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2010 | Buch

Metrics on the Phase Space and Non-Selfadjoint Pseudo-Differential Operators

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This book is devoted to the study of pseudo-di?erential operators, with special emphasis on non-selfadjoint operators, a priori estimates and localization in the phase space. We have tried here to expose the most recent developments of the theory with its applications to local solvability and semi-classical estimates for non-selfadjoint operators. The?rstchapter,Basic Notions of Phase Space Analysis,isintroductoryand gives a presentation of very classical classes of pseudo-di?erential operators, along with some basic properties. As an illustration of the power of these methods, we give a proof of propagation of singularities for real-principal type operators (using aprioriestimates,andnotFourierintegraloperators),andweintroducethereader to local solvability problems. That chapter should be useful for a reader, say at the graduate level in analysis, eager to learn some basics on pseudo-di?erential operators. The second chapter, Metrics on the Phase Space begins with a review of symplectic algebra, Wigner functions, quantization formulas, metaplectic group and is intended to set the basic study of the phase space. We move forward to the more general setting of metrics on the phase space, following essentially the basic assumptions of L. H¨ ormander (Chapter 18 in the book [73]) on this topic.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Basic Notions of Phase Space Analysis
Abstract
A differential operator of order m on ℝ n can be written as
$$ a(x,D) = \sum\limits_{\left| \alpha \right| \leqslant m} {a_a (x)D_x^\alpha } , $$
where we have used the notation (4.1.4) for the multi-indices. Its symbol is a polynomial in the variable ζ and is defined as
$$ a(x,\xi ) = \sum\limits_{\left| \alpha \right| \leqslant m} {a_a (x)\xi ^\alpha } , \xi ^\alpha = \xi _1^{\alpha _1 } \ldots \xi _n^{\alpha _n } . $$
We have the formula
$$ (a(x,D)u)(x) = \int_{\mathbb{R}^n } {e^{2i\pi x \cdot \xi } a(x,\xi )\hat u(\xi )d\xi } , $$
(1.1)
where û is the Fourier transform as defined in (4.1.1). It is possible to generalize the previous formula to the case where a is a tempered distribution on ℝ2n.
Nicolas Lerner
Chapter 2. Metrics on the Phase Space
Abstract
In this chapter, we describe a general version of the pseudo-differential calculus, due to L. Hörmander ([69], Chapter 18 in [73]). That version followed some earlier generalizations due to R. Beals and C. Fefferman ([8]) and to R. Beals ([6]). It was followed by some other generalizations due to A. Unterberger [145] and to a joint work of J.-M. Bony and N. Lerner ([20]), whose presentation we follow. We also give a precised version of the Fefferman-Phong inequality, following [100] where we provide an upper bound for the number of derivatives sufficient to obtain that inequality. Finally, we study the Sobolev spaces naturally attached to an admissible metric on the phase space, essentially following the paper by J.-M. Bony and J.-Y. Chemin [19].
Nicolas Lerner
Chapter 3. Estimates for Non-Selfadjoint Operators
Abstract
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
$$ L = D_t + a(t,x,\xi )^w + ib(t,x,\xi )^w , where a,b \in S_{1,0}^1 , b \geqslant 0 (or b \leqslant 0) $$
(3.1.1)
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)
https://static-content.springer.com/image/chp%3A10.1007%2F978-3-7643-8510-1_3/978-3-7643-8510-1_3_Equ2_HTML.gif
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.
Nicolas Lerner
Chapter 4. Appendix
Abstract
Let n≥1 be an integer. The Schwartz space S(ℝ n ) is defined as the space of C functions u from ℝ n to ℂ such that, for all multi-indices1 α, β∈ℕ n ,
$$ \mathop {\sup }\limits_{x \in \mathbb{R}^n } \left| {x^\alpha \partial _x^\beta u(x)} \right| < + \infty . $$
Nicolas Lerner
Backmatter
Metadaten
Titel
Metrics on the Phase Space and Non-Selfadjoint Pseudo-Differential Operators
verfasst von
Nicolas Lerner
Copyright-Jahr
2010
Verlag
Birkhäuser Basel
Electronic ISBN
978-3-7643-8510-1
Print ISBN
978-3-7643-8509-5
DOI
https://doi.org/10.1007/978-3-7643-8510-1