Hàm suy rộng - Hay

Graduate Texts in Mathematics 25
Editorial Board
S. Axler
K.A. Ribet
2
Graduate Texts in Mathematics
1 Takeuti/Zaring. Introduction to Axiomatic
Set Theory. 2nd ed.
2 Oxtoby. Measure and Category. 2nd ed.
3 Schaefer. Topological Vector Spaces.
2nd ed.
4 Hilton/Stammbach. A Course in
Homological Algebra. 2nd ed.
5 Mac Lane. Categories for the Working
Mathematician. 2nd ed.
6 Hughes/Piper. Projective Planes.
7 J P. Serre. A Course in Arithmetic.
8 Takeuti/Zaring. Axiomatic Set Theory.
9 Humphreys. Introduction to Lie Algebras
and Representation Theory.
10 Cohen. A Course in Simple Homotopy
Theory.
11 Conway. Functions of One Complex
Variable I. 2nd ed.
12 Beals. Advanced Mathematical Analysis.
13 Anderson/Fuller. Rings and Categories
of Modules. 2nd ed.
14 Golubitsky/Guillemin. Stable Mappings
and Their Singularities.
15 Berberian. Lectures in Functional Analysis

and Operator Theory.
16 Winter. The Structure of Fields.
17 Rosenblatt. Random Processes. 2nd ed.
18 Halmos. Measure Theory.
19 Halmos. A Hilbert Space Problem Book.
2nd ed.
20 Husemoller. Fibre Bundles. 3rd ed.
21 Humphreys. Linear Algebraic Groups.
22 Barnes/Mack. An Algebraic Introduction
to Mathematical Logic.
23 Greub. Linear Algebra. 4th ed.
24 Holmes. Geometric Functional Analysis
and Its Applications.
25 Hewitt/S
tromberg. Real and Abstract
Analysis.
26 Manes. Algebraic Theories.
27 Kelley. General Topology.
28 Zariski/Samuel. Commutative Algebra. Vol. I.
29 Zariski/Samuel. Commutative Algebra. Vol. II.
30 Jacobson. Lectures in Abstract Algebra I.
Basic Concepts.
31 Jacobson. Lectures in Abstract Algebra II.
Linear Algebra.
32 Jacobson. Lectures in Abstract Algebra III.
Theory of Fields and Galois Theory.
33 Hirsch. Differential Topology.
34 Spitzer. Principles of Random Walk. 2nd ed.
35 Alexander/Wermer. Several Complex
Variables and Banach Algebras. 3rd ed.

36 Kelley/Namioka et al. Linear Topological
Spaces.
37 Monk. Mathematical Logic.
38 Grauert/Fritzsche. Several Complex
Variables.
39 Arveson. An Invitation to C-Algebras.
40 Kemeny/Snell/Knapp. Denumerable
Markov Chains. 2nd ed.
41 Apostol. Modular Functions and Dirichlet
Series in Number Theory. 2nd ed.
42 J P. Serre. Linear Representations of Finite
Groups.
43 Gillman/Jerison. Rings of Continuous
Functions.
44 Kendig. Elementary Algebraic Geometry.
45 Loève. Probability Theory I. 4th ed.
46 Loève. Probability Theory II. 4th ed.
47 Moise. Geometric Topology in Dimensions
2 and 3.
48 Sachs/Wu. General Relativity for
Mathematicians.
49 Gruenberg/Weir. Linear Geometry.
2nd ed.
50 Edwards. Fermat’s Last Theorem.
51 Klingenberg. A Course in Differential
Geometry.
52 Hartshorne. Algebraic Geometry.
53 Manin. A Course in Mathematical Logic.
54 Graver/Watkins. Combinatorics with
Emphasis on the Theory of Graphs.

55 Brown/Pearcy. Introduction to Operator
Theory I: Elements of Functional Analysis.
56 Massey. Algebraic Topology: An
Introduction.
57 Crowell/Fox. Introduction to Knot Theory.
58 Koblitz. p-adic Numbers, p-adic Analysis,
and Zeta-Functions. 2nd ed.
59 Lang. Cyclotomic Fields.
60 Arnold. Mathematical Methods in
Classical Mechanics. 2nd ed.
61 Whitehead. Elements of Homotopy
Theory.
62 Kargapolov/Merizjakov. Fundamentals
of the Theory of Groups.
63 Bollobas. Graph Theory.
64 Edwards. Fourier Series. Vol. I. 2nd ed.
65 Wells. Differential Analysis on Complex
Manifolds. 2nd ed.
66 Waterhouse. Introduction to Affi ne Group
Schemes.
67 Serre. Local Fields.
68 Weidmann. Linear Operators in Hilbert
Spaces.
69 Lang. Cyclotomic Fields II.
70 Massey. Singular Homology Theory.
71 Farkas/Kra. Riemann Surfaces. 2nd ed.
72 Stillwell. Classical Topology and
Combinatorial Group Theory. 2nd ed.
73 Hungerford. Algebra.
74 Davenport. Multiplicative Number Theory.

