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A Gårding Inequality on a Manifold with Boundary

  • Nicolas Lerner
  • Xavier Saint Raymond
Chapter
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 46)

Abstract

Let a ∈ C (ℝ n × ℝ n ) and m ∈ ℝ. The function a is said to be a symbol of order m if one has estimates
$$ \left| {\partial _x^\partial \partial _\xi ^\beta a\left( {x,\xi } \right)} \right| \leqslant C_{\alpha ,\beta } \left( {1 + \left| \xi \right|} \right)^{m - \left| \beta \right|} $$
uniformly on ℝ n × ℝ n for all multi-indices α, β ∈ ℤ + n . With such a symbol we associate the pseudo-differential operator (of order m) a(x, D): S(ℝ n ) → S(ℝ n ) defined by
$$ a\left( {x,D} \right)u\left( x \right) = \left( {2\pi } \right)^{ - n} \int {e^{i\left\langle {x,\xi } \right\rangle } a\left( {x,\xi } \right)\hat u\left( \xi \right)d\xi } . $$
.

Keywords

Weight Function Cutoff Function Principal Symbol Carleman Estimate Plancherel Formula 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 2001

Authors and Affiliations

  • Nicolas Lerner
    • 1
  • Xavier Saint Raymond
    • 2
    • 3
  1. 1.Département de Mathématiques IrmarUniversité de Rennes 1Rennes CedexFrance
  2. 2.Département de MathématiquesUniversité de NantesNantes Cedex 3France
  3. 3.UMR 6629 du CNRSFrance

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