Regularity Properties of Functional Equations in Several Variables (Advances in Mathematics)

REGULARITY PROPERTIES OF FUNCTIONAL EQUATIONS IN SEVERAL VARIABLES

Advances in Mathematics VOLUME 8 Series Editor: J. Szep, Budapest University of Economics, Hungary

Advisory Board: S.-N. Chow, Georgia Institute of Technology, U.S.A. G. Erjaee, Shiraz University, Iran W. Fouche, University of South Africa, South Africa

P. Grillet, Tulane University, U.S.A.

H.J. Hoehnke, Institute of Pure Mathematics of the Academy of Sciences, Germany F. Szidarovszky, University of Airzona, U.S.A. P.G. Trotter, University of Tasmania, Australia

P. Zecca, Universitb di Firenze, Italy

REGULARITY PROPERTIES OF FUNCTIONAL EQUATIONS IN SEVERAL VARIABLES

ANTAL JARAI Eotvos Lorand University, Budapest, Hungary

- Springer

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e-ISBN 0-387-24414-X

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SPIN 11377986

This book is dedicated to JBnos Aczkl, my "mathematical grandfather", the teacher of several of us in the field of functional equations, and to my teacher ZoltAn Dar6czy who introduced me to functional equations.

TABLE OF CONTENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix . . . . . . . . . . . . . . . . . . 1 Chapter I . PRELIMINARIES $1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 1 $ 2. Notation and terminology . . . . . . . . . . . . . . . . . . 40

Preface

Chapter I1. STEINHAUS TYPE

. . . . . . . . . $ 3. Generalizations of a theorem of Steinhaus . . . . . . . . 5 4. Generalizations of a theorem of Piccard . . . . . . . . . . Chapter I11. BOUNDEDNESS AND CONTINUITY OF SOLUTIONS . . . . . . . . . . . . . $ 5. Measurability and boundedness . . . . . . . $ 6 . Continuity of bounded measurable solutions 5 7. On a problem of Mazur . . . . . . . . . . . . . . . . 5 8. Continuity of measurable solutions . . . . . . . . . . . $ 9. Continuity of solutions having Baire property . . . . . . $10. Almost solutions . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. 73 . 73 . 76 . 81 . 86 . 94

Chapter IV . DIFFERENTIABILITY AND ANALYTICITY . 5 11. Local Lipschitz property of continuous solutions 5 12. Holder continuity of solutions . . . . . . . . $ 13. Solutions of bounded variation . . . . . . . . $ 14. Differentiability . . . . . . . . . . . . . . 5 15. Higher order differentiability . . . . . . . . . 5 16. Analyticity . . . . . . . . . . . . . . . .

. . . . . . .

109 109 128 132 137 141 144

THEOREMS

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . 53 . . . 53 . . . 66

. . . . . . .

100

Chapter V . REGULARITY THEOREMS O N MANIFOLDS . . . . . . 157 5 17. Local and global results on manifolds . . . . . . . . . . . . 157 Chapter VI . REGULARITY RESULTS

W I T H FEWER VARIABLES

5 18. ~ w i a t a k ' smethod . . . . . . . . . . . § 19. Between measurability and continuity . . . 5 20 . Between Baire property and continuity . . $ 21. Between continuity and differentiability . .

. . . .

. . . .

. . . .

. . . .

Chapter VII . APPLICATIONS . . . . . . . . . . . 5 22 . Simple applications . . . . . . . . . . . . . . . $23. Characterization of the Dirichlet distribution . . . . $24. Characterization of Weierstrass's sigma function . .

. . . .

. . . .

. . . . . . . .

. . . . . . . . . .

. . . . .

169 170 174 204 218

. . . .

. . . .

231 231 275 285

. . . .

viii

Table of contents

References . . . . . . . . . . . . . . . . . . . . . . . . . . Index . . . . . . . . . . . . . . . . . . . . . . . . . . . .

335 347

PREFACE This book is about regularity properties of functional equations. It contains, in a unified fashion, most of the modern results about regularity of non-composite functional equations with several variables. It also contains several applications including very recent ones. I hope that this book makes these results more accessible and easier to use for everyone working with functional equations. This book could not have been written without the stimulating atmosphere of the International Symposium on Functional Equations conference series and thus I am grateful to all colleagues working in this field. This series of conferences was created by JBnos Aczitl. I am especially grateful to him for inviting me to the University of Waterloo, Canada, which provided a peaceful working environment. I started this book in September 1998 during my stay in Waterloo. I thank Mikl6s Laczkovich, the referee of my C.Sc. dissertation very much for his remarks and suggestions. Between 1974 and 1997 I worked at Kossuth Lajos University, Debrecen, Hungary. Naturally, I am grateful to all members of the Debrecen School of functional equations. Finally, I would like to thank those of my colleagues, joint pieces of work with whom in one way or another are contained in this book: JAnos Aczkl, ZoltBn Darbczy, Roman Ger, Gyula Maksa, Zsolt PAles, Wolfgang Sander, and LAsz16 Szitkelyhidi. I thank my sons, Antal JArai Jr. and ZoltAn JBrai for correcting several errors of my "Hunglish"; all the remaining errors are mine. Finally, I wish to express my gratitude to Kluwer Academic Publishers, in particular, Professor Jen6 Sz6p and John C. Martindale for the first rate and patient technical help. Financial support for this book was provided mainly by Szitchenyi Scholarship for Professors, Hungary and partly by OTKA TO31995 grant. Budapest, May 25, 2003. Antal JBrai Department of Computer Algebra Eotvos LorAnd University PBzmAny Pitter sittAny 1/C H-1117 Budapest, Hungary e-mail: [email protected] http://compalg.inf.elte. hu/-ajarai

Chapter I.

PRELIMINARIES

In this chapter, starting with simple examples, we describe the problems with which we will deal in this book. We also present simple examples of our methods. First we formulate the fundamental problem, then analyse its conditions and explore its applicability. We then formulate theorems that follow from our results as corollaries to that fundamental problem. Then we survey possibilities for generalization. We close this chapter by summarizing our notation and terminology, including the formulation of theorems not readily available in the literature or usually formulated in a different way.