3rd ed.
75 Hochschild. Basic Theory of Algebraic
Groups and Lie Algebras.
(continued after index)
Gerd Grubb
Distribution ands peratorsO

@ .
Editorial Board
S. Axler
Mathematics Department
San Francisco State University
San Francisco, CA 94132
USA

K.A. Ribet
Mathematics Department
University of California at Berkeley
Berkeley, CA 94720-3840
USA

ISBN: 978-0-387-84894-5 e-ISBN: 978-0-387-84895-2
Library of Congress Control Number: 2008937582
Mathematics Subject Classifi cation (2000): xx
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Gerd Grubb
University of Copenhagen
Universitetsparken
5
2100 Koebenhavn
Denmark
grubb math ku.dk
47B 47B06:
Department of Mathematical Sciences
Preface
This textbook gives an introduction to distribution theory with emphasis on
applications using functional analysis. In more advanced parts of the book,
pseudodifferential methods are introduced.
Distribution theory has been developed primarily to deal with partial (and
ordinary) differential equations in general situations. Functional analysis in,
say, Hilbert spaces has powerful tools to establish operators with good map-
ping properties and invertibility properties. A combination of the two allows
showing solvability of suitable concrete partial differential equations (PDE).
When partial differential operators are realized as operators in L
2
(Ω) for
an open subset Ω of R
n
, they come out as unbounded operators. Basic courses
in functional analysis are often limited to the study of bounded operators, but

we here meet the necessity of treating suitable types of unbounded operators;
primarily those that are densely defined and closed. Moreover, the emphasis
in functional analysis is often placed on selfadjoint or normal operators, for
which beautiful results can be obtained by means of spectral theory, but
the cases of interest in PDE include many nonselfadjoint operators, where
diagonalization by spectral theory is not very useful. We include in this book a
chapter on unbounded operators in Hilbert space (Chapter 12), where classes
of convenient operators are set up, in particular the variational operators,
including selfadjoint semibounded cases (e.g., the Friedrichs extension of a
symmetric operator), but with a much wider scope.
Whereas the functional analysis definition of the operators is relatively
clean and simple, the interpretation to PDE is more messy and complicated.
It is here that distribution theory comes in useful. Some textbooks on PDE
are limited to weak definitions, taking place e.g. in L
2
(Ω). In our experience
this is not quite satisfactory; one needs for example the Sobolev spaces with
negative exponents to fully understand the structure. Also, Sobolev spaces
with noninteger exponents are important, in studies of boundary conditions.
Such spaces can be very nicely explained in terms of Fourier transformation
of (temperate) distributions, which is also useful for many further aspects of
the treatment of PDE.
v
vi Preface
In addition to the direct application of distribution theory to interpret
partial differential operators by functional analysis, we have included some
more advanced material, which allows a further interpretation of the solu-
tion operators, namely, the theory of pseudodifferential operators, and its
extension to pseudodifferential boundary operators.
The basic part of the book is Part I (Chapters 1–3), Chapter 12, and

Part II (Chapters 4–6). Here the theory of distributions over open sets is
introduced in an unhurried way, their rules of calculus are established by
duality, further properties are developed, and some immediate applications
are worked out. For a correct deduction of distribution theory, one needs a
certain knowledge of Fr´echet spaces and their inductive limits. We have tried
to keep the technicalities at a minimum by relegating the explanation of such
spaces to an appendix (Appendix B), from which one can simply draw on the
statements, or go more in depth with the proofs if preferred. The functional
analysis needed for the applications is explained in Chapter 12. The Fourier
transformation plays an important role in Part II, from Chapter 5 on.
The auxiliary tools from functional analysis, primarily in Hilbert spaces,
are collected in Part V. Besides Chapter 12 introducing unbounded operators,
there is Chapter 13 on extension theory and Chapter 14 on semigroups.
Part III is written in a more compact style.WehereextendthePDEtheory
by the introduction of x-dependent pseudodifferential operators (ψdo’s), over
open sets (Chapter 7) as well as over compact C
∞
manifolds (Chapter 8).
This is an important application of distribution theory and leads to a very
useful “algebra” of operators including partial differential operators and the
solution operators for the elliptic ones. Fredholm theory is explained and used
to establish the existence of an index of elliptic operators.
Pseudodifferential operators are by many people regarded as a very so-
phisticated tool, and indeed it is so, perhaps most of all because of the im-
precisions in the theory: There are asymptotic series that are not supposed
to converge, the calculus works “modulo operators of order −∞”, etc. We
have tried to sum up the most important points in a straightforward way.
Part IV deals with boundary value problems. Homogeneous boundary con-
ditions for some basic cases were considered in Chapter 4 (with reference to
the variational theory in Chapter 12); this was supplied with the general

G˚arding inequality at the end of Chapter 7. Now we present an instructive
example in Chapter 9, where explicit solution operators for nonhomogeneous
Dirichlet and Neumann problems are found, and the role of half-order Sobolev
spaces over the boundary (also of negative order) is demonstrated. Moreover,
we here discuss some other Neumann-type conditions (that are not always el-
liptic), and interpret the abstract characterization of extensions of operators
in Hilbert space presented in Chapter 13, in terms of boundary conditions.
Whereas Chapter 9 is “elementary”, in the sense that it can be read di-
rectly in succession to Parts I and II, the next two chapters, based on Part III,
contain more heavy material. It is our point of view that a modern treatment
Preface vii
of boundary value problems with smooth coefficients goes most efficiently by a
pseudodifferential technique related to the one on closed manifolds. In Chap-
ter 10 we give an introduction to the calculus of pseudodifferential boundary
operators (ψdbo’s) initiated by Boutet de Monvel, with full details of the
explanation in local coordinates. In Chapter 11 we introduce the Calder´on
projector for an elliptic differential operator on a manifold with boundary,
and show some of its applications.
The contents of the book have been used frequently at the University of
Copenhagen for a one-semester first-year graduate course, covering Chapters
1–3, 12 and 4–6 (in that order) with some omissions. Chapters 7–9 plus
summaries of Chapters 10, 11 and 13 were used for a subsequent graduate
course. Moreover, Chapters 12–14, together with excursions into Chapters 4
and 5 and supplements on parabolic equations, have been used for a graduate
course.
The bibliography exposes the many sources that were consulted while the
material was collected. It is not meant to be a complete literature list of the
available works in this area.
It is my hope that the text is relatively free of errors, but I will be interested
to be informed if readers find something to correct; then I may possibly set

