nonlinear superposition operators

239 Pages·1990·1.604 MB·English

by  Appell J. Zabrejko, P.P

JU¨RGEN APPELL
Universita¨t Wu¨rzburg, Fakulta´t fu´r Mathematik, Am Hubland,
D-8700 Wu¨rzburg, WEST GERMANY
and
PETR P. ZABRE˘IKO
Belgosuniversitet, Matematicheski˘ı Fakul’tet, Pl. Lenina 4,
SU-220080 Minsk, SOVIET UNION
NONLINEAR SUPERPOSITION OPERATORS
1989
CONTENTS
Preface 1
Chapter 1 The superposition operator in the space S 1
1.1. The space S 1
1.2. The superposition operator 1
1.3. Sup-measurable functions 1
1.4. Caratheodory and Shragin functions 1
1.5. Boundedness conditions 1
1.6. Continuity conditions 1
1.7. Compactness conditions 1
1.8. Special classes of functions 1
1.9. Notes, remarks and references 1
Chapter 2 The superposition operator in ideal spaces 1
2.1. Ideal spaces 1
2.2. The domain of definition of the superposition operator 1
2.3. Local and global boundedness conditions 1
2.4. Special boundedness properties 1
2.5. Continuity conditions 1
2.6. Lipschitz and Darbo conditions 1
2.7. Differentiability conditions 1
2.8. Higher derivatives and analyticity 1
2.9. Notes, remarks and references 1
Chapter 3 The superposition operator in Lebesgue spaces 1
3.1. Lebesgue spaces 1
3.2. Acting conditions 1
3.3. The growth function 1
3.4. Absolute boundedness and uniform continuity 1
3.5. Lipschitz and Darbo conditions 1
3.6. Differentiability conditions and analyticity 1
3.7. The case p = or q = 1
∞ ∞
3.8. The -characteristic 1
L
3.9. Notes, remarks and references 1
Chapter 4 The superposition operator in Orlicz spaces 1
4.1. Orlicz spaces 1
4.2. Acting conditions 1
4.3. Boundedness conditions 1
4.4. Continuity conditions 1
4.5. Lipschitz and Darbo conditions 1
4.6. Differentiability conditions and analyticity 1
4.7. Notes, remarks and references 1
I
Chapter 5 The superposition operator in symmetric spaces 1
5.1. Symmetric spaces 1
5.2. Lorentz and Marcinkiewicz spaces 1
5.3. Acting conditions in symmetric spaces 1
5.4. Some properties in symmetric spaces 1
5.5. Notes, remarks and references 1
Chapter 6 The superposition operator in the spaces C and BV 1
6.1. The space C 1
6.2. Some properties of Caratheodory functions 1
6.3. The superposition operator in the space C 1
6.4. The superposition operator between C and S 1
6.5. The superposition operator in the space BV 1
6.6. Notes, remarks and references 1
Chapter 7 The superposition operator in Ho¨lder spaces 1
7.1. Ho¨Ider spaces 1
7.2. Acting conditions 1
7.3. Boundedness conditions 1
7.4. Continuity conditions 1
7.5. Lipschitz and Darbo conditions 1
7.6. Differentiability conditions 1
7.7. The superposition operator in the space 1
φ,p
J
7.8. Notes, remarks and references 1
Chapter 8 The superposition operator in spaces of smooth functions 1
8.1. The spaces Ck and Hk 1
φ
8.2. The superposition operator in the space Ck 1
8.3. The superposition operator in the space Hk 1
φ
8.4. The superposition operator in the space R (L) 1
µ
8.5. The superposition operator in Roumieu classes 1
8.6. Notes, remarks and references 1
Chapter 9 The superposition operator in Sobolev spaces 1
9.1. Sobolev spaces 1
9.2. Sufficient acting conditions in W1 1
p
9.3. Necessary acting conditions in W1 1
p
9.4. Boundedness and continuity conditions in W1 1
p
9.5. Boundedness and continuity conditions in Wk 1
p
9.6. Degeneracy results 1
9.7. The superposition operator in Sobolev-Orlicz spaces 1
9.8. Notes, remarks and references 1
Bibliography 1
List of Symbols 1
Subject Index 1
II
Preface
The present monograph is concerned with a thorough study of the nonlinear
operator
Fx(s) = f(s,x(s)). (1)
Here f = f(s,u) is a given function which is defined on the Cartesian product of
some set Ω, which in most cases is either a metric space or a measure space or both,
with the set R of real or the set C of complex numbers, and takes values in R or C,
respectively. By definition, the operator F associates to each real (or complex) function
x(s) on Ω the real (or complex) function f(s,x(s)) on Ω; therefore F is usually called a
superposition operator (sometimes also composition operator, substitution operator, or
Nemytskij operator).