1. INTRODUCTION 1.1. G e n e r a l considerations a n d simple examples. As a first, illustrative example let us consider the best-known functional equation, Cauchy's equation

with unknown function f . In a wider sense differential equations, integral equations, variational problems, etc. are also functional equations, but here we will use this expression in a more restrictive sense for functional equations

2

Chapter I. Preliminaries

without infinitesimal operations such as integration and differentiation. For a more formal definition, see Aczd [3], 0.1. To formulate a functional equation exactly we have to give the set of functions in which we look for solutions. We also have to give the domain of the functional equation. In the above example this is the set of the pairs (x, y) of the variables x and y for which equality has to be satisfied. For example, we may look for all measurable functions f : R -+ R such that (1) is satisfied for all (x, y) E [0, m[ x R. Conditions such as measurability, Baire property, continuity everywhere or in a point, boundedness, differentiability, analyticity, etc. are called regularity conditions. If this kind of conditions are imposed on the solution, then we say that we look for regular solutions. Otherwise, if we look for solutions among all maps from a given set into another given set, then we say that we look for the general solution of the functional equation. Usually, the domain of the functional equation is the set of all tuples of the variables for which both sides are defined. For example, if we say that f : R -+ R is a solution of Cauchy's functional equation, then it is implicitly understood that (1) is satisfied for all (x, y) E R x R. If the domain of the equation is not the largest possible for which both sides are defined, then we speak about an equation with restricted domain; the term conditional equation is also used, especially if the domain of the equation also depends on the solution or solutions. Cauchy's equation is a functional equation with two variables; the variables denoted by x and y in (1). Equations like f (x) = f (-x), f (x) = -f (-x), f (22) = f ( x ) ~or , difference equations are called functional equations in a single variable. The "single variable" may also be a vector variable; it is understood that there are no more variables in the equation than the number of places in the unknown function or the minimal number of places in the unknown functions - if there is more than one. Otherwise we speak about a functional equation in several variables. This distinction is very useful in practice. There is a large difference between functional equations with a single variable and several variables: the methods used in the two cases are quite different. In this book we deal with functional equations in several variables. About equations in a single variable see the books Kuczma [I261 and Kuczma, Choczewski, Ger [128]. The distinction between functional equations in a single variable and in several variables and what we have said about variables, domain, regular and general solutions also apply to systems of functional equations. Further simple examples of functional equations are Cauchy's exponen-

§ 1. Introduction

3

tial equation

Cauchy's power equation

and Cauchy's logarithmic equation

Observe, that solutions f of (2) mapping 10, GO[ into the normed algebra of all bounded linear operators on a Banach space gives operator semigroups. The usefulness of semigroups in the study of evolution equations such as the heat equation or Schrodinger's equation is well known, see for example Hille and Phillips [72]. The overall importance of equations (1)-(4) is due to the fact that they describe homomorphisms. We move toward a general theory of functional equations, and we do not intend to study specific functional equations, except as examples, even if they are very important. It is a well-known phenomenon that one functional equation can determine several unknown functions. This is the situation, for example, for the analogue of Cauchy's functional equation with several unknown functions which is called Pexider's equation:

Indeed, if f 1, f 2 ,f 3 : R -+ R, by putting y = 0 and x = 0 in (5) we may express f 2 and f 3 by f l , respectively. By putting x = 0 and y = 0 simultaneously in (5) and using the resulting relation we obtain that f = f l - f1(0) satisfies Cauchy's equation (1). Hence (5) can be reduced to (1). Similar phenomena occur often when different occurrences of the unknown function f are replaced by f l , f 2 , etc., a process sometimes called "Pexiderization". Jensen 's equation

can be considered as a special case of Pexider's equation, and we obtain that an f : R -+ R function is a solution of (6) if and only if the function f - f (0) satisfies (1).

4

Chapter I. Preliminaries

It is also possible to consider functional inequalities. Functional inequality

related to Cauchy's equation describes subadditive functions and functions satisfying

are the so-called Jensen convex functions. We will use the above simple functional equations as illustrative examples. Their detailed study can be found in the book of Acz6l [3] or in the book of Acz6l and Dhombres [20]. 1.2. Simple examples: smooth solutions. Let us suppose that a solution f : IR + R of Cauchy's equation f (x y) = f (x) f (y)is analytic. Substituting y = x we obtain the equation f (22)= 2f (x)in a single variable x E R. Analyticity is such a strong regularity condition that even this single variable equation has not too many analytic solutions. For the solution we obtain f (x)= co clx

+

+

+

+

in a neighborhood of the origin, and hence that the solution can only be f (x)= cx with an arbitrary constant c = cl. Substitution shows that this is indeed a solution of Cauchy's equation. The case of Cauchy's exponential equation f (x y) = f (x)f (y)is much more interesting. As above, we obtain the single variable equation f (22)= f ( x ) ~and, , if f : R + R is analytic, f (x)= co clx , then

+ + +

3

.

.

Hence co = ci. There are two possibilities. The first is that co = 0, which implies that c, = 0 for each n, and hence f = 0. The second is that co = 1. In this case cl could be arbitrary, and from the equation

$1. Introduction

5

using the notation c = cl, we obtain by induction that cn = cn/n!. Hence all analytic solutions are given by f (x) = C r = o cnxn/n! = exp(cx). The same method gives complex analytic solutions f : C + C,too. Let us observe that this is a nice way to introduce exponential functions (and hence the related functions sin, cos, sinh, and cosh) using only the most important property of exponentiation. Note that this was Cauchy's original motivation to investigate functional equations 1.1.(1)-1.1. (4): he wanted to avoid "circulus vitiosus" by studying power functions; see the historical remarks in the book of Acz6l and Dhombres [20], pp. 365-371. Now let us only suppose that the solution f : R + R is twice differentiable. In the case of Cauchy's equation, f (x y) = f (x) f (y), let us differentiate both sides with respect to y. This "kills" the first term on the right-hand side, and we obtain that f l ( x y) = f '(y) for every x, y E R. Differentiating again, but with respect to x we can "kill" the other term on the right-hand side, too, and we obtain f"(x y) = 0. Substituting y = 0 we have f "(x) = 0, a differential equation. All solutions of this equation have the form f (x) = co + cx, co, c E R. Substituting this into the original functional equation we see that co = 0, and we obtain that twice differentiable solutions are exactly the functions f (x) = cx. This simple example illustrates a general method to get "smooth enough" solutions. The general tactic is to "kill" some terms by applying appropriate differential operators, and to obtain differential equations by appropriate substitutions. Usually, appropriate substitutions or use of certain symmetries of the equation results in a differential equation with lower degree. For example, substituting y = 0 in the equation f l ( x y) = fl(y) we obtain that f l ( x ) = c with c = fl(0), a first order equation. Cauchy's exponential equation, f (x y) = f (x)f (y), similarly yields f '(x y) = f (x)f '(y), and after substituting y = 0 we obtain f '(x) = cf (x), where c = f '(0). Let us observe that in both cases, the general once differentiable solution f : R + R is the same as the general analytic solution.