up a homepage with the information.
Besides drawing on many books and papers, as referred to in the text, I
have benefited from the work of colleagues in Denmark, reading their earlier
notes for related courses, and getting their advice on my courses. My thanks
go especially to Esben Kehlet, and to Jon Johnsen, Henrik Schlichtkrull,
Mogens Flensted-Jensen and Christian Berg. I also thank all those who have
helped me improve the text while participating in the courses as graduate
students through the years. Moreover, my thanks go to Mads Haar and Jan
Caesar for creating the figures, and to Jan Caesar for his invaluable help in
adapting the manuscript to Springer’s style.
Copenhagen, June 2008 Gerd Grubb
Contents
Preface v
Part I Distributions and derivatives
1 Motivation and overview 3
1.1 Introduction 3
1.2 Onthedefinition ofdistributions 6
2 Function spaces and approximation 9
2.1 Thespaceoftestfunctions 9
2.2 Someotherfunction spaces 16
2.3 Approximationtheorems 17
2.4 Partitionsofunity 23
ExercisesforChapter2 24
3 Distributions. Examples and rules of calculus 27
3.1 Distributions 27
3.2 Rulesofcalculusfordistributions 31
3.3 Distributionswith compactsupport 36
3.4 Convolutionsandcoordinatechanges 40
3.5 The calculation rules and the weak
∗

topology on D

46
ExercisesforChapter3 50
Part II Extensions and applications
4 Realizations and Sobolev spaces 57
4.1 Realizationsofdifferentialoperators 57
4.2 Sobolevspaces 61
4.3 Theone-dimensionalcase 68
4.4 Boundary value problems in higher dimensions . . . . . . . . . . . . . 77
ExercisesforChapter4 86
ix
x Contents
5 Fourier transformation of distributions 95
5.1 Rapidlydecreasingfunctions 95
5.2 Temperatedistributions 103
5.3 The Fourier transform on S

105
5.4 Homogeneity 110
5.5 Application totheLaplaceoperator 114
5.6 Distributions associated with nonintegrable functions . . . . . . . 116
ExercisesforChapter5 120
6 Applications to differential operators. The Sobolev theorem123
6.1 Differential and pseudodifferential operators on R
n
123
6.2 Sobolev spaces of arbitrary real order. The Sobolev theorem . 127
6.3 Dualities between Sobolev spaces. The Structure theorem . . . . 132
6.4 Regularity theory for elliptic differential equations . . . . . . . . . . 138

ExercisesforChapter6 144
Miscellaneousexercises(exam problems) 147
Part III Pseudodifferential operators
7 Pseudodifferential operators on open sets 163
7.1 Symbolsandoperators,mappingproperties 163
7.2 Negligibleoperators 172
7.3 Compositionrules 178
7.4 Elliptic pseudodifferential operators . . . . . . . . . . . . . . . . . . . . . . . 183
7.5 Strongly elliptic operators, the G˚arding inequality . . . . . . . . . . 189
ExercisesforChapter7 193
8 Pseudodifferential operators on manifolds, index of
elliptic operators 195
8.1 Coordinate changes 195
8.2 Operatorsonmanifolds 197
8.3 Fredholmtheory,the index 208
ExercisesforChapter8 215
Part IV Boundary value problems
9 Boundary value problems in a constant-coefficient case . . . 219
9.1 Boundary maps for the half-space . . . . . . . . . . . . . . . . . . . . . . . . 219
9.2 The Dirichlet problem for I − Δonthehalf-space 228
9.3 The Neumann problem for I − Δ on the half-space . . . . . . . . . 234
9.4 Other realizations of I − Δ 236
9.5 Variable-coefficient cases, higherorders,systems 246
ExercisesforChapter9 248
Contents xi
10 Pseudodifferential boundary operators 251
10.1 Therealformulation 251
10.2 Fourier transform and Laguerre expansion of S
+
266

10.3 Thecomplexformulation 275
10.4 Compositionrules 281
10.5 Continuity 289
10.6 Elliptic ψdbo’s 296
ExercisesforChapter10 300
11 Pseudodifferential methods for boundary value problems . 305
11.1 The Calder´onprojector 305
11.2 Application to boundary value problems . . . . . . . . . . . . . . . . . . . 317
11.3 The solution operator for the Dirichlet problem . . . . . . . . . . . . 320
ExercisesforChapter11 326
Part V Topics on Hilbert space operators
12 Unbounded linear operators 337
12.1 Unbounded operators in Banach spaces . . . . . . . . . . . . . . . . . . . 337
12.2 Unbounded operators in Hilbert spaces . . . . . . . . . . . . . . . . . . . . 340
12.3 Symmetric, selfadjoint and semibounded operators . . . . . . . . . . 343
12.4 Operators associated with sesquilinear forms . . . . . . . . . . . . . . . 350
12.5 TheFriedrichsextension 359
12.6 Moreonvariationaloperators 361
ExercisesforChapter12 364
13 Families of extensions 373
13.1 A universalparametrizationofextensions 373
13.2 Thesymmetriccase 383
13.3 Consequencesforresolvents 393
13.4 Boundary triplets and M-functions 397
ExercisesforChapter13 402
14 Semigroups of operators 405
14.1 Evolutionequations 405
14.2 Contraction semigroups in Banach spaces . . . . . . . . . . . . . . . . . . 409
14.3 ContractionsemigroupsinHilbert spaces 416
14.4 Applications 419

