Elliptic Pseudo-Differential Operators — An Abstract Theory

338 Pages·1979·4.38 MB·English

by  H. O. Cordes (auth.)

Lecture Notes ni
Mathematics
Edited by .A Dold and .B Eckmann
756
.H .O Cordes
Elliptic Pseudo-Differential
Operators -
An Abstract Theory
galreV-regnirpS
Berlin Heidelberg New kroY 1979
Author
H. O. Cordes
Department of Mathematics
University of California
Berkeley, CA 94720/USA
AMS Subject Classifications (1970): 35-02, 35A25, 35J30, 35J40,
35J45, 35J 70, 35S99, 45 E05, 45 K05, 46 F10, 46 L05, 47 B30,
47 C10, 47F05
ISBN 3-540-09704-X Springer-Verlag Berlin Heidelberg NewYork
ISBN 0-387-09704-X Springer-Verlag NewYork Heidelberg Berlin
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To the memory
of F.Rellich
P=r=e=f=a=c=e=
The present volume essentially is an edited version of the notes after the
first two quarters of a lecture on partial differential eouations held in 1976/77
at Berkeley. It represents an approach to theory of linear elliptic partial
differential equations with emphasis on the abstract link between normal solvabili-
ty of a linear operator and the structure of certain commutative C~-algebras,
which are isomorphic to function algebras, by the Gel'fand-~lafmark theorem.
We will require a rather large amount of functional analysis. In particular
we use distribution theory (Chapters I and II), theory of Fredholm operators
(Appendix AI) and theory of commutative C~-algebras (Appendix AII). To be self-
contained we also perhaps should have included a discussion of the spectral theorem
for unbounded linear operators in Hilbert space, which is needed occasionally.
In chapter III we study an algebra of linear differential operators with
C -coefficients over ~n, acting either on L2_Sobolev spaces s' s c ~, or on the
locally convex spaces ~, ~_~. This investigation is rather formal. In particular
one seeks to adjoin convolution operators, like (I-A) -r , r > O, with the n-dimen-
sional Laplace operator A, and will have to derive a generalized Leibnitz formula
for products a(M)b(D) of a multiplication operator a(M) and a convolution
operator b(D). This is accomplished in form of an asyr~iptotic expansion which is a
finite sum whenever b({) is a polynomial. We introduce an extension algebra
of finitely generated pseudo differential operators for which a calculus is
derived. A variety of compactness results are derived, preparatory for chapter IV.
In chapter IV we study the Laplace comparison algebras Q, -~<s~ ~, of L2@Rn),
defined as C~-operator-algebras with unit over ~s generated by multiplications by
bounded continuous functions with oscillation vanishing at infinity, and by the
special convolutions D.(I-A) -I/2, j=l .... n. For an operator A e ~s' s E ~, an
3
abstract symbol is defined as the continuous function CA(X,~) associated to the
coset v A of A rood ~(~s ) by the Gelfand-Na'imark isometry ~s/K(~s ~ ) C~4 s) with
the maximal ideal space ~[s" It proves that {Z = ~s is independent of s, and the
symbol ~ of course is closely related to the 'conventional' symbol of a differen-
tial- (or pseudo-differential-) operator. An operator A ~ ~s is Fredholm if and
only if its symbol does never vanish. This results in a variety of very sharp
n
criteria regarding normal solvability of differential operators over ~ , as well as
on the spectrum, regularity etc. of such operators.
These results are extended to the case s =~ as well. In particular some effort
is spent to answer questions about the image of ~ in C )I~ under 0. This proves
important for derivation of a sharp Garding inequality, for example.
While chapter IV in effect is devoted to singular elliptic problems over R ,-°
with differential operators 'comparable' to the Laplace operator, we then turn to
genuine elliptic boundary problems in chapter V. The approach there is two-fold.
First one may apply Garding's inequality for results about the Dirichlet problem,
using a conventional approach via apriori estimates. Second one again may generate
n
a Laplace comparison algebra, for a subdomain of ~ , and use it in a manner
similar as in chapter IV. This is accomplished for a half space - and for s=O only.
The general elliptic boundary problem is discussed, for an even order elliptic
equation, with Lopatinskij-Shapiro type boundary conditions. It is no obstacle that
a half space is noncompact. In fact the method seems to apply more easily for the
singular problems.
The Laplace comparison algebra of the half space does not have compact
commutators, but gives raise to an ideal chain ~ <~C K(~), with a/~ ~ C~l)
and ~IK(~) A C~{2,K(~)), ~= L2((O,~)), ~= L2(~+I .)
2
For reason of simplicity we are restricting the discussion to L -theory only,
n
and only to the singular elliptic problem over ~ , and to the half space for the
elliptic boundary problem. Let us notice that far more general results have been
established by the author and his associates and students. In particular )~ R. Illner
i; I discussed LP-theory for n. R. McOwen 2 discussed Laplace comparison algebras
on complete Riemannian manifolds. M. Taylor 3 discussed a hypo-elliptic compari-
son algebra over n and E. Tomer ~4~ used a second order elliptic comparison
operator different from £ (the harmonic oscillator in quantum theory).