+

+

+

+

+

+ +

1.3. Simple examples: regularity properties. How to obtain solutions of the above examples, Cauchy's equation and Cauchy's exponential equation under much weaker regularity assumptions? A general way is to prove that weak regularity conditions, say continuity or measurability of solutions implies much stronger regularity conditions, their differentiability or even analyticity. For example, let us observe that in both of the above cases the differential equation obtained for the solutions in the previous point implies directly that the solutions are analytic (see Dieudonnh [49], 10.5.3). If we have a continuous solution f : R + R, then integrating Cauchy's

6

Chapter I. Preliminaries

equation over an interval [a,b] of positive length yields

Substituting a new variable u = x

+ y we obtain

The right-hand side is differentiable, so we obtain that f is differentiable. If we want to deduce that f is twice differentiable, we can apply the same obtained by differentiation with reasoning to the equation f '(x y) = f '(x) respect to x from the original. Higher order differentiability can be obtained analogously. In the case of Cauchy's exponential equation f 0 is one of the continuous solutions f : R + R. If f (yo) # 0, then we can choose a neighborhood [a,b] of yo such that f (y)/f (yo) 2 112 for each y E [a,b]. Integrating we obtain

+

-

and hence that

This implies that f is differentiable. Here, again, applying the same method for the equation fl(x y) = fl(x)f (y) obtained from the original equation by differentiation with respect to x gives that the solutions are twice differentiable, etc. Now, let us consider a measurable solution f : R -+ R of Cauchy's equation. Let [a, b] be an interval with positive length 7. Let xo E R be arbitrary. By Lusin's theorem there exists a compact set C1contained in [xo a , xo b] and having Lebesgue measure greater than 3714 such that f lC1is continuous. If /x- x o /< 7718, then the set C1- x is contained in C = [a - 718, b 7/81. Since the Lebesgue measure of C \ (C1- x) and C \ (C1 - xo)are less than 712, they cannot cover C.Hence the intersection (C1- x) n (C1- xo) is nonvoid. Now, let E > 0 be arbitrary. Since f lCl is uniformly continuous, there exists a S > 0 such that if u,u' E C1then If (u)- f ( .')I < E. Hence, if Ix - xol < min{71/8, S) then for any y E (C1- x) n (CI- xo) we obtain

+

+

+

+

7

'$1.Introduction

i. e., f is continuous at xo. Since x0 was arbitrary, f is continuous everywhere. The same method can be applied to Cauchy's exponential equation after introducing the new variable t = x y instead of x, i. e., to the equation

+

f (4= f (t - df (Y).

+

Note that Cauchy's logarithmic equation f (xy) = f (x) f (y) has no other solution f : R + R than f = 0;this follows by substituting y = 0. In the case of Cauchy's power equation f (xy) = f (x)f (y) there are solutions f : R + R which are measurable but non-continuous, continuous but non-differentiable, etc. Indeed, the functions x e /x1 and x ci\ 1x1 sgn x are solutions for any c E R if OC is understood as 0. 1.4. Hilbert's fifth problem. In his celebrated address to the 1900 International Congress of Mathematicians, in his fifth problem Hilbert ([70] p. 304) asked1

". . . how far Lie's concept of continuous groups of transformations is approachable in our investigations without the assumption of differentiability of the functions" More precisely2:

". . . hence there arises the question whether, through the introduction of suitable new variables and parameters, the group can always be transformed into one whose defining functions are differentiable . . . >,.

Explaining that the group property is connected to a system of functional equations, in the second part of his fifth problem Hilbert goes on as follows3:

". . . inwieweit der Liesche Begriff der kontinuierlichen Transformationsgruppe auch ohne Annahme der Differenzierbarkeit der Funktionen unserer Untersuchung zuganglich st." ". . . es ensteht mithin die Rage, ob nicht etwa durch Einfuhrung geeigneter neuer Verandertlicher und Parameter die Gruppe stets in eine solche iibergefiihrt werden kann, fiir welche die definierenden Funktionen differenzierbar sind, . . . " . " ~ b e r h a uwerden ~t wir auf das weite und nicht uninteressante Feld der Funktionalgleichungen gefuhrt, die bisher meist nur unter der Voraussetzung der Differenzierbarkeit der auftretenden Funktionen untersucht worden sind. Insbesondere die von ABEL (Werke, Bd. 1, S. 1,61, 389) mit so vielem Scharfsinn behandelten Funktionalgleichungen, die Differenzengleichungen und andere in der Literatur vorkommende Gleichungen weisen an sich nichts auf, was zur Forderung der Differenzierbarkeit der auftretenden Funktionen zwingt, und bei gewissen Existenzbeweisen in der Variationsrechnung fie1 mir direkt die Aufgabe zu, aus dem Bestehen einer Differenzengleichung die Differenzierbarkeit der betrachteten Funktionen beweisen zu mussen. In allen diesen Fallen erhebt sich daher die Frage, inwieweit etwa die Aussagen, die wir im Falle der Annahme differenzierbarer Funktionen machen konnen, unter geeigneten Modifikationen ohne diese Voraussetzung giiltig sind."

8

Chapter I. Preliminaries

"Moreover, we are thus led to the wide and interesting field of functional equations which have been heretofore investigated usually only under the assumption of the differentiability of the functions involved. In particular the functional equations treated by Abel (Oeuvres, vol. 1, pp. 1, 61, 389) with so much ingenuity, the difference equations, and other equations occurring in the literature of mathematics, do not directly involve anything which necessitates the requirements of the differentiability of the accompanying functions. In the search for certain existence proofs in the calculus of variations I came directly upon the problem: To prove the differentiability of the function under consideration from the existence of a difference equation. In all these cases, then, the problem arises: In how far are the assertions which we can make in the case of differentiable functions true under proper modifications without this assumption ?" (Hilbert's emphases.) After this Hilbert quotes a result of Minkowski which states that under certain conditions the solutions of the functional inequality

are partially differentiable, and remarks that certain functional equations, for example the system of functional equations

where a, p are given real numbers, may have solutions f which are continuous but non-differentiable, even if the given function g is analytic. In our present-day language, it is customary to formulate the fifth problem of Hilbert as the question whether a locally Euclidean topological group is a Lie group. However, in the second part of his fifth problem, Hilbert draws attention to more general problems which today are called regularity problems. They require to prove that differentiability assumptions for functional equations, differential equations, and other equations can be replaced by much weaker assumptions (possibly with appropriate modifications of the problem). This idea returns in problems nineteen and twenty of Hilbert concerning calculus of variation and partial differential equations. See the book of Zeidler [209],II/A, pp. 86-93. As a general reference about Hilbert's problems, see the book [26] edited by Alexandrov.

§ 1. Introduction

9

Concerning Abel's work on functional equations mentioned by Hilbert see the short note [12] and the paper [14] of Aczkl. These initiated intense research to solve Abel's equations under as weak conditions as possible. Two of these equations will be used here as illustration to our methods among applications (see 22.10 and 24); concerning a third one see the papers of Sablik 11741, 11751, [176]. 1.5. A general scheme t o solve functional equations. The simple examples above, as well as the second part of Hilbert's fifth problem, suggest a general scheme to obtain regular solutions of functional equations: (1) Prove - using regularity theorems - that each solution of the functional equation satisfying some "weak" regularity property - say measurability, the Baire property, etc. - satisfies "strong" regularity conditions for example, local integrability, continuity, differentiability or analyticity.