ExercisesforChapter14 421
A Some notation and prerequisites 423
ExercisesforAppendix A 429
xii Contents
B Topological vector spaces 431
B.1 Fr´echetspaces 431
B.2 Inductive limits of Fr´echetspaces 441
ExercisesforAppendix B 443
C Function spaces on sets with smooth boundary 447
ExercisesforAppendix C 450
Bibliography 451
Index 457
Part I
Distributions and derivatives
Chapter 1
Motivation and overview
1.1 Introduction
In the study of ordinary differential equations one can get very far by using
just the classical concept of differentiability, working with spaces of continu-
ously differentiable functions on an interval I ⊂ R:
C
m
(I)={u : I → C |
d
j
dx
j
u exists and is continuous on I for 0 ≤ j ≤ m }.
(1.1)
The need for more general concepts comes up for example in the study

of eigenvalue problems for second-order operators on an interval [a, b]with
boundary conditions at the endpoints a, b, by Hilbert space methods. But
here it usually suffices to extend the notions to absolutely continuous func-
tions, i.e., functions u(x)oftheform
u(x)=

x
x
0
v(y) dy + c, v locally integrable on I. (1.2)
Here c denotes a constant, and “locally integrable” means integrable on com-
pact subsets of I. The function v is regarded as the derivative
d
dx
u of u,and
the fundamental formula
u(x)=u(x
0
)+

x
x
0
d
dy
u(y) dy (1.3)
still holds.
But for partial differential equations one finds when using methods from
functional analysis that the spaces C
m

are inadequate, and there is no good
concept of absolute continuity in the case of functions of several real variables.
One can get some ways by using the concept of weak derivatives:Whenu
and v are locally integrable on an open subset Ω of R
n
,wesaythatv =
∂
∂x
j
u
in the weak sense, when
3
4 1 Motivation and overview
−

Ω
u
∂
∂x
j
ϕdx=

Ω
vϕ dx, for all ϕ ∈ C
∞
0
(Ω); (1.4)
here C
∞
0

(Ω) denotes the space of C
∞
functions on Ω with compact support
in Ω. (The support supp f of a function f is the complement of the largest
open set where the function is zero.) This criterion is modeled after the fact
that the formula (1.4) holds when u ∈ C
1
(Ω), with v =
∂
∂x
j
u.
Sometimes even the concept of weak derivatives is not sufficient, and the
need arises to define derivatives that are not functions, but are more general
objects. Some measures and derivatives of measures will enter. For example,
thereistheDiracmeasureδ
0
that assigns 1 to every Lebesgue measurable set
in R
n
containing {0}, and 0 to any Lebesgue measurable set not containing
{0}.Forn =1,δ
0
is the derivative of the Heaviside function defined in (1.8)
below. In the book of Laurent Schwartz [S61] there is also a description of
the derivative of δ
0
(on R) — which is not even a measure — as a “dipole”,
with some kind of physical explanation.
For the purpose of setting up the rules for a general theory of differention

where classical differentiability fails, Schwartz brought forward around 1950
the concept of distributions: a class of objects containing the locally integrable
functions and allowing differentiations of any order.
This book gives an introduction to distribution theory, based on the work
of Schwartz and of many other people. Our aim is also to show how the
theory is combined with the study of operators in Hilbert space by methods
of functional analysis, with applications to ordinary and partial differential
equations. In some chapters of a more advanced character, we show how the
distribution theory is used to define pseudodifferential operators and how
they are applied in the discussion of solvability of PDE, with or without
boundary conditions. A bibliography of relevant books and papers is collected
at the end.
Plan
Part I gives an introduction to distributions.
In the rest of Chapter 1 we begin the discussion of taking derivatives in the
distribution sense, motivating the study of function spaces in the following
chapter.
Notation and prerequisites are collected in Appendix A.
Chapter 2 studies the spaces of C
∞
-functions (and C
k
-functions) needed
in the theory, and their relations to L
p
-spaces.
The relevant topological considerations are collected in Appendix B.
In Chapter 3 we introduce distributions in full generality and show the
most prominent rules of calculus for them.
1.1 Introduction 5

Part II connects the distribution concept with differential equations and
Fourier transformation.
Chapter 4 is aimed at linking distribution theory to the treatment of par-
tial differential equations (PDE) by Hilbert space methods. Here we intro-
duce Sobolev spaces and realizations of differential operators, both in the
(relatively simple) one-dimensional case and in n-space, and study some ap-
plications.
Here we use some of the basic results on unbounded operators in Hilbert
space that are collected in Chapter 12.
In Chapter 5, we study the Fourier transformation in the framework of
temperate distributions.
Chapter 6 gives a further development of Sobolev spaces as well as appli-
cations to PDE by use of Fourier theory, and shows a fundamental result on
the structure of distributions.
Part III introduces a more advanced tool, namely, pseudodifferential opera-
tors (ψdo’s), a generalization of partial differential operators containing also
the solution operators for elliptic problems.
Chapter 7 gives the basic ingredients of the local calculus of pseudodiffer-
ential operators. Applications include a proof of the G˚arding inequality.
Chapter 8 shows how to define ψdo’s on manifolds, and how they in
the elliptic case define Fredholm operators, with solvability properties mod-
ulo finite-dimensional spaces. (An introduction to Fredholm operators is in-
cluded.)
Part IV treats boundary value problems.
Chapter 9 (independent of Chapter 7 and 8) takes up the study of bound-
ary value problems by use of Fourier transformation. The main effort is spent
on an important constant-coefficient case which, as an example, shows how
Sobolev spaces of noninteger and negative order can enter. Also, a connec-
tion is made to the abstract theory of Chapter 13. This chapter can be read
directly after Parts I and II.

