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2010 | Buch

Introduction Kapitel lesen Erstes Kapitel lesen

Pseudo-Differential Operators and Symmetries

Background Analysis and Advanced Topics

verfasst von: Michael Ruzhansky, Ville Turunen

Verlag: Birkhäuser Basel

Buchreihe : Pseudo-Differential Operators

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This monograph is devoted to the development of the theory of pseudo-di?erential n operators on spaces with symmetries. Such spaces are the Euclidean space R ,the n torus T , compact Lie groups and compact homogeneous spaces. The book consists of several parts. One of our aims has been not only to present new results on pseudo-di?erential operators but also to show parallels between di?erent approaches to pseudo-di?erential operators on di?erent spaces. Moreover, we tried to present the material in a self-contained way to make it accessible for readers approaching the material for the ?rst time. However, di?erent spaces on which we develop the theory of pseudo-di?er- tial operators require di?erent backgrounds. Thus, while operators on the - clidean space in Chapter 2 rely on the well-known Euclidean Fourier analysis, pseudo-di?erentialoperatorsonthetorusandmoregeneralLiegroupsinChapters 4 and 10 require certain backgrounds in discrete analysis and in the representation theory of compact Lie groups, which we therefore present in Chapter 3 and in Part III,respectively. Moreover,anyonewhowishestoworkwithpseudo-di?erential- erators on Lie groups will certainly bene?t from a good grasp of certain aspects of representation theory. That is why we present the main elements of this theory in Part III, thus eliminating the necessity for the reader to consult other sources for most of the time. Similarly, the backgrounds for the theory of pseudo-di?erential 3 operators on S and SU(2) developed in Chapter 12 can be found in Chapter 11 presented in a self-contained way suitable for immediate use.

Inhaltsverzeichnis

Frontmatter

Introduction

Introduction
Abstract
Pseudo-differential operators (ΨDO) can be considered as natural extensions of linear partial differential operators, with which they share many essential properties. The study of pseudo-differential operators grew out of research in the 1960s on singular integral operators; being a relatively young subject, the theory is only now reaching a stable form.
Michael Ruzhansky, Ville Turunen

Foundations of Analysis

Frontmatter
Chapter A. Sets, Topology and Metrics
Abstract
First, we present the basic notations and properties of sets, used elsewhere in the book. The set theory involved is “naive”, sufficient for our purposes; for a thorough treatment, see, e.g., [46]. The sets of integer, rational, real or complex numbers will be taken for granted, we shall not construct them.
Michael Ruzhansky, Ville Turunen
Chapter B. Elementary Functional Analysis
Abstract
We assume that the reader already has knowledge of (complex) matrices, determinants, etc. In this chapter, we shall present basic machinery for dealing with vector spaces, especially Banach and Hilbert spaces. We do not go into depth in this direction as there are plenty of excellent specialised monographs available devoted to various aspects of the subject, see, e.g., [11, 35, 53, 59, 63, 70, 87, 89, 90, 116, 134, 146, 153]. However, we still make an independent presentation of a collection of results which are indispensable for anyone working in analysis, and which are useful for other parts of this book.
Michael Ruzhansky, Ville Turunen
Chapter C. Measure Theory and Integration
Abstract
This chapter provides sufficient general information about measures and integration for the purposes of this book. The starting point is the concept of an outer measure, which “measures weights of subsets of a space”. We should first consider how to sum such weights, which are either infinite or non-negative real numbers. For a finite set K, notation
$$ \sum\limits_{j \in K} {a_j } $$
abbreviates the usual sum of numbers a j ∈ [0, ∞] over the index set K. The conventions here are that a+∞=∞ for all a ∈ [0, ∞], and that
$$ \sum\limits_{j \in 0/} {a_j = 0.} $$
Infinite summations are defined by limits as follows: Definition C.0.1. The sum of numbers a j ∈ [0, ∞] over the index set J is
$$ \sum\limits_{j \in J} {a_j : = } \sup \left\{ {\sum\limits_{j \in K} {a_j :} K \subset J is finite} \right\}. $$
Exercise C.0.2. Let 0 < a j < ∞ for each j ∈ J. Suppose
$$ \sum\limits_{j \in J} {a_j } < \infty . $$
Show that J is at most countable.
Michael Ruzhansky, Ville Turunen
Chapter D. Algebras
Abstract
An algebra is a vector space endowed with a multiplication, satisfying some compatibility conditions. In the sequel, we are going to deal with spectral properties of algebras under various additional assumptions.
Michael Ruzhansky, Ville Turunen