(2) Obtain the "strongly" regular solutions by deriving another kind of equation from the functional equation (differential equation, integral equation, an equation system for the coefficients of the power series expansion, etc.) or by using special methods of the theory of functional equations. Hundreds of papers illustrate this time-honored practice. In step (2) we use special properties of the functional equations and ad hoc considerations: special substitutions, symmetries of the equations, integral transforms, "killing" some terms by applying appropriate differential operators, etc. This is practiced in several books and papers about functional equations. The reader can find some examples among the applications in the last chapter of this book. It is impossible to refer here to all the enormous literature. We refer the reader to books, survey papers, and research papers. The following books contain introductory material about functional equations in several variables: Aczd [2], [5]; Hille [71]; Kuczma [127]; Saaty [173], section 3. The following are monographs about functional equations in several variables: Acz4l [I], and the revised English edition [3]; Aczd-Dhombres [20]. Special topics are dealt with in Acz6l [lo] (social and behavioral sciences); Aczkl-Dar6czy 1191 (measures of information); Aczkl-Golqb [22] (geometry); Dhombres [45] (conditional Cauchy equations); Eichhorn [51] (economics); Szekelyhidi [I931 (convolution-type functional equations). See moreover the survey papers [8], [14], [16], [120], the survey papers in the volume [15], the volumes of Aequationes Mathematicae, and the bibliography of papers on functional equations in the volumes of Aequationes Mathematicae.

10

Chapter I. Preliminaries

The main subject of this book is to deal with step (1). We will obtain general regularity results for a large class of functional equations, called non-composite functional equations with several variables. Our results give answers to most of the questions of the second part of Hilbert's fifth problem for this class. Here it is also impossible to refer to all the papers dealing with the regularity problem of special functional equations. We refer the reader to the book and survey papers cited above and to the volumes of Aequationes Mathematicae and the bibliography therein. Part of the regularity results dealing with general types of functional equations in several variables will be cited in the introduction of the appropriate sections of this book. Before starting to deal with regularity questions, we give a short overview of and some references about general methods to get "smooth" solutions of functional equations. 1.6. General methods of determining "smooth" solutions of functional equations. In 1.2 we have seen two methods. One of them was to find analytic solutions by obtaining a system of equations for the coefficients from the functional equation. The other one was to reduce the functional equation to a differential equation. This second method, which is used extensively, usually gives the most comfortable way to determine "smooth" solutions. Examples can be found in the book of Aczd [3], see especially 4.2.1, 4.2.4. The deduction of the differential equation usually uses the tactic of "killing some terms" by differentiation, and using special substitutions and special symmetries of the equation. Actually this method goes back to Abel. He wrote (quote from Aczkl [9]):4 "Actually, as many equations can be found by repeated differentiations with respect to the two independent variables as are necessary to eliminate arbitrary functions. In this manner, an equation is obtained which contains only one of these functions and which will generally be a differential equation of some order. Thus it is generally possible to find all the functions by means of a single equation. "In der Tath lassen sich durch wiederholte Differentiationen nach den beiden verandertlichen Grossen, so vie1 Gleichungen finden, als notig sind, um belibige Functionen zu eliminieren, so dass man zu einer Gleichung gelangt, welche nur noch eine dieser Functionen enthalt und welche im Allgemeinen eine Differential-Gleichung von irgend einer Ordnung sein wird. Man kann also im Allgemeinen alle die Functionen vermittelst einer einzigen Gleichung finden. Daraus folgt, dass eine solche Gleihung nur selten moglich sein wird. Denn d a die Form einer beliebigen Function die in der gegebenen Bedingungs-Gleichung vorkommt, vermoge der Gleichung selbst, von der Formen der andern abhangig sein soll, so ist offenbar, dass man im Allgemeinen keine dieser Functionen als gegeben annehmen kann."

5 1. Introduction

11

From this it follows that such an equation can exist only rarely. Indeed, since the form of an arbitrary function appearing in the given conditional equation, by virtue of the equation itself, is to be dependent on the forms of others, it is obvious that, in general, one cannot assume any of these functions to be given." For functional equations with several variables and several unknown functions PAles [163] developed this method into a general algorithm. We will treat his algorithm in detail in 5 16 and we will illustrate it in 22.11. Presently this is the most general method to find smooth solutions. Another important method is to use an integral transform to get a simpler functional equation. This method is illustrated in the book of Aczkl [3], 4.1.1. Szkkelyhidi developed this idea into a very general method. His books [191], [I941 contain several examples, mostly functional equations on commutative groups, where mean periodic functions, Fourier transform and some refinements and spectral synthesis play an important r d e ; see also our joint survey paper [120]. With Szkkelyhidi's methods various special cases of the general functional equation

can be treated, where functions f j , gi, hi are unknown and functions cpj, y ! ~ ~ are given. In general, cpj, $j are homomorphisms, or in some sense structurepreserving mappings of the common domain of the unknown functions. G is a commutative topological group or semigroup. For simplicity we restrict ourselves here to the case where the range of the unknown functions is in C, although more general ranges can be considered as well. Equation (1) can be roughly characterized as a "linear equation with linear arguments", although it is linear only in the sense that linear combinations of solution vectors (fl , . . . , f,) also satisfy a similar equation with another n. Although the methods and results depend heavily on the commutative structure of the underlying groups or semigroups, in some cases they can be applied also in the noncommutative situation. An important special case is n = 0, where regular solutions are continuous (generalized) polynomials, i. e., they can be represented as a finite sum A,+(x,x, . . . ,x) where A,+ : G " C is a k-additive function, i. e., x ++ additive in each variable. The exact characterization of the solutions depends on some algebraic conditions. Other important special cases are the Levi-CivitA type equation (case m = 1, cpl = $1 = id) and the D'Alembert-type equation (case m = 2,

Ck

Chapter I. Preliminaries

12

~1 = $1 = ( ~ = 2 -g2 = id). In these cases regular solutions f l and f l , f2, respectively, are continuous exponential polynomials, i. e., can be represented pkmk, where each p,+ : G -+ C is a (generalized) polynomial as a finite sum and each mk : G -+ C is a multiplicative function. All these special cases are related to systems of convolution-type functional equations f *,u = 0, where p runs in a given set of compactly supported complex measures. The essence of the method for solving such convolutiontype systems is to prove that the continuous solutions form a translation invariant closed linear subspace, and there are "sufficiently many" exponential polynomials in this subspace. The continuous solutions can be constructed from exponential polynomial solutions. All these methods work for general solutions, too, if we consider G with the discrete topology.