In Chapter 10 we present the basic ingredients in a pseudodifferential
theory of boundary value problems introduced originally by L. Boutet de
Monvel; this builds on the methods of Chapters 7 and 8 and the example in
Chapter 9, introducing new operator types.
Chapter 11 shows how the theory of Chapter 10 can be used to discuss
solvability of elliptic boundary value problems, by use of the Calder´on projec-
tor, that we construct in detail. As a special example, regularity of solutions
of the Dirichlet problem is shown. Some other boundary value problems are
taken up in the exercises.
Part V gives the supplementing topics needed from Hilbert space theory.
Chapter 12, departing from the knowledge of bounded linear operators
in Hilbert spaces, shows some basic results for unbounded operators, and
develops the theory of variational operators.
6 1 Motivation and overview
Chapter 13 gives a systematic presentation of closed extensions of adjoint
pairs, with consequences for symmetric and semibounded operators; this is of
interest for the study of boundary value problems for elliptic PDE and their
positivity properties. We moreover include a recent development concerning
resolvents, their M -functions and Kre˘ın formulas.
Chapter 14 establishes some basic results on semigroups of operators, rel-
evant for parabolic PDE (problems with a time parameter), and appealing
to positivity and variationality properties discussed in earlier chapters.
Finally, there are three appendices. In Appendix A, we recall some basic rules
of calculus and set up the notation.
Appendix B gives some elements of the theory of topological vector spaces,
that can be invoked when one wants the correct topological formulation of
the properties of distributions.
Appendix C introduces some function spaces, as a continuation of Chapter
2, but needed only later in the text.
1.2 On the definition of distributions

The definition of a weak derivative ∂
j
u was mentioned in (1.4) above. Here
both u and its weak derivative v are locally integrable functions on Ω. Observe
that the right-hand side is a linear functional on C
∞
0
(Ω), i.e., a linear mapping
Λ
v
of C
∞
0
(Ω) into C, here defined by
Λ
v
: ϕ → Λ
v
(ϕ)=

Ω
vϕ dx. (1.5)
The idea of Distribution Theory is to allow much more general functionals
than this one. In fact, when Λ is any linear functional on C
∞
0
(Ω) such that
−

Ω

u∂
j
ϕdx=Λ(ϕ) for all ϕ ∈ C
∞
0
(Ω), (1.6)
we shall say that
∂
j
u = Λ in the distribution sense, (1.7)
even if there is no function v (locally integrable) such that Λ can be defined
from it as in (1.5).
Example 1.1. Here is the most famous example in the theory: Let Ω = R
and consider the Heaviside function H(x); it is defined by
H(x)=

1forx>0,
0forx ≤ 0.
(1.8)
1.2 On the definition of distributions 7
It is locally integrable on R. But there is no locally integrable function v such
that (1.4) holds with u = H:
−

R
H
d
dx
ϕdx=


vϕ dx, for all ϕ ∈ C
∞
0
(R). (1.9)
For, assume that v were such a function, and let ϕ ∈ C
∞
0
(R)withϕ(0) = 1
and set ϕ
N
(x)=ϕ(Nx). Note that max |ϕ(x)| =max|ϕ
N
(x)| for all N ,and
that when ϕ issupportedin[−R, R], ϕ
N
issupportedin[−R/N, R/N]. Thus
by the theorem of Lebesgue,

R
vϕ
N
dx → 0forN →∞, (1.10)
but on the other hand,
−

R
H
d
dx
ϕ

N
dx = −

∞
0
Nϕ

(Nx) dx = −

∞
0
ϕ

(y) dy = ϕ(0) = 1. (1.11)
So (1.9) cannot hold for this sequence of functions ϕ
N
, and we conclude that
a locally integrable function v for which (1.9) holds for all ϕ ∈ C
∞
0
(R) cannot
exist.
A linear functional that does match H in a formula (1.6) is the following
one:
Λ:ϕ → ϕ(0) (1.12)
(as seen by a calculation as in (1.11)). This is the famous delta-distribution,
usually denoted δ
0
. (It identifies with the Dirac measure mentioned earlier.)
There are some technical things that have to be cleared up before we can

define distributions in a proper way.
For one thing, we have to look more carefully at the elements of C
∞
0
(Ω).
We must demonstrate that such functions really do exist, and we need to
show that there are elements with convenient properties (such as having the
support in a prescribed set and being 1 on a smaller prescribed set).
Moreover, we have to describe what is meant by convergence in C
∞
0
(Ω),
in terms of a suitable topology. There are also some other spaces of C
∞
or
C
k
functions with suitable support or integrability properties that we need
to introduce.
These preparatory steps will take some time, before we begin to introduce
distributions in full generality. (The theories that go into giving C
∞
0
(Ω) a
good topology are quite advanced, and will partly be relegated to Appendix
B. In fact, the urge to do this in all details has been something of an obsta-
cle to making the tool of distributions available to everybody working with
PDE — so we shall here take the point of view of giving full details of how
one operates with distributions, but tone down the topological discussion to
some statements one can use without necessarily checking all proofs.)