In an implicit form, the superposition operator (1) can be found in the first pages
of any calculus textbook (in the old terminology, as "composite function"function of a
functionetc.), where some of its elementary properties are described. Typical examples
of such properties are the continuity of the superposition of continuous functions, the
differentiability of the superposition of differentiable functions, and similar statements.
Many other results of this type are scattered, mostly as lemmas or auxiliary results, in
a vast literature on mathematical analysis, functional analysis, differential and integral
equations, probability theory and statistics, variational calculus, optimization theory, and
otherfields of contemporarymathematics – thesuperpositionoperatoroccurs everywhere.
In many situations, the investigation of the basic properties of the operator (1)
is quite straightforward and does not involve any particular difficulties. But this is not
always so. In fact, at the beginning of nonlinear analysis it was often tacitly assumed
that "nice"properties of the function f carry over to the corresponding operator F; this
turned out to be false even in well-known classical function spaces. A typical example of
this phenomenon is the behaviour of the superposition operator in Lebesgue spaces. For
instance, the smoothness (and even the analyticity) of the function f does not imply the
smoothness of F, considered as an operator between two Lebesgue spaces. Moreover, just
the fact that F acts from the Lebesgue space L into the Lebesgue space L , say, leads
p q
to the very restrictive growth condition f(s,u) = O( u p/q). Further, if F is (Frechet-)
| |
differentiable between L and L , and the partial derivative f′ of f with respect to u
p q u
exists, then necessarily f′(s,u) = O( u (p−q)/q) if p q, and f′ (s.u) 0 if p < q. Finally,
| | ≥ u ≡
if F is analytic between L and L , then the function f reduces to a polynomial in u (of
p q
degree at most p/q). All these facts are rather surprising; they show that many of the
important properties of the function f do not imply analogous properties of the operator
F, or vice versa.
III
Classical mathematical analysismainly dealt withspaces ofcontinuous ordifferenti-
able functions] already Lebesgue spaces arose only in special fields, e.g. Fourier series,
approximation theory, probability theory. In modern nonlinear analysis, however, the
arsenalofavailablefunctionspaces hasbeenconsiderably enlarged.Inthisconnection, one
should mention Sobolev spaces and their generalizations which are simply indispensable
for the study of partial differential equations, Orlicz spaces which are the natural tool
in the theory of both linear and nonlinear integral equations, Ho¨lder spaces and their
generalizations which arebasic for the investigation ofsingular integral equations, Lorentz
andMarcinkiewiczspaceswhicharewidelyusedininterpolationtheoryforlinearoperators,
and special classes of spaces of differentiable or smooth functions which frequently occur
in the theory of ordinary or partial differential equations and variational calculus. The
usefulness of all these spaces in various fields of mathematical analysis emphasizes the
need for a systematic study of the superposition operator (1), considered as an operator
from one such space into another.
In this connection, there are still many open problems. In particular, for many of
these spaces one does not even know acting conditions for F, by means of conditions on f,
which are both necessary and sufficient (sufficient conditions are often easily formulated).
On the other hand, many special facts regarding the elementary properties of F, such
as continuity, boundedness, or compactness, are well-known in, say, Orlicz spaces, Ho¨lder
spaces, or Sobolev spaces. Unfortunately, all these results are scattered in research papers
and special monographs. We therefore conclude that it would be useful to collect the basic
factsonthesuperpositionoperator,topresent themainideaswhichhavebeenshowntobe
useful in studying its properties, and to provide a comparison of its behaviour in different
spaces. This is the purpose of the present monograph.