The elementary method of SGto is based on the following observation: Let X, Y be sets, F; : X + IF, gi : Y + IF (i = 1 , 2 , .. . ,n ) be mappings into a field. Suppose, that the functional equation

x n

0=

(2)

fi (x)gi(y)

for all x E X , y E Y

i=l

is satisfied. I f f l , f 2 , , . . . , f, are linearly independent and f j = El='=, ci,jfi for j = r 1,.. . ,n with some ci,j E IF, then gi = - Cy=r+l ci,jgj for i = 1 , 2 , . . . ,r . Especially,

+

Indeed,

and since f l , . . . ,f, are linearly independent, their coefficients have to be zero for any y E Y.

-

We remark that the above statement is a generalization of the frequently used trivial observations that if f (x)g(y) 0, then f (a) G 0 or g(y) 0 and that if f (x) = g(y) for all x E X, y E Y, then there exists a constant c such

-

3 1. Introduction

13

that f (x) c and g(y) = c. These correspond to the case n = 1 and n = 2, respectively. Of course, the above statement is a special case of more general theorems. We mention it here explicitly only because several nonlinear or even composite functional equations can be solved by Suto's method, reducing them - by differentiation or by other tricks - to equation (2). See for some recent applications of S ~ t o ' smethod the papers [144-1501 of Lundberg. We will give a simple example of how to apply Suto's method in 22.11. Certain functional equations can be reduced to integral equations. We do not illustrate this method here; see the book of Aczd [3], 4.1.2. Another method, which needs only weak regularity conditions, the socalled "distribution method" was initiated by Fenyo [54]. The main idea is that continuous (or locally integrable) solutions are considered as Schwartz distributions and the equation is viewed as an equation for distributions. Differential equations (in the distribution sense) are obtained using similar tricks as for smooth solutions, but usually with more technical problems. Finally, distribution solutions turn out to be regular distributions belonging to (smooth) functions. See the papers of Baker [31], [32], [33], [34] about this method. The applicability of the distribution method is restricted, because no multiplication among Schwartz distributions is defined. By Schwartz's impossibility theorem, this cannot be done in a satisfying way. It is even more hopeless to substitute distributions into general Cm functions with several variables. The distribution method has to be restricted to functional equations that are not very far from being linear. Sometimes it is possible to find the solution of a functional equation directly, on a dense set. In such cases continuous solutions are determined by solutions on this dense set. For example, for Cauchy's equation, by substitution x = y = 0 we obtain f (0) = 0 and by induction f (nx) = n f (x) for n E N. Substitution y = -x shows that f (-x) = -f (x), hence the above formula holds for all n E Z.Now substituting x l n for x shows that f ( x l n ) = f (%)In,whenever 0 # n E Z.Hence f (rx) = r f (x), whenever r E Q. This shows that f (1) determines the values of f at rational points and by approximating real numbers with rational numbers we obtain that the only continuous solutions are f (x) = cx where c = f (1). For further examples see the book of Aczdl [3], section 2.

A further method is to reduce the functional equation to another functional equation which is already solved. We have seen an example for this in 1.1. In this more or less algebraic way we can usually find the general

Chapter I. Preliminaries

14

solutions of the equation. Usually, regular solutions of the original equation are obtained from regular solutions of the other equation, hence we may determine the regular solutions, too. The reader can find several examples for this kind of reduction of a functional equation to another functional equation in Aczkl [3] and in Aczkl-Dhombres [20]. We note that merely the knowledge of the general solution does not always make it possible to deduce the regular solutions easily. For example, the general solution of Cauchy's equation is the following: let us fix an arbitrary base B of R over Q,a Hamel base. Then every real number x can be uniquely written as finite linear combinaribi of base elements hi with rational coefficients ri. Any function tions f : B -+ R can be uniquely extended to a solution of Cauchy's equation by f (x) = Ci ri f ( h i ) Let us observe that even in this very simple case it is not clear - without regularity results - which solutions are measurable.

xi

We have to mention here the method of Vincze. It is based on the following observation: Suppose that functions filj, i = 1 , 2 , . . . ,m, j = 1 , 2 , . . . ,n m a p an arbitrary set X into a field IF and they satisfy the functional equation

Then for any function f o : X

+ IF

the functional equation

is satisfied, too, where fi,0 = f o for all i = 1 , 2 , . . . , m . The statement can be easily verified by expanding all determinants in (4) with respect to f O ( x k ) k, = 0, I , . . . ,n. By choosing an appropriate f o some terms may become zero and we get a simpler functional equation. For further information on how to apply this method and how to reduce certain functional equations to the form (3) see the paper of Vincze [205]. We conclude this section with some general remarks. For general purposes usually the simplest way is to use general regularity results to prove that the solutions are a few times differentiable, and then to obtain a differential equation for the solutions. We usually cannot completely avoid using regularity results with the other methods either. Methods that use

5 1. Introduction

15

less regularity are usually more complicated and cannot be applied to the most general cases. Despite general methods, we are very far from having algorithmic methods to solve a large part of the functional equations that are of recent interest.

1.7. Composite and non-composite functional equations with several variables. The main topic of the present book is to prove general regularity results. For several functional equations regularity theorems do not hold. Besides functional equations in a single variable - like those mentioned by Hilbert, see 1.4 - there are also several composite functional equations (in which unknown functions are substituted into each other) that have continuous, but non-differentiable solutions. For example, the Acz& Benz functional equation

i(n:*

first studied in [17], has the solution x H 1x1). All continuous solutions of this equation were given by Dar6czy [41]. The functional equation system describing group axioms in the first part of Hilbertls fifth problem also needs special treatment (re-coordinating). Although, for example, in the paper of Acz6l [4] under very general conditions it is proved that the real function solutions f of the functional equation

are monotonic, it seems to us that there is no general regularity theory yet for such equations. (We will summarize some recent results about composite equations in 1.16.) However, for those functional equations with several variables where the unknown functions are not substituted into themselves or into each other, general regularity theorems can be proven. Such theorems constitute the main body of this book. Our purpose is to investigate the "most general" non-composite functional equation with several variables having the form

for all (X, Y) from a set E. Here f , fo, fl l . . . , fn are unknown functions, all other functions are given. All functions are vector-vector functions. Introducing the new variable x = G(X, Y) and solving the above equation for