The reader is urged to consult Appendix A (with notation and prerequi-
sites) before starting to read the next chapters.
Chapter 2
Function spaces and approximation
2.1 The space of test functions
Notation and prerequisites are collected in Appendix A.
Let Ω be an open subset of R
n
.ThespaceC
∞
0
(Ω), consisting of the C
∞
-
functions on Ω with compact support in Ω, is called the space of test functions
(on Ω). The support supp u of a function u ∈ L
1,loc
(Ω) is defined as the
complement of the largest open set where u vanishes; we can write it as
supp u =Ω\


{ω open in Ω | u|
ω
=0}

. (2.1)
We show first of all that there exist test functions:
Lemma 2.1. 1
◦

Let R>r>0. There is a function χ
r,R
(x) ∈ C
∞
0
(R
n
) with
the properties: χ
r,R
(x)=1for |x|≤r, χ
r,R
(x) ∈ [0, 1] for r ≤|x|≤R,
χ
r,R
(x)=0for |x|≥R.
2
◦
There is a function h ∈ C
∞
0
(R
n
) satisfying:
supp h = B
(0, 1),h(x) > 0 for |x| < 1,

h(x) dx =1. (2.2)
Proof. 1
◦

. The function
f(t)=

e
−1/t
for t>0,
0fort ≤ 0,
is a C
∞
-function on R.Fort = 0 this is obvious. At the point t =0wehave
that f(t) → 0fort  0, and that the derivatives of f(t)fort =0areofthe
form
∂
k
t
f(t)=

p
k
(1/t)e
−1/t
for t>0,
0fort<0,
9
10 2 Function spaces and approximation
for certain polynomials p
k
, k ∈ N
0
. Since any polynomial p satisfies

p(1/t)e
−1/t
→ 0fort  0, f and its derivatives are differentiable at 0.
From f we construct the functions (see the figure)
f
1
(t)=f(t −r)f(R − t) ,f
2
(t)=

∞
t
f
1
(s) ds .
t
f
t
r
R
f
1
t
r
R
f
2
Here we see that f
2
(x) ≥ 0 for all x,equals0fort ≥ R and equals

C =

R
r
f
1
(s) ds > 0
for t ≤ r. The function
χ
r,R
(x)=
1
C
f
2
(|x|) ,x∈ R
n
,
then has the desired properties.
2
◦
. Here one can for example take
h(x)=
χ
1
2
,1
(x)

χ

1
2
,1
(x) dx
.

Note that analytic functions (functions defined by a converging Taylor
expansion) cannot be in C
∞
0
(R) without being identically zero! So we have
to go outside the elementary functions (such as cos t, e
t
, e
−t
2
, etc.) to find
nontrivial C
∞
0
-functions. The construction in Lemma 2.1 can be viewed from
a “plumber’s point of view”: We want a C
∞
-function that is 0 on a certain
interval and takes a certain positive value on another; we can get it by twisting
the graph suitably. But analyticity is lost then.
For later reference we shall from now on denote by χ a function in C
∞
0
(R

n
)
satisfying
2.1 The space of test functions 11
χ(x)
⎧
⎪
⎨
⎪
⎩
=1 for |x|≤1 ,
∈ [0, 1] for 1 ≤|x|≤2 ,
=0 for |x|≥2 ,
(2.3)
one can for example take χ
1,2
constructed in Lemma 2.1. A C
∞
0
-function that
is 1 on a given set and vanishes outside some larger given set is often called a
cut-off function. Of course we get some other cut-off functions by translating
the functions χ
r,R
around. More refined examples will be constructed later
by convolution, see e.g. Theorem 2.13. These functions are all examples of
test functions, when their support is compact.
We use throughout the following convention (of “extension by zero”) for
test functions: If ϕ ∈ C
∞

0
(Ω), Ω open ⊂ R
n
, we also denote the function
obtained by extending by zero on R
n
\ Ωbyϕ;itisinC
∞
0
(R
n
). When
ϕ ∈ C
∞
0
(R
n
) and its support is contained in Ω, we can regard it as an
element of C
∞
0
(Ω) and again denote it ϕ. Similarly, we can view a C
∞
-
function ϕ with compact support in Ω ∩ Ω

(Ω and Ω

open) as an element
of C

∞
0
(Ω) or C
∞
0
(Ω

), whatever is convenient.
Before we describe the topology of the space C
∞
0
(Ω) we recall how some
other useful spaces are topologized. The reader can find the necessary infor-
mation on topological vector spaces in Appendix B and its problem session.
When we consider an open subset Ω of R
n
, the compact subsets play an
important role.
Lemma 2.2. Let Ω be a nonempty open subset of R
n
.Thereexistsasequence
of compact subsets (K
j
)
j∈N
such that
K
1
⊂ K
◦

2
⊂ K
2
⊂···⊂K
◦
j
⊂ K
j
⊂ ,

j∈N
K
◦
j
=Ω. (2.4)
Ω
0
K
j
1
j
j
Proof. We can for example take
K
j
=

x ∈ Ω ||x|≤j and dist (x, Ω) ≥
1
j


; (2.5)
12 2 Function spaces and approximation
the interior of this set is defined by the formula with ≤ and ≥ replaced by <
and >.(IfΩ=∅, the condition dist (x, Ω) ≥
1
j
is left out.) If necessary, we
can omit the first, at most finitely many, sets with K
◦
j
= ∅ and modify the
indexation. 
When K is a compact subset of Ω, it is covered by the system of open sets
{K
◦
j
}
j∈N
and hence by a finite subsystem, say with j ≤ j
0
.ThenK ⊂ K
j
for
all j ≥ j
0
.
Recall that when [a, b] is a compact interval of R, C
k
([a, b]) (in one of

the versions C
k
([a, b], C)orC
k
([a, b], R)) is defined as the Banach space of
complex resp. real functions having continuous derivatives up to order k,
provided with a norm
f

C
k
=

0≤j≤k
sup
x
|f
(j)
(x)|, or the equivalent norm
f
C
k =sup{|f
(j)
(x)||x ∈ [a, b], 0 ≤ j ≤ k } .
(2.6)
In the proof that these normed spaces are complete one uses the well-known
theorem that when f
l
is a sequence of C
1