The following represents the author's personal approach to the subject, which
exists for some time while the attempts to publish a comprehensive version have
remained incomplete. We are indebted to E. Herman for joint efforts in the late
1960-s. To Lars G~rding we owe the chance of presenting the material in a lecture
at Lund in 1970-71. We are indebted to J. Frehse and to Alexander v. Humboldt
foundation for a free year to work on the subject. - Finally we are grateful to Mrs.
.I Kreuder and the staff of SFB 72, Bonn for organizing the t!fping of the manuscript
)~ i I R. Illner, On algebras of pseudo-differential operators in LP~n); Commun. on
partial diff. equ. 2 (1977) 359-393; 12 H.O. Cordes and R. McOwen, Remarks on
singular elliptic theory for complete Riemannien manifolds; Pacific. Journ.
Math. 70 (1977) 133-141; ~3E M.E. Taylor, Gel'land theor!7 of pseudo-differential
operators and hypo- elliptic operators; Trans. Amer. Math. Soc. 153 (1971) 495-
510; ~4 E. Tomer, On the C~-algebra of the Hermite Operator; Thesis at
Berkeley, to appear.
C=o=n=t=e=n=t=s= page
Chapter I. A survey of distribution theory. 1
i. Con~non facts about function spaces and functions. 4
2. Definition of distributions and elementary properties. ii
3. The convolution algebra Ll~n); the Fourier integral and 2O
the space S.
4. Temperate distributions and temperate functions; Fourier 27
transform of distributions and Fourier multipliers.
5. Distributions with compact support. 34
6. Order of a distribution and topology in ~', S', ~'. 38
7. Tensor products of distributions. 42
8. Convolution product of distributions. 47
9. On (Fourier) distribution kernels and kernel 53
multiplication.
iO. References. 57
Chapter II. Distributions with rational singularities. 58
.i Rational points and pseudo-functions. 61
2. Positive homogeneous functions and distributions; 66
finite parts.
3. Asymptotic expansions modulo D. 70
4. Fourier transform of distributions with rational 73
singularities.
5. Fourier equivalence of asymptotic expansions 77
mod D and mod Ix[.
6. References. 82
Chapter III. Finitely generated pseudo-differential operators. 83
i. Definition of L2-Sobolev spaces. 85
2. Some more simple properties of Sobolev spaces. 89
~s-bOunded operators.
3. 91
4. Operators of [(~) and their true order function. 96
o
5. The spaces 0(r),-~ ~ r ~ + ~ and the algebras [ , ~ 0. 99
6. A Leibnitz formula for finitely generated ~do's. 103
7. Calculus of pseudo-differential operators. iiO
8. A compactness result. 114
9. Commutator relations between multiplications 117
and differentiations.
o
iO. The algebra 0~ of finitely generated ~do-s with 120
zero oscillation at =.
11. References. 126
page
2
Chapter IV. On the Laplace comparison algebras for L .)n.?@ 127
.I The Laplace comparison algebras. 130
.2 Herman's Lemma and the proof of theorem 1.6. 137
3. The algebra ~ and the algebra TD of ~do-s. 141
4. Fredholm theory in the algebra ~ ; 149
Green inverse.
5. Some more auxiliary results. 153
6. Con~nuting ~do-s with differentiations. 157
.7 Symbols of operators in~ , and a Leibnitz formula. 165
8. More smoothness conditions for the symbol. 169
9. Derivatives of a linear operator in (S,S'). 171
10. About infitely differentiable operators of (~0 ) . 177
ii. References. 180
Chapter V. Elliptic boundary problems. 181
.i On Garding's inequality. Elliptic and strongly 184
elliptic %do-s.
.2 On the Dirichlet problem for strongly elliptic e_quations. 187
.3 The general elliptic boundary problem on a half-space. 193
4. The Laplace comparison algebra 01of L20R~ 'I)- 196
5. Wiener-Hopf convolutions. 201
6. Mellin convolutions. 204
.7 The algebra ~and its structure. 207
8. C~-algebras with matrix-valued or compact-operator- 217
valued symbols.
9. Fredholm and structure theory of C~-algebras 224
with compact(operator valued) symbol.
10. The Laplace comparison of ~ = L2~gn+1),+ 228
continued.
11. Proof of the left-over auxiliary results. 234
12. General boundary conditions, for a constant 237
coefficient equation.