16

Chapter I. Preliminaries

f (x) we usually get a somewhat simpler form, which much better suits our purposes. In several cases introducing another new variable y = Go(X,Y ) , we can get the form

where in one of the unknown functions only the variable y appears. Since for our purposes this "explicit" equation is the most convenient one, we will use this form in general and we do not deal with the conditions under which the above transformations can be (at least locally) performed. In this equation it seems to be unnecessary to include fa - and hence also to introduce the new variable y = Go(X,Y) -, but f o has a distinguished r61e among the unknown functions on the right-hand side. As a problem of Sander [I821 shows (see 22.2), it is often enough to suppose much weaker conditions (or none at all) for f than for the other unknown functions. For the given functions we mainly have to prescribe smoothness conditions: simple examples show that these conditions cannot be omitted. Of course, we may suppose, that the right-hand side of the equation really depends on y, and so the equation is not an equation in a single variable. This is certainly agi the case if we suppose that the rank of the matrices - is maximal. We 89 usually suppose that the set D is an open set. A general problem of regularity is to investigate if Lebesgue measurable solutions or solutions having the Baire property are analytic. Most of our regularity theorems state that, if the given functions are "sufficiently nice" and the unknown functions f l , f P ,. . . , fn have a regularity property, then the function f has a regularity property which is "one better". To obtain an affirmative answer to the regularity problem, the following steps can be used: (I)

measurability implies continuity;

(11)

solutions having the Baire property are continuous;

(111) continuous solutions are almost everywhere differentiable; (IV) almost everywhere differentiable solutions are continuously differentiable;

+ 1 times con-

(V)

all p times continuously differentiable solutions are p tinuously differentiable;

(VI)

infinitely many times differentiable solutions are analytic.

5 1. Introduction

17

If f = f o = f l = . . . = f n r then we obtain step by step that the measurable solutions, or solutions having the Baire property, are analytic. If this is not the case, then expressing other unknown functions from the equation might help to draw a similar conclusion. Let us remark that the regularity problem of composite and non-composite functional equations overlap. Indeed, calling functions "unknown" and "known" in regularity theorems is done only to distinguish those functions for which very weak regularity assumptions (say measurability) are supposed from those for which much stronger assumptions (say continuous differentiability) are supposed. Hence weakening regularity assumptions on "known" functions yields results much more useful for composite equations. To better understand why the above steps (I)-(VI) are chosen, what the main difficulties are, and what kind of conditions we have to impose on the given functions, we first show some basic ideas. 1.8. The classical method. The following general method goes back to Andrade (1900) and Kac (1937) and is well known among functional equationists. It is described (in a somewhat less general form) in the book of Aczdl [3], 4.2.2, 4.2.3. It can be applied to functional equations of the form

c n

h(x, ~ )(x) f = ~ o ( xY),

+

hi (x1Y, fi ( ~ i ( xY,) ) ) ( 2 1 Y) t D; i=l where f , f l , . . . , fn are the unknown functions defined on some open sets X, X I , . . . ,Xn c R and taking values in some open sets Z, Z1,. . . , Zn c R. Let us suppose that Y C R is an open set, D C X x Y is also open, h, ho : a h aho ahi ahi D 4 R and hi : D x Zi + R are continuous, and --, - - - fi, a x a x ' dx ' du ' i = 1 , 2 , . . . ,n are also continuous. Let us suppose that the functions are twice continuously differentiable. Let xo E X and suppose that there exists agi a yo E Y such that (x0,yo) E D , ~ ( X Oyo) , # 0, and -(xo,yo) # 0 for ay i = 1 , 2 , . . . ,n. Then f is continuously differentiable in a neighborhood of 20. To prove this, let us observe that for some open bounded interval Xo and for an interval [a,b] c Y we have that Xo x [a, b] C D, (xO,yo) E Xo x ]a, b[, and J~~ h(x, y) dy # 0 whenever x E Xo. Integrating both sides of equation (1) with respect to y over [a, b] we obtain

(1)

ii

18

Chapter I. Preliminaries

Using the notation gi,,(y) = gi(x, y) let us introduce the new variable u = i = 1 , 2 , . . . ,n separately in each of the integrals on the right-hand side. For this we substitute Xo with a smaller neighborhood of xo and [a,b] with a smaller neighborhood of yo, if necessary. Then we obtain gi,%(y),

From this equation it follows that f is continuously differentiable. Indeed, a parametric integral of the type

is continuously differentiable if 2,Y c R are bounded open sets, 3 : x x [a, b] + ? and ?2. : x x ? + R are continuous functions, and the partial derivatives and and let us denote

2

are continuous, too. To see this, let yl E 8 ( x , y) =

? be fixed,

l:

L(x, u) du.

Then H is continuously differentiable and hence H (x, g(x, a ) ) is also continuously differentiable.

f (x)

= H (x, g(x, b)) -

ah

Let us observe that we only need the continuity of and z. Therefore, h(x, y) can have the form h* (x, y, fo(y)) with continuously differentiable h* but merely continuous (maybe unknown) f o which can even be vector valued. hi (x, y) f j (y) with m < 0 and with In particular, we may take h(x, y) = (possibly unknown) continuous functions f j , m 5 j < 0 and continuously differentiable functions hi. Moreover, let us observe that equations like

may be reduced to (1) introducing new variables x = G(Z, Q ) , y = Go(2,Q ) .

§ 1. Introduction

19

The generalization of this method to complex or even vector-valued unknown functions is immediate, but the generalization to vector-vector unknown functions is non-trivial. We can integrate over a simplex S with nonvoid interior, and we obtain the parametric integral

Of course, the differentiability of such parametric integrals is classic, if 3 and ?L are smooth, but we only know that 3 is continuously differentiable and h together with its partial derivative with respect to x is continuous. We investigate these questions and the generalization of the above classic method to vector-vector functions in 5 11. 1.9. Higher order differentiability. The above classical method can be used to prove higher order differentiability, if we use the fact that a parametric integral

I"(x) =

+

with smooth 3 is p 1 times differentiable if h is p times continuously differentiable and its p t h derivative is continuously partially differentiable with respect to x. An even better trick, which works also for the more general nonlinear functional equation 1.7.(1) is to take the p t h derivative of both sides of the equation. What we get is of type (1) from the previous point, and so can be treated by the above mentioned generalization of the classical method. This trick will be applied in 5 15.

1.10. Continuity of solutions. The classical method (and its generalization too) can also be applied to derive continuity of solutions from their local integrability. Hence, if we can obtain local boundedness of solutions from their measurability, then we can obtain continuity, too. This was the usual way to obtain continuity from measurability. For example, by Cauchy's equation f (x y) = f (x) f (y), if a solution f : R + R is measurable on a set K having positive Lebesgue measure, then it is automatically bounded on a smaller set C still having positive measure, because the measure of the sets K, = {x E K : (x)l 5 n ) goes to the measure of K as n + m. Now by a classical theorem of Steinhaus, the set C - C contains a neighborhood of the origin. Function f is bounded on this neighborhood because if x E C - C , then x can be represented as x = z - y , z , y E C, i. e., there exists a y E C such that x y E C , too. But hence

+

+

If

+

20

Chapter I. Preliminaries

i. e., f is bounded on a neighborhood of the origin. Now by integrating over [-E, E] we obtain