Here the key problem is, as already mentioned, to find conditions on the function
f which imply certain properties of the corresponding operator F. In this connection, the
mainpropertiesweareinterested inare:boundedness andcompactness oncertainsubsets,
continuity and differentiability at single points, continuity and continuous differentiability
onopensubsets,specialcontinuityproperties(suchasLipschitz,uniform,orweakcontinu-
ity), analyticity, and related properties. These are just the properties which occur most
frequentlyintheapplicationofmethodsofnonlinearanalysis,suchasfixed-pointprinciples,
degreetheory,bifurcationmethods,variationaltechniques,tononlinearequationsinvolving
superposition operators. Thus, the reader may typically find answers to questions of the
following type: what are necessary and sufficient conditions for the function f such that
the corresponding operator F maps the Lebesgue space L into the Lebesgue space L , or
p q
is continuous between two Orlicz spaces L and L , or differentiable between two Ho¨lder
M N
spaces H and H , or bounded between two Sobolev spaces Wk and Wm, or Lipschitz
φ ψ p q
continuous in the space BV?
Whenpreparingthematerialforthismonograph,weintentionallyconfinedourselves
to the scalar case. The vector case, i.e. when the superposition operator F maps Rm- (or
Cm-) valued functions on Ω into Rm- (or Cm-) valued functions on Ω (and f is defined, of
course,onΩ Rm orΩ Cm withvaluesinRn orCn,respectively), isatleastasimportant
× ×
as the scalar case. However, much less is known in this case, and the development of a
"higher-dimensional"theory would be beyond the scope of the present work and would
probably require us to increase the size of this survey at least twofold. In large parts of
the monograph, Ω may also be the set of all natural numbers, equipped with the counting
IV
measure; consequently, our results cover superposition operators in sequence spaces as
well. The main emphasis is put, however, on "usual"functions, i.e. the case when Ω is
some domain in Euclidean space.
Apart from the superposition operator (1), the related operator
Φx(s) = x(φ(s)) (2)
is sometimes also called superposition operator in the literature, where φ is some bijection
ofΩontoitself;moreprecisely,operatorsofthistypeshouldbecalled"inner"superposition
operators, in contrast to the "outer"superposition operator (1). In spite of the similar
structure of the operators (1) and (2), their properties are quite different; this is clear, for
instance, from the fact that the operator (2) is linear, while the major difficulty in the
study of the operator (1) lies in its nonlinearity. Throughout this monograph, we shall be
concerned only with the outer superposition operator (1).
Anotheroperatorwhichiscloselyrelatedtotheoperator(1)istheintegralfunctional
Φx = f(s,x(s))ds (3)
Z
Ω
, which is of fundamental importance, for example, in variational problems of nonlinear
analysis. We shall be concerned with the operator (3) only marginally and refer to the
vast literature on variational methods.
The monograph consists of nine chapters. Each chapter is divided into a number
of sections and provides a self-contained systematic study of the superposition operator
in some class of function (or sequence) spaces. We have tried to make the exposition
as complete and explicit as possible, including proofs, examples, and counterexamples.
The last section of each chapter is devoted to possible generalizations, special cases, open
problems, related fields and detailed bibliographical references. Each theorem, lemma, or
formula is indexed within the corresponding chapter; thus, for example, Lemma 1.2 is the
second lemma of the first chapter. By and we denote the beginning and the end,
⇒ ⇐
respectively, of a proof.
The contents of the monograph go as follows. The first chapter is entirely devoted
to the study of the superposition operator in the space S = S(Ω) of measurable functions
on Ω, where Ω is an arbitrary nonempty set with measure. Here a basic problem is that of
finding conditions on the function f which ensure that the operator F maps measurable
functions into measurable functions. Surprisingly enough, this turns out to be a highly
nontrivial problem.
As a matter of fact, the space S is a complete metric linear space, but not normable.
Most fundamental principles of linear and nonlinear functional analysis are formulated,
however, in a Banach space setting. Consequently, it is desirable to study the properties
of the superposition operator not only in S, but also in normed subspaces of S. It turns
out that the most appropriate class of Banach spaces of measurable functions is that of
so-called ideal spaces (or Banach lattices), which were considered by many authors for
different purposes. General properties of the superposition operator in ideal spaces are
described in detail in the second chapter.
The third and fourth chapters are concerned with the superposition operator in
Lebesgue and Orlicz spaces, respectively. Here the theory is most complete and advanced,
V
and one can characterize all basic properties of the operator F (in particular, acting
conditions) in terms of the generating function f.
