Estimates for Non-Selfadjoint Operators

  • Nicolas Lerner
Part of the Pseudo-Differential Operators book series (PDO, volume 3)


We have seen in the first chapter that it is very easy to prove a priori estimates for pseudo-differential operators of real principal type (see, e.g., the proof of Theorem 1.2.38), and also for principal-type operators whose imaginary part is either always non-negative or always non-positive (Theorem 1.2.39). Thanks to Remark 1.2.36, it boils down to proving an L2L2 injectivity estimate |Lu|0∼|u|0 for
$$ L = D_t + a(t,x,\xi )^w + ib(t,x,\xi )^w , where a,b \in S_{1,0}^1 , b \geqslant 0 (or b \leqslant 0) $$
depending continuously on t∈ℝ. Note that Lemma 4.3.21 in the appendix provides an even more precise form of such an estimate.1 At any rate, proving local solvability or an L2L2 injectivity estimate for that class of examples does not require much: for instance in the case b≥0, just conjugate the operator L by eλt so that
$$ L_\lambda = e^{2\pi \lambda t} Le^{ - 2\pi \lambda t} = D_t + a(t,x,\xi )^w + i(b(t,x,\xi )^w + \lambda ), $$
choose λ large enough so that, thanks to Gå;rding’s inequality (Theorem 1.1.26), b w +λ≥0 as an operator, and apply Lemma 4.3.21. Note in particular that for an evolution equation (t ∈ ℝ is the time-variable) where ReQ(t) is a pseudo-differential operator with real-valued Weyl symbol, the inequality (4.3.47) shows that the forward Cauchy problem is well-posed whenever ReQ(t) is bounded from below, e.g., is non-negative. Most evolution equations or systems of mathematical physics are essentially of the type (3.1.2) with a non-negative ReQ(t) together with very significant complications coming from rough coefficients or nonlinearities.


Symplectic Manifold Principal Symbol Principal Type Local Solvability Weyl Symbol 
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© Birkhäuser Verlag AG 2010

Authors and Affiliations

  • Nicolas Lerner
    • 1
  1. 1.Projet Analyse fonctionnelle Institut de Mathématique de JussieuUniversité Pierre et Marie Curie (Paris VI)Paris cedex 05France

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