This implies that f is continuous near the origin, and by applying (1) again we get that f is differentiable near the origin. But by Cauchy's equation, fixing any y we see that f around y is merely a constant plus a translate of f around the origin. Hence if a solution is continuous, differentiable, analytic, etc. near the origin, then the same holds everywhere. This way to get all measurable solutions f : R 4 R also works for Cauchy's exponential equation. Today this method is much less important as was earlier, and we use it only if the measures in question are not Radon measures. Some application of this method for invariant expansions of the Haar measure and for covariant expansions of the Lebesgue measure will be given in $5 5, 6. There is a much more appropriate method, which works for Radon measures, specially for the Lebesgue measure. It is based on using Lusin's theorem: an idea which seems to have first appeared in McKiernan [I591 and in Baker [29]. We have seen an example of this method in 1.3. The main idea works also for the general case

i. e., for equation 1.7.(2): Let us choose some compact set C with positive measure. Choosing a "large" compact subset Kiof gi(xo,C) on which fi is continuous, if we are able to prove that the set

is nonvoid whenever x is close enough to xo, then by substituting this y into the equation we obtain that f (x) is close to f ( x 0 ) We will use this method to obtain continuity directly from measurability: see § 8. An important observation here is that it is enough if the functional equation is satisfied almost everywhere (see $ 10). The importance of this observation lies in that if the solutions are differentiable almost everywhere, then the derivative (which is automatically a Bore1 function, see 14.1) satisfies also a functional equation, and utilizing the above observation we obtain that the derivative is continuous. This idea, namely that "differentiability

§ 1. Introduction

21

almost everywhere implies continuous differentiability" is treated in 5 14. Let us observe that this means that from having locally bounded variation (for real variable solutions) or from having local Lipschitz continuity (for vector variable solutions) we can conclude continuous differentiability. An interesting, but out-of-the-main-stream question is treated in 5 7. Problem 24 of Mazur in the Scottish Book [I571 asks whether additive functional~on a Banach space that are Lebesgue measurable along any curve are continuous. This is some kind of "measurability implies continuity" problem. We will briefly discuss it, proving that being Lebesgue measurable along any curve is equivalent to being universally measurable. 1.11. Steinhaus type theorems. As we have seen in the previous point, the theorem of Steinhaus [I891 is a useful tool to treat measurable solutions of functional equations. Another version asserts that, for any Lebesgue measurable sets Al, A2 C IR with positive Lebesgue measure, Al A2 has an interior point. This theorem allows various generalizations and modifications; IR may be replaced by other topological measure spaces and the addition may be replaced by a two variable function. A large part of these generalizations are based on Weil's idea [206] that the convolution

+

is continuous. From this, by setting fi to the characteristic function of Ai the theorem of Steinhaus is easily obtained. Let us observe that the p measure of set 1.10.(1) in the previous point is

where fi is the characteristic function of Ki.This shows that what we need to prove "measurability implies continuity" type theorems is some kind of a generalized Steinhaus theorem, and the two topics are strongly connected. In 5 3 we first generalize Weil's theorem proving the continuity of a

Using this result, we generalize the theorem of Steinhaus for the case of continuously differentiable functions of more than two variables. Several previously known results follow as special cases. This generalization will be used implicitly in $5 5, 8, 10, and 19 and explicitly in 5 6 and in the application given in 5 23.

22

Chapter I. Preliminaries

1.12. Baire category. There is a strong analogy between measure and Baire category: the book of Oxtoby [I611 is a nice introduction. Based on this, we may expect that most of the measure theoretical results have the category analogue. The category analogue of Steinhaus' theorem is the theorem of Piccard, stating that if C C R is of second category and has the Baire property, then C - C contains a neighborhood of the origin. This has similar generalizations as the theorem of Steinhaus: see $ 4. "Baire property implies continuity" type results are analogous to ''measurability implies continuity" type results; they are treated in $ 9.

1. l 3 . Differentiability of continuous solutions. No doubt, the hardest step is to prove differentiability of solutions from continuity. As we have seen in 1.10, it is enough to derive differentiability almost everywhere; this is implied if we prove local Lipschitz property. The generalization of the classical method 1.8 for vector-vector functions is useful but far from being satisfying. We will investigate some properties between continuity and local Lipschitz property: Holder continuity (5 12) and essential bounded variation (5 13). Recently with some additional compactness conditions the author obtained a theorem, which is strong enough to use in most of the practical cases: see 11.6. The first application of this new result was the solution of the equation of "the duplication of the cube" 22.15. But I think the best illustration of the power of this result is that the recent results of the "Habilitationsschrift" of M. Bonk about the characterization of Weierstrass's sigma function by functional equation (a problem which goes back to Abel) can be obtained, even in a more general setting; see $24. 1.14. Analyticity. The final step could be to prove that infinitely many times differentiable solutions are analytic. In this step there are only weak results; see $ 16. We will consider there a result from JBrai [95]. Often it is possible to derive &om the functional equation a differential equation. In such cases, regularity theory of differential equations can be used. See Dieudonni! [49], 10.5.3 for ordinary differential equations and the references in Zeidler [209], II/A, pp. 86-93 and the books [73], [74] of Hormander for partial differential equations. Such type of method is used in the paper of Lawruk and ~ w i a t a k[I381 to prove analyticity of the solutions of a generalized mean value type equation. PBles [I631 also used differential equations to deduce analyticity on a "large" set; see the details in $ 16. 1.15. Other regularity properties. As we have seen above, steps (I)(VI) from 1.7 give a natural "stairway" to climb up from measurability or Baire property to analyticity. Of course, several other regularity properties could be imagined. Let us briefly mention some of them.

§ 1. Introduction

23

Often used for simpler cases, but not so useful for more general equations is to suppose that the solution of the functional equation satisfies some property at a point or on a neighborhood of a point. The most popular condition is that there exists a point, where the solution is continuous. As we mentioned, by Cauchy's equation f (x y) = f (x) f (y) fixing a y we see that the solution around y is merely a constant plus a translate of the solution around the origin. Hence by Cauchy's equation local continuity implies continuity everywhere. (The same is true for several other properties.) Recently Matkowski [I541 generalized this consideration to the Cauchy-type equation f (&,Y)) = h ( x , y , f ( 4 , f(Y))