Some other classes of ideal spaces which include, for example, the classical Lorentz
and Marcinkiewicz spaces are dealt with in the fifth chapter. In this connection, only very
few elementary results are presently known.
The sixth chapter is devoted to the superposition operator in the space C = C(Ω)
of continuous functions on Ω, where Ω is a compact domain without isolated points in
Euclidean space. Here the basic facts are well-known "folklore"; however, we shall also
discuss some special problems which have not been studied yet. Moreover, we briefly
discuss the superposition operator in the space BV offunctions of bounded variation.
In the seventh chapter we shall present a systematic study of the superposition
operator in Ho¨lder-type spaces. It turns out that the behaviour of the superposition
operator in such spaces is quite different from that in spaces of measurable functions.
The eighth chapter will be concerned with the superposition operator in spaces of
functions which are characterized by certain differentiability or smoothness properties.
Moreover, we shall consider the operator F in various spaces of finitely or infinitely
differentiable functions, including Roumieu, Beurling and Gevrey classes.
Some results on the superposition operator in Sobolev spaces are given in the
ninth chapter. Unfortunately, in spite of the importance of these spaces in the theory of
distributions and partial differential equations, they have been given very little attention
in the literature.
Some remarks on the bibliography are in order. We hope to present a rather
exhaustive listofreferences onthesuperpositionoperatorinfunctionandsequence spaces.
The bibliography at the end covers the period from 1918 to 1988 and contains about 400
items, half of them in Russian; thus, it may also serve as a guide to the Soviet literature.
For the reader’s convenience, we have added English translations (if there are any) of
Russian books and major journal papers, and, beginning with 1960, the corresponding
review numbers of Zentralblatt fu¨r Mathematik (Zbl.), Referativnyj Zhurnal Matematika
(R.Zh.), and Mathematical Reviews (M.R.). We are indebted to Nguyˆen˜ Hoˆn`g Tha´i,
Heinz-Willi Kro¨ger and Reiner Welk for computer-aided help in finding many review
numbers.
It is a great pleasure to thank all colleagues and friends who sent us reprints,
preprints, and unpublished manuscripts, and helped us to make the list of references
more complete by suggesting new (or simply forgotten) entries. In particular, we are
indebted to Nguyˆen˜ Hoˆn`g Tha´i , Jevgenij M. Semjonov, and Marek Z. Berkolajko for
severalhelpfuldiscussions onChapter4,Chapter5,andChapter7,respectively.Moreover,
we are grateful to the publishers, especially to David Tranah and Mark Hendy, for
fruitful collaboration and useful advice. Last but not least, our special thanks go to Fritzi
Stegmuller for her excellent typing of the manuscript with extraordinary patience, and
for never grumbling at a lot of changes.
This book could not have been realized without the possibility of travels and
meetings in both Germany and the Soviet Union, generously supported by the Deutscher
Akademischer Austauschdienst (Bonn) and the Ministry of Higher Education (Moscow).
The first author acknowledges the hospitality of the Belorusskij Gosudarstvennyj Univer-
VI
sitetatMinsk,thesecondauthorthatoftheUniversita¨tAugsburgandtheRuhruniversita¨t
Bochum.
Spring 1989 Ju¨rgen Appell, Petr P. Zabre˘ıko
VII
Chapter 1
The superposition operator in the space S
In this chapter we study the superposition operator Fx(s) = f(s,x(s)) in the
complete metric space S of measurable functions over some measure space S. First, we
consider some classes of functions f which generate a superposition operator F from S
into S; a classical example is the class of Carath´eodory functions, a more general class
that of Shragin functions.
As a matter of fact, there exist functions f, called "monsterswhich generate the
zero operator Fx = 0, but are not measurable on Ω R and hence are not Caratheodory
×
functions; this disproves the old-standing Nemytskij conjecture. On the other hand, we
show that a function which generates a continuous superposition operator (in measure)
is "almost"a Caratheodory function.
Wegiveanecessary andsufficient conditionforthefunctionf togenerateabounded
superposition operator F in the space S. In particular, this conditions holds always if f
is a Caratheodory function. On the other hand, we show that the superposition operator
F is "never"compact in the space S, except for the trivial case when F is constant.
