥ s 2 + ∥ u ∥ s + ϵ 2 ) for all u in the Schwartz space, where ∥ ∥ t is the usual H t norm. We use microlocalization of levels I, II and III in the spirit of Fefferman’s SAK principle.

How to cite

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Hérau, Frédéric. "Fefferman's SAK principle in one dimension." Annales de l'institut Fourier 50.4 (2000): 1229-1264. <http://eudml.org/doc/75455>.

@article{Hérau2000,
abstract = {In this article we give a complete proof in one dimension of an a priori inequality involving pseudo-differential operators: if $a$ and $b$ are symbols in $S^2_\{1,0\}$ such that $\vert a\vert \le b$, then for all $\epsilon &gt;0$ we have the estimate $\Vert a^wu\Vert ^2_s\le C_\epsilon (\Vert b^wu\Vert ^2_s +\Vert u\Vert ^2_\{s+\epsilon \})$ for all $u$ in the Schwartz space, where $\Vert \;\Vert _t$ is the usual $H_t$ norm. We use microlocalization of levels I, II and III in the spirit of Fefferman’s SAK principle.},
author = {Hérau, Frédéric},
journal = {Annales de l'institut Fourier},
keywords = {pseudo-differential operators; microlocal analysis; uncertainty principle; Weyl-Hörmander calculus; Gårding inequality; Fefferman-Phong inequality; SAK principle},
language = {eng},
number = {4},
pages = {1229-1264},
publisher = {Association des Annales de l'Institut Fourier},
title = {Fefferman's SAK principle in one dimension},
url = {http://eudml.org/doc/75455},
volume = {50},
year = {2000},
}

TY - JOUR
AU - Hérau, Frédéric
TI - Fefferman's SAK principle in one dimension
JO - Annales de l'institut Fourier
PY - 2000
PB - Association des Annales de l'Institut Fourier
VL - 50
IS - 4
SP - 1229
EP - 1264
AB - In this article we give a complete proof in one dimension of an a priori inequality involving pseudo-differential operators: if $a$ and $b$ are symbols in $S^2_{1,0}$ such that $\vert a\vert \le b$, then for all $\epsilon &gt;0$ we have the estimate $\Vert a^wu\Vert ^2_s\le C_\epsilon (\Vert b^wu\Vert ^2_s +\Vert u\Vert ^2_{s+\epsilon })$ for all $u$ in the Schwartz space, where $\Vert \;\Vert _t$ is the usual $H_t$ norm. We use microlocalization of levels I, II and III in the spirit of Fefferman’s SAK principle.
LA - eng
KW - pseudo-differential operators; microlocal analysis; uncertainty principle; Weyl-Hörmander calculus; Gårding inequality; Fefferman-Phong inequality; SAK principle
UR - http://eudml.org/doc/75455
ER -

References

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