Commutative Symmetries

Frontmatter
Chapter 1. Fourier Analysis on ℝ n
Abstract
In this chapter we review basic elements of Fourier analysis on ℝ n . Consequently, we introduce spaces of distributions, putting emphasis on the space of tempered distributions S′(ℝ n ). Finally, we discuss Sobolev spaces and approximation of functions and distributions by smooth functions. Throughout, we fix the measure on ℝ n to be Lebesgue measure. For convenience, we may repeat a few definitions in the context of ℝ n although they may have already appeared in Chapter C on measure theory. From this point of view, the present chapter can be read essentially independently. The notation used in this chapter and also in Chapter 2 is 〈ξ〉 = (1 + |ξ|2)1/2 where |ξ| = (ξ12 + ξ n 2)1/2, ξ ∈ ℝ n .
Michael Ruzhansky, Ville Turunen
Chapter 2. Pseudo-differential Operators on ℝ n
Abstract
The subject of pseudo-differential operators on ℝ n is well studied and there are many excellent monographs on the subject, see, e.g., [27, 33, 55, 71, 101, 112, 130, 135, 152], as well as on the more general subject of Fourier integral operators, microlocal analysis, and related topics in, e.g., [30, 56, 45, 81, 113]. Therefore, here we only sketch main elements of the theory. In this chapter, we use the notation 〈ξ〉 = (1 + |ξ|2)1/2.
Michael Ruzhansky, Ville Turunen
Chapter 3. Periodic and Discrete Analysis
Abstract
In this chapter we will review basics of the periodic and discrete analysis which will be necessary for development of the theory of pseudo-differential operators on the torus in Chapter 4. Our aim is to make these two chapters accessible independently for people who choose periodic pseudo-differential operators as a starting point for learning about pseudo-differential operators on ℝ n . This may be a fruitful idea in the sense that many technical issues disappear on the torus as opposed to ℝ n . Among them is the fact that often one does not need to worry about convergence of the integrals in view of the torus being compact. Moreover, the theory of distributions on the torus is much simpler than that on ℝ n , at least in the form required for us. The main reason is that the periodic Fourier transform takes functions on \( \mathbb{T}^n \) = ℝ n /ℤ n to functions on ℤ n where, for example, tempered distributions become pointwise defined functions on the lattice ℤ n of polynomial growth at infinity. Also, on the lattice ℤ n there are no questions of regularity since all the objects are defined on a discrete set. However, there are many parallels between Euclidean and toroidal theories of pseudo-differential operators, so looking at proofs of similar results in different chapters may be beneficial. In many cases we tried to avoid overlaps by presenting a different proof or by giving a different explanation.
Michael Ruzhansky, Ville Turunen
Chapter 4. Pseudo-differential Operators on $$ \mathbb{T}^n $$
Abstract
Pseudo-differential operators on the torus \( \mathbb{T}^n \) = ℝ n /ℤ n , or the periodic pseudo-differential operators, are studied next. The presentation is written in a way for a reader to be able to compare and to contrast it to the general theory of pseudo-differential operators on the Euclidean space from Chapter 2.
Michael Ruzhansky, Ville Turunen
Chapter 5. Commutator Characterisation of Pseudo-differential Operators
Abstract
On a smooth closed manifold the pseudo-differential operators can be characterised by taking commutators with vector fields, i.e., first-order partial derivatives. This approach is due to Beals ([12], 1977), Dunau ([32], 1977), and Coifman and Meyer ([23], 1978); perhaps the first ones to consider these kind of commutator properties were Calderón and his school [21]. For other contributions, see also [26], [133] and [80].
Michael Ruzhansky, Ville Turunen

Representation Theory of Compact Groups

Frontmatter
Chapter 6. Groups
Abstract
Loosely speaking, groups encode symmetries of (geometric) objects: if we consider a space X with some specific structure (e.g., a Riemannian manifold), a symmetry of X is a bijection f: X→X preserving the natural involved structure (e.g., the Riemannian metric) — here, the compositions and inversions of symmetries yield new symmetries. In a handful of assumptions, the concept of groups captures the essential properties of wide classes of symmetries, and provides powerful tools for related analysis.
Michael Ruzhansky, Ville Turunen
Chapter 7. Topological Groups
Abstract
A topological group is a natural amalgam of topological spaces and groups: it is a Hausdorff space with continuous group operations. Topology adds a new flavour to representation theory. Especially interesting are compact groups, where group-invariant probability measures exist. Moreover, nice-enough functions on a compact group have Fourier series expansions, which generalise the classical Fourier series of periodic functions.
Michael Ruzhansky, Ville Turunen
Chapter 8. Linear Lie Groups
Abstract
In this chapter we study linear Lie groups, i.e., Lie groups which are closed subgroups of GL(n, ℂ). But first some words about the general Lie groups: Definition 8.0.1 (Lie groups). A Lie group is a C∞-manifold which is also a group such that the group operations are C∞-smooth.
Michael Ruzhansky, Ville Turunen
Chapter 9. Hopf Algebras
Abstract
Instead of studying a compact group G, we may consider the algebra C(G) of continuous functions C → ℂ. The structure of the group is encoded in the function algebra, but we shall see that this approach paves the way for a more general functional analytic theory of Hopf algebras, which possess nice duality properties.
Michael Ruzhansky, Ville Turunen

