Book contents
- Frontmatter
- Contents
- Preface
- Chapter 0 Introductory discussions
- Chapter 1 Calculus of pseudodifferential operators
- Chapter 2 Elliptic operators and parametrices in ℝn
- Chapter 3 L2-Sobolev theory and applications
- Chapter 4 Pseudodifferential operators on manifolds with conical ends
- Chapter 5 Elliptic and paraolic problems
- Chapter 6 Hyperbolic first order systems
- Chapter 7 Hyperbolic differential equations
- Chapter 8 Pseudodifferential operators as smooth operators of L(H)
- Chapter 9 Particle flow and invariant algebra of a semi-strictly hyperbolic system; coordinate invariance of Opψxm.
- Chapter 10 The invariant algebra of the Dirac equation
- References
- Index
Chapter 1 - Calculus of pseudodifferential operators
Published online by Cambridge University Press: 09 February 2010
- Frontmatter
- Contents
- Preface
- Chapter 0 Introductory discussions
- Chapter 1 Calculus of pseudodifferential operators
- Chapter 2 Elliptic operators and parametrices in ℝn
- Chapter 3 L2-Sobolev theory and applications
- Chapter 4 Pseudodifferential operators on manifolds with conical ends
- Chapter 5 Elliptic and paraolic problems
- Chapter 6 Hyperbolic first order systems
- Chapter 7 Hyperbolic differential equations
- Chapter 8 Pseudodifferential operators as smooth operators of L(H)
- Chapter 9 Particle flow and invariant algebra of a semi-strictly hyperbolic system; coordinate invariance of Opψxm.
- Chapter 10 The invariant algebra of the Dirac equation
- References
- Index
Summary
Introduction.
In this chapter we deal with the details of pseudo-differential operator calculus. We follow a presentation in a lecture of 1974/75 [CP], inspired by the local approach of Hoermander [Hr2]dealing with operators on ℝn, for didactical reasons. Replacing asymptotic expansions of [CP] by Leibniz formulas with integral reminder of 1.5 is an improvement we learned from R.Beals [B1] who uses ‘weight functions’ more general than our ‘x’m2 ‘x’m1 of (3.2). Still asymptotic expansions are needed, and will be studied in 1.6.
We will discuss 4 different representations of ψdo', referenced as a(x,D)=a(M1D), a(Mr,D), a (Ml, Mr, D), and a(Mw,D), the first two corresponding to the left and right multiplying of Kohn and Nirenberg [KN], and the others to a representation of Friedrichs [Fr3], and the Weyl representation.
The reader who dislikes the infinitely repeated formal discussions has our sympathy. For other approaches to the same subject cf. ch.7 where the ψdo's of certain symbol classes are identified as operators on H=L2 (φn), smooth under action of certain Lie subgroups of U(H). Or else, cf. [C1] and [C2, where regular and singular elliptic boundary problems are approached with tools of C* -algebras, avoiding entirely the ψdo-calculus.
The calculus presented generalizes formal calculus of differential operators. We get a collection of Frechet algebras containing differential operators, with formulas for product and adjoint like Leibniz formulas, containing generalized inverses (so-called Green inverses) of their elliptic and hypo-elliptic operators, as seen in ch.II. The algebras are ‘graded’: Each of their operators has a differentiation (m1) and a multiplication (m2) order.
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- The Technique of Pseudodifferential Operators , pp. 52 - 80Publisher: Cambridge University PressPrint publication year: 1995