13. Relation between ~n+l and ~, and its effect on symbols. 243
--O
14. The operator A = <a>T and its normal solvability. 245
15. References. 251
Appendix AI. Preparations on linear operators and operator algebras. 253
.i Definition of Fredholm map; historical examples. 254
.2 Algebraic theory of Fredholm operators. 258
.3 Bounded Fredholm operators on Banach spaces. 263
4. Compact operators and Rellich Criteria. 268
page
5. Fredholm closed algebras and Riesz-ideals. 273
6. Unbounded linear operators on Banach spaces. 276
.7 Fredholm pairs and unbounded operators. 284
8. References. 289
Appendix AII. Preparations on Commutative Banach algebras. 29O
.I Fundamentals of B-algebras. 291
.2 Spectrum and spectral radius. 296
3. The maximal ideal space. 303
4. The Gel'fand theorem. 3O8
.5 The associate dual map. 313
6. The Stone-Weierstrass theorem. 317
7. ~-algebras. 319
8. Closed ideals of a C -algebra. 324
9. References 327
Subject index 328
O. Introduction.
In the present chapter we will learn about the basics of the most important
kind of generalized functions, called distributions. Perhaps it will be an aid
for understanding the concept if we start with the remark that the emphasis is not
so much on generalizing the topological function concept, but rather on a new
function-integration theory, called distribution and distribution integral. The
rigorous theory of Riemann and Lebesgue integral, while being a most satisfying
logical complex has left some incompatibilities with practical needs. Some of these
will be removed.
Physicists have known and used distributions (of a special kind) for a long
time. For example, in Physics, mass or electrical charge, in the macroscopic
world, will never be concentrated at single points. One will have a 'density
function', a 'distribution function' ~(x) = Z(xl,x2,x )3 defined and (non-negative)
real-valued in (xl,x2,x3)-space 3. The mass of a (measurable) subset QC 3 is
given by the integral m~ = ]~(x)dx. However, for a simple mathematical descrip-
tion of dynamics it is very useful to consider the idealization of point mass,
which does not occur in nature but can be approximated arbitrarily by assigning
large values to the distribution function Z near a given point ° x and having it
zero elsewhere.
A 'mass point' is a physical reality. A rigid body can be replaced in many
respects, by a mass point at its center of gravity, etc.. However there is no
distribution function for a mass point, if the integral in above formula for m~ is
interpreted as Riemann or Lebesgue integral. Hence one introduces a generalized
distribution function called Dirac delta function @. It takes the values 6(0) = ~,
@(x) = O, x%O. But the most important fact is that we have ~@(x)dx = O, as O}~,
= ,i as Og~, for open sets .~ More generally,
(O.i) ~ 6(x)~(x)dx = ~(O)
3
for every continuous function ~ over ~ .
Here the function concept does not have to be generalized but a new integral
concept must be introduced. Things get worse with the charge distribution of an
electrical dipole. Physically, a dipole consists of a pair of point charges of
equal magnitude but opposite sign, at distance zero. It is approximated by a
pair of magnitude q at distance d. In the limit d + O, keeping the direction and
the dipole moment M=qd constant, an electrical field is reached which is not zero.
The charge distribution UD of a l-dimensional dipole, for simplicity, of dipole
moment ,i pointing into the positive x-direction should have the property that
f
(0.2) JgD(X)~(x)dx = %'(O) = d%/dx(O) ,
for every continuously differentiable ~ over ~. Again this is incompatible with
Riemann or Lebesgue integration. No assignment of values of the 'function' PD(X) at
x=O is reasonnable.
Another such incompatibility between integration theory and practical needs
arises in theory of hyperbolic second order partial differential equations, as
first explored by J. Hadamard 4. There it was necessary to introduce a certain
very general singular integral concept, called finite part. A finite part integral,
from our later view point of chapter II, will be a special kind of distribution
integral, just as well as the so-called Cauchy principal value which also occurs in
other parts of analysis. The underlying generalized functions perhaps are not as
simple as the above delta-function and the distribution DD" Thus we shall make no
attempt of a description until later on (chapter II).
The above examples may serve as guide into the new concept: The name
'distribution' may be derived from the fact that the two generalized functions 6
and PD above are modelled after mass or charge distributions of idealized physical
objects. In abstract mathematical definition (in section )2 a distribution T no
n
longer will be given by its values at every point xeR , but by the values of the
integral
(O.3) T~dx ,
with testing functions }, infinitely differentiable and zero for large Ixl, where
the 'distribution integral' (0.3) simply is only an abstract linear functional over
the space D of testing functions, continuous in a manner to be specified. Not all
measurable functions are also distributions but only these which are locally
integrable, and these are uniquely determined, modulo a nul set by their Lebesgue
integrals (0.3).
The main reason we have here of presenting the material again, in view of a
large number of existing books on the subject, is that of self-containedness and
preselection of material most useful for us. We have tried to focus on the aspects
of classical analysis, and refer to Laurent schwartz 71 for all more sophisticated
matters of topology. There perhaps the most natural introduction into distribution
spaces, as a class of locally convex topological vector spaces can be found. Also
there are detailed results of almost every kind, which had to be omitted here.
As another book emphasizing classical analysis we mention Gelfand-Silov 3~ .
Other works more oriented towards locally convex spaces are Treves EI2 and Edwards
2~. It would be impossible to mention or even read all publications on the subject
There are other kinds of generalized functions, like the hyper functions of
Sato and the ultradistributions. None of these can be discussed.