-functions such that f
l
and f

l
converge uniformly to f resp. g for l →∞,thenf is C
1
with derivative
f

= g. There is a similar result for functions of several variables:
Lemma 2.3. Let J =[a
1
,b
1
]×···×[a
n
,b
n
] be a closed box in R
n
and let f
l
be
a sequence of functions in C
1
(J) such that f
l
→ f and ∂
j

f
l
→ g
j
uniformly
on J for j =1, ,n.Thenf ∈ C
1
(J) with ∂
j
f = g
j
for each j.
Proof. For each j we use the above-mentioned theorem in situations where
all but one coordinate are fixed. This shows that f has continuous partial
derivatives ∂
j
f = g
j
at each point of J. 
So C
k
(J) is a Banach space with the norm
u
C
k
(J)
=sup{|∂
α
u(x)||x ∈ J, |α|≤k }. (2.7)
We define

C
∞
(J)=

k∈N
0
C
k
(J). (2.8)
This is no longer a Banach space, but can be shown to be a Fr´echet space
with the family of (semi)norms p
k
(f)=f
C
k
(J)
, k ∈ N
0
,byarguments
as in Lemma 2.4 below. (For details on Fr´echet spaces, see Appendix B, in
particular Theorem B.9.)
For spaces of differentiable functions over open sets, the full sup-norms are
unsatisfactory since the functions and their derivatives need not be bounded.
We here use sup-norms over compact subsets to define a Fr´echet topology.
Let Ω be open and let K
j
be an increasing sequence of compact subsets as
in Lemma 2.2. Define the system of seminorms
2.1 The space of test functions 13
p

k,j
(f)=sup{|∂
α
f(x)|||α|≤k, x∈ K
j
}, for j ∈ N. (2.9)
Lemma 2.4. 1
◦
For each k ∈ N
0
, C
k
(Ω) is a Fr´echet space when provided
with the family of seminorms {p
k,j
}
j∈N
.
2
◦
The space C
∞
(Ω) =

k∈N
0
C
k
(Ω) is a Fr´echet space when provided
with the family of seminorms {p

k,j
}
k∈N
0
,j∈N
.
Proof. 1
◦
. The family {p
k,j
}
j∈N
is separating, for when f ∈ C
k
(Ω) is =0,
then there is a point x
0
where f(x
0
) =0,andx
0
∈ K
j
for j sufficiently large;
for such j, p
k,j
(f) > 0. The seminorms then define a translation invariant
metric by Theorem B.9. We just have to show that the space is complete
under this metric. Here we use Lemma 2.3: Let (f
l

)
l∈N
be a Cauchy sequence
in C
k
(Ω). Let x
0
= {x
01
, ,x
0n
}∈Ω and consider a box J = J
x
0
,δ
= {x |
|x
m
−x
0m
|≤δ, m =1, ,n} around x
0
,withδ taken so small that J ⊂ Ω.
Since J ⊂ K
j
0
for a certain j
0
, the Cauchy sequence property implies that f
l

defines a Cauchy sequence in C
k
(J), i.e., f
l
and its derivatives up to order k
are Cauchy sequences with respect to uniform convergence on J.Sothereis
a limit f
J
in C
k
(J). We use similar arguments for other boxes J

in Ω and
find that the limits f
J

and f
J
are the same on the overlap of J and J

.In
this way we can define a C
k
-function f that is the limit of the sequence in
C
k
(J) for all boxes J ⊂ Ω. Finally, p
j,k
(f
l

− f ) → 0 for all j, since each K
j
can be covered by a finite number of box-interiors J
◦
.Thenf
l
has the limit
f in the Fr´echet topology of C
k
(Ω).
2
◦
. The proof in this case is a variant of the preceding proof, where we
now investigate p
j,k
for all k also. 
The family (2.9) has the max-property (see Remark B.6), so the sets
V (p
k,j
,ε)={f ∈ C
∞
(Ω) ||∂
α
f(x)| <ε for |α|≤k, x ∈ K
j
} (2.10)
constitute a local basis for the system of neighborhoods at 0. One could in
fact make do with the sequence of seminorms {p
k,k
}

k∈N
, which increase with
k.
For any compact subset K of Ω we define
C
∞
K
(Ω) = {u ∈ C
∞
(Ω) | supp u ⊂ K } , (2.11)
the space of C
∞
-functions with support in K (cf. (2.1)); this space is provided
with the topology inherited from C
∞
(Ω).
The space C
∞
K
(Ω) is a closed subspace of C
∞
(Ω) (so it is a Fr´echet space).
The topology is for example defined by the family of seminorms {p
k,j
0
}
k∈N
0
(cf. (2.9)) with j
0

taken so large that K ⊂ K
j
0
. This family has the max-
property.
In the theory of distributions we need not only the Fr´echet spaces C
∞
(Ω)
and C
∞
K
(Ω) but also the space
C
∞
0
(Ω) = {ϕ ∈ C
∞
(Ω) | supp ϕ is compact in Ω } . (2.12)
14 2 Function spaces and approximation
As already mentioned, it is called the space of test functions, and it is also
denoted D(Ω).
If we provide this space with the topology inherited from C
∞
(Ω), we get
an incomplete metric space. For example, if Ω is the interval I =]0, 3[ and
ϕ(x)isaC
∞
-function on I with supp ϕ =[1, 2], then ϕ
l
(x)=ϕ(x − 1+