Finally, we consider superposition operators which are generated by functions f
with special properties (e.g. monotonicity), and characterize the points of discontinuity
of such operators.
1.1 The space S
Let Ω be an arbitrary set, some (σ-algebra of subsets of Ω (which will be called
M
measurable in what follows), and µ a countably additive and σ-finite measure on . By
M
λ we denote some normalized ("probability") measure on which is equivalent to µ (i.e.
M
has the same null sets); one possible choice of λ could be, for instance,
λ(D) = n(s)dµ,
Z
D
where n is any positive function on Ω with
n(s)dµ = 1
Z
Ω
In most examples, we shall deal with either some bounded perfect set Ω with nonempty
interior in some finite dimensional space, together with the algebra of Borel- or
M
Lebesgue-measurable subsets and the Lebesgue measure µ, or the set of natural numbers,
together with the algebra of all subsets and the counting measure. More complicated
1
examples,ofcourse,arealsopossible:ΩbeinganarbitraryLebesgueorBorelsubset,andµ
theLebesgue or Borelmeasure, or Ω being a "nice"subset ofa finite dimensional manifold,
together with a suitable algebra of subsets and some measure µ. Such examples will
M
be considered only in quite exceptional cases. We point out that we do not suppose the
σ-algebra to be complete with respect to the measure µ.
M
Recall (Saks’ lemma) that the set Ω can be divided, uniquely up to null sets, into
two parts Ω and Ω such that µ is atomic-free ("continuous") on Ω (i.e. any subset of Ω
c d c c
canbedivided into two partsofequal measure), andµ ispurely atomic("discrete") onΩ ,
d
i.e. Ω is a finite or countable union of atoms of positive measure. In "natural"examples,
d
one of the sets Ω or Ω is usually empty, and thus one deals with real "function spaces"or
c d
"sequence spaces".
Asusual,wedenotebyS = S(Ω, ,µ)thesetofall(realorcomplexvalued)almost
M
everywhere finite µ-measurable functions on Ω; more precisely, S consists of equivalence
classes ofsuch functions, where two functions x andy are called equivalent if they coincide
almost everywhere on Ω. The set S can be equipped with the usual algebraic operations,
where the zero element is the function θ(s) = 0 almost everywhere, as well as with the
metric ρ(x,y) = [x y], where
−
[z] = inf h+λ( s : s Ω, z(s) > h ) (1.1)
0<h<∞{ { ∈ | | } }
or
z(s)
[z] = | | dλ (1.2)
1+ z(s)
Z | |
Ω
Withrespecttothismetric,S becomesacompletemetricspace,andconvergenceρ(x ,x)
n
→
0 is equivalent to convergence of x in measure to x, i.e. λ( s : x (s) x(s) > h ) 0
n n
{ | − | } →
as n , for any h > 0.
→ ∞
It is convenient to introduce also a partial ordering in the space S: we write x
≤
y (x,y S) if x(s) y(s) for almost all sΩ. In this way, S becomes an ordered linear
∈ ≤
space,i.e.x y impliesthatx+z y+z forz S,andthatλx λy forλ 0;moreover,
≤ ≤ ∈ ≤ ≥
if x and y are two sequences in S which converge to x S and y S, respectively, then
n n
∈ ∈
x y implies that also x y.Finally, S is a K-space (in the sense of L.V.Kantorovich),
n n
≤ ≤
which means thatany set which is boundedfromabove(respectively below) admitsaleast
upperbound(respectively greatestlower bound),wherethese notionsaredefined asusual.
As in every ordered linear space, one can consider convergence with respect to the
above ordering in S. A sequence x in S is order convergent to x S if liminfx =
n n
∈ n→∞
limsupx = x, where
n
n→∞
liminfx = sup inf x limsupx = inf supx .
n m n m
n→∞ k m≥k n→∞ k m≥k
InthespaceS,thistypeofconvergencecoincideswithconvergencealmosteverywhere. Itis
wellknown(Lebesgue’stheorem) thatconvergence almosteverywhere impliesconvergence
inmeasure; the converse is true only if the measure µ is discrete (i.e. Ω = ).Nevertheless
c
∅
(Riesz’ theorem), each sequence which is convergent in measure admits a subsequence
which converges almost everywhere (to the same limit, of course). We still mention a well
2