A Gårding Inequality on a Manifold with Boundary

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


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
$$ a\left( {x,D} \right)u\left( x \right) = \left( {2\pi } \right)^{ - n} \int {e^{i\left\langle {x,\xi } \right\rangle } a\left( {x,\xi } \right)\hat u\left( \xi \right)d\xi } . $$


Weight Function Cutoff Function Principal Symbol Carleman Estimate Plancherel Formula 
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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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