1
l
),
l ∈ N, is a sequence of functions in C
∞
0
(I) which converges in C
∞
(I)tothe
function ϕ(x − 1) ∈ C
∞
(I) \C
∞
0
(I).
We prefer to provide C
∞
0
(Ω) with a stronger and somewhat more compli-
cated vector space topology that makes it a sequentially complete (but not
metric) space. More precisely, we regard C
∞
0
(Ω) as
C
∞
0
(Ω) =
∞


j=1
C
∞
K
j
(Ω) , (2.13)
where K
j
is an increasing sequence of compact subsets as in (2.4) and the
topology is the inductive limit topology, cf. Theorem B.17 (also called the LF-
topology). The spaces C
∞
K
j
(Ω) are provided with Fr´echet space topologies by
families of seminorms (2.9).
The properties of this space that we shall need are summed up in the fol-
lowing theorem, which just specifies the general properties given in Appendix
B (Theorem B.18 and Corollary B.19):
Theorem 2.5. The topology on C
∞
0
(Ω) has the following properties:
(a) Asequence(ϕ
l
)
l∈N
of test functions converges to ϕ
0
in C

∞
0
(Ω) if and
only if there is a j ∈ N such that supp ϕ
l
⊂ K
j
for all l ∈ N
0
,and
ϕ
l
→ ϕ
0
in C
∞
K
j
(Ω):
sup
x∈K
j
|∂
α
ϕ
l
(x) − ∂
α
ϕ
0

(x)|→0 for l →∞, (2.14)
for all α ∈ N
n
0
.
(b) AsetE ⊂ C
∞
0
(Ω) is bounded if and only if there exists a j ∈ N such that
E is a bounded subset of C
∞
K
j
(Ω).Inparticular,if(ϕ
l
)
l∈N
is a Cauchy
sequence in C
∞
0
(Ω), then there is a j such that supp ϕ
l
⊂ K
j
for all l,
and ϕ
l
is convergent in C
∞

K
j
(Ω) (and then also in C
∞
0
(Ω)).
(c) Let Y be a locally convex topological vector space. A mapping T from
C
∞
0
(Ω) to Y is continuous if and only if T : C
∞
K
j
(Ω) → Y is continuous
for each j ∈ N.
(d) A linear functional Λ:C
∞
0
(Ω) → C is continuous if and only if there is
an N
j
∈ N
0
and a c
j
> 0 for any j ∈ N, such that
|Λ(ϕ)|≤c
j
sup{|∂

α
ϕ(x)||x ∈ K
j
, |α|≤N
j
} (2.15)
for all ϕ ∈ C
∞
K
j
(Ω).
2.1 The space of test functions 15
Note that (a) is a very strong assumption on the sequence ϕ
l
. Convergence
in C
∞
0
(Ω) implies convergence in practically all the other spaces we shall
meet. On the other hand, (d) is a very mild assumption on the functional Λ;
practically all the functionals that we shall meet will have this property.
We underline that a sequence can only be a Cauchy sequence when there
is a j such that all functions in the sequence have support in K
j
and the
sequence is Cauchy in C
∞
K
j
(Ω). Then the sequence converges because of the

completeness of the Fr´echet space C
∞
K
j
(Ω). In the example mentioned above,
the sequence ϕ
l
(x)=ϕ(x − 1+
1
l
)inC
∞
0
(]0, 3 [) is clearly not a Cauchy
sequence with respect to this topology on C
∞
0
(]0, 3[).
It is not hard to show that when (K

j
)
j∈N
is another sequence of compact
subsets as in (2.4), the topology on C
∞
0
(Ω) defined by use of this sequence is
the same as that based on the first one (Exercise 2.2).
We now consider two important operators on these spaces. One is differen-

tiation, the other is multiplication (by a C
∞
-function f); both will be shown
to be continuous. The operators are denoted ∂
α
(with D
α
=(−i)
|α|
∂
α
)resp.
M
f
.WealsowriteM
f
ϕ as fϕ — and the same notation will be used later
for generalizations of these operators.
Theorem 2.6. 1
◦
The mapping ∂
α
: ϕ → ∂
α
ϕ is a continuous linear opera-
tor in C
∞
0
(Ω). The same holds for D
α

.
2
◦
For any f ∈ C
∞
(Ω), the mapping M
f
: ϕ → fϕ is a continuous linear
operator in C
∞
0
(Ω).
Proof. Clearly, ∂
α
and M
f
are linear operators from C
∞
0
(Ω) to itself. As for
the continuity it suffices, according to 2.5 (c), to show that ∂
α
resp. M
f
is
continuous from C
∞
K
j
(Ω) to C

∞
0
(Ω) for each j. Since the operators satisfy
supp ∂
α
ϕ ⊂ supp ϕ, supp M
f
ϕ ⊂ supp ϕ, (2.16)
for all ϕ, the range space can for each j be replaced by C
∞
K
j
(Ω). Here we have
for each k:
p
k,j
(∂
α
ϕ)=sup{|∂
β
∂
α
ϕ|||β|≤k, x∈ K
j
}
≤ sup{|∂
γ
ϕ|||γ|≤k + |α|,x∈ K
j
} = p

k+|α|,j
(ϕ) ,
(2.17)
which shows the continuity of ∂
α
. The result extends immediately to D
α
.By
the Leibniz rule (A.7) we have for each k:
p
k,j
(fϕ)=sup{|∂
α
(fϕ)|||α|≤k, x∈ K
j
} (2.18)
≤ sup{

β≤α
|c
α,β
∂
β
f∂
α−β
ϕ|||α|≤k, x∈ K
j
}
≤ C
k

p
k,j
(f)p
k,j
(ϕ),
for a suitably large constant C
k
; this shows the continuity of M
f
. 