Non-commutative Symmetries

Frontmatter
Chapter 10. Pseudo-differential Operators on Compact Lie Groups
Abstract
In this chapter we develop a global theory of pseudo-differential operators on general compact Lie groups. As usual, S1,0 m (ℝ n × ℝ n ) ⊂ C∞ (ℝ n × ℝ n ) refers to the Euclidean space symbol class, defined by the symbol inequalities
$$ \left| {\partial _\xi ^\alpha \partial _x^\beta p(x,\xi )} \right| \leqslant C(1 + \left| \xi \right|)^{m - \left| \alpha \right|} , $$
(10.1)
for all multi-indices α, β ∈ ℕ0 n , ℕ0 = {0}∪ℕ where the constant C is independent of x ξ ∈ ℝ n but may depend on α, β, p, m. On a compact Lie group G we define the class Ψ m (G) to be the usual Hörmander class of pseudo-differential operators of order m. Thus, the operator A belongs to Ψ m (G) if in (all) local coordinates operator A is a pseudo-differential operator on ℝ n with some symbol p(x, ξ) satisfying estimates (10.1), see Definition 5.2.11. Of course, symbol p depends on the local coordinate systems.
Michael Ruzhansky, Ville Turunen
Chapter 11. Fourier Analysis on SU(2)
Abstract
In this chapter we develop elements of Fourier analysis on the group SU(2) in a form suitable for the consequent development of the theory of pseudo-differential operators on SU(2) in Chapter 12. Certain results from this chapter can be found in [148], which, together with [154] we can recommend for further reading, including for some instances of explicitly calculated Clebsch-Gordan coefficients. However, on this occasion, with pseudo-differential operators in mind and the form of the analysis necessary for us and adopted to Chapter 10, we present an independent exposition of SU(2) with considerably more direct proofs and different arguments compared to, e.g., [148].
Michael Ruzhansky, Ville Turunen
Chapter 12. Pseudo-differential Operators on SU(2)
Abstract
In this chapter we carry out the analysis of operators on SU(2) with an application to the operators on the three-dimensional sphere \( \mathbb{S}^3 \). In particular, we derive a much simpler symbolic characterisation of pseudo-differential operators on SU(2) than the one given in Definition 10.9.5. In turn, this will also yield a characterisation of full symbols of pseudo-differential operators on the 3-sphere \( \mathbb{S}^3 \). We note that this approach works globally on the whole sphere, since the version of the Fourier analysis that we use is different from the one in, e.g., [110, 122, 111] which covers only a hemisphere, with singularities at the equator. For a general introduction and motivation for the analysis on SU(2) we refer the reader to the introduction in Part IV where the cases of SU(2) and \( \mathbb{S}^3 \) were put in a perspective.
Michael Ruzhansky, Ville Turunen
Chapter 13. Pseudo-differential Operators on Homogeneous Spaces
Abstract
In this chapter we discuss pseudo-differential operators on homogeneous spaces. The main question addressed here is how operators on such a space are related to pseudo-differential operators on the group that acts on the space. Once such a correspondence is established, one can use it to map the whole construction developed earlier from the group to the homogeneous space. We also note that among other things, this chapter provides an application to the characterisation of pseudo-differential operators in terms of Σ m -classes in Theorem 10.9.6. An important class of examples to keep in mind here are the spheres \( \mathbb{S}^n \cong SO(n)\backslash SO(n + 1) \cong SO(n + 1)/SO(n) \).
Michael Ruzhansky, Ville Turunen
Backmatter
Metadaten
Titel
Pseudo-Differential Operators and Symmetries
verfasst von
Michael Ruzhansky
Ville Turunen
Copyright-Jahr
2010
Verlag
Birkhäuser Basel
Electronic ISBN
978-3-7643-8514-9
Print ISBN
978-3-7643-8513-2
DOI
https://doi.org/10.1007/978-3-7643-8514-9

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