+

+

with unknown function f . But in the case of other equations it seems to be not so useful to suppose continuity at a point. As an illustrative example, we call the attention of the reader to the fact that those solutions f : ]0,1[ + IR of the fundamental equation of information

that are continuous at a point, are not so easy to obtain. See the papers of Diderrich [46], [47], [48], and Maksa [152]. Compare this with 22.8 to see how easily measurable solutions are obtained by applying our general results. (The detailed study of this equation can be found in the book of Aczd and Dar6czy [19]. A very good survey paper on newer results about functional equations and information measures is Aczkl [Ill.) Another popular condition is that the solution is bounded, bounded from above or from below, or locally bounded. For example, if f : R + R is a solution of Cauchy's functional equation which is bounded from, say, above on a neighborhood of a point xo, then f (x) = x f (1)for all x E R. Indeed, let us observe that because of f ( r x sy) = r f (x) sf (y) whenever x, y E R and r, s E 0, the graph of f is a linear subspace of IR2 over Q.If f (x) # x f (1) for some x E R, then (1,f (1)) and (x, f (x)) are (over R) linearly independent elements of the graph, and hence the graph of f is dense in R2. Even much less than local boundedness is enough to prove that for solutions f : IR -+ R of Cauchy's equation we have f (x) = xf (1) everywhere. If we suppose only that f is bounded (say, from above by K ) on a measurable set C with positive Lebesgue measure (a condition equivalent to the existence of a measurable majorant function of f on C), then

+

+

Chapter I. Preliminaries

24

+

whenever x, y E C , i. e., f is bounded from above by 2K on the set C C. But by Steinhaus' theorem (1.10) this set contains a nonvoid open set. Again, such boundedness or local boundedness conditions work well as regularity assumptions for simpler equations but are not so useful for more complicated equations. As an example let us mention again the above fundamental equation (1) of information. Locally bounded solutions (and hence solutions continuous at some point) were obtained by Diderrich, [46], [47]. See also the papers of Diderrich [48] and of Maksa [152]. There was a longstanding open question whether all nonnegative solutions of this equation are continuous. Finally, Dar6czy and Maksa [44] gave a counterexample. But as a much simpler example let us observe that among f : R -+ C solutions of Cauchy's exponential equation there are bounded but non-continuous functions: any function f : R -+ C defined by f (x) = e i f l ( " ) where f l : R i R is a solution of Cauchy's functional equation, is a solution of Cauchy's exponential equation, and has absolute value 1 everywhere. Furthermore we remark that even for very nice functional equations a complex solution is non-differentiable as a complex function, although it is even analytic as a function of two real variables. A trivial example is Cauchy's functional equation. The most general continuous solution f : C i C is f (2) = cz dz with complex constants c, d. It is not differentiable as a complex function if d # 0. (However, there are some exceptions, see Hille [71], pp. 20-22, and Bonk's regularity theorem, 24.2.) The above examples support our view that it is better to omit studying continuous at a point, locally bounded or globally bounded solutions from our general considerations, and not try to obtain results stating complex differentiability.

+

1.16. Regularity of composite equations. Of course, several composite equations have been studied under weak regularity conditions. A triumph in this field has already been mentioned in 1.4: the solution by M. Sablik [174], [175], [I761 of Abel's functional equation

with three unknown real functions under continuity assumptions. Of course, it is not possible to list here all results concerning solutions of composite equations under mild conditions. As we explained in 1.7, presently it is not clear for which general type of composite equations we may expect general regularity results. Nevertheless, for real function solutions, besides the general result of Acz6l mentioned in 1.7, recently Pdes [I651 suggested the following scheme to find solutions:

§ 1. Introduction

25

(I) Use monotonicity and the functional equation to prove Jensen convexity of the internal functions; (11) Berstein-Doetsch theorem shows that Jensen convexity implies convexity, and hence differentiability of the internal functions at all but countably many points; (1II)Lebesgue theorem on almost everywhere differentiability of monotone functions yields differentiability of the external functions everywhere, and this can be used to prove continuous differentiability of the internal functions; (IV) Differentiating the equation, the composite parts can be eliminated and a non-composite functional equation for the derivatives of the internal functions can be obtained. In his survey papers [165], [166] PBles gives some recent examples of composite equations solved by this scheme. He also formulated there several open problems. Here we restrict ourselves to cite briefly some examples - only to give some impression which kind of composite equations can be solved presently. For the rest of this point, all functions are unknown. The equations

and

are related. Continuous and strictly monotone decreasing solutions of (1) were found by Aczkl, Maksa, and PBles [23] and all strictly monotonic solutions of (2) by PBles [164]. The equation

was solved without any assumptions by Aczkl, Ger, and JBrai in [21]. See 22.18 for further details. The strictly monotonic solutions of the more general equation

26

Chapter I. Preliminaries

and the related equation

were found by Aczd, Maksa, Ng, and PBles [25]. A further related equation is

Strictly monotonic solutions were found by Aczd, Maksa, and PBles in [24]. In the paper of Maksa, Marley, and PBles [I531 some strictly monotonic solutions of the functional equation

and the related equation

are found. A further related equation is

All strictly monotonic solutions of this equation are determined in PBles[164]. For the more general equation

it is proved by Gilanyi and PBles [59] that if the functions are strictly monotonic (all pi, Gi, and gi in the same way), then the functions pi and gi are differentiable, and other functions are strictly concave or strictly convex, moreover, f is continuously differentiable.

5 1. Introduction

27

Several problems concerning functional means leads to interesting composite equations. See the summary of some recent results about such type of questions in PBles [165]. Here we consider as an example only the MatkowskiSfito equation

All C2 solutions were found by Matkowski [155]. Quite recently Dar6czy and PBles found all strictly monotonic solutions of (11).

1.17. A general problem. The methods treated above suggest that equation 1.7.(2), the "most general non-composite explicit equation'' may be the central object of our investigations. If we suppose that f = f o = f l = f 2 = . . . = f,, then we can use steps (I)-(VI) of the "bootstrap" method of 1.7. Here there is no need to include the term fo(y) = f (y); if we omit it we do not restrict generality. The condition that the right-hand side really depends on y will be ensured by supposing that the rank of the matrices agi - is maximal. In this way we obtain the following fundamental regularity dy problem of non-composite functional equations: 1.18. P r o b l e m . Let X, Y, and Z be open subsets of R', RS, and Rt, respectively, and let D be an open subset of X x Y and let W be an open subset of D x Zn. Let f : X + Z, gi : D + X (i = 1 , 2 , . . . ,n ) , and h : W + Z be functions. Suppose that

(2) h is analytic;

(3) gi is analytic and for each x E X there exists a y for which (x, y) E D 8s. y) has rank r (i = 1 , 2 , .. . , n). and -(x,

dy Is it true that every f , which is measurable or has the Baire property is analytic?

1.19. R e m a r k : variants of t h e problem. It is possible to substitute the word "analytic" by 'Y!03"in the above problem. Clearly, this problem is even more important because most often regular solutions of functional

Chapter I. Preliminaries

28

equations are obtained by reduction to differential equations. More generally, we can investigate the "CP-problem", where 1 5 p 5 cc or p = w, where CWmeans the class of analytic functions (w > oo). Since it is not a strong restriction to suppose that the given functions are smoother than we expect the solution to be, it can be useful to investigate the "CP-Cqproblem" : We suppose that the given functions are in CP, and ask whether the solution f is in Cq? Here 0 5 q 5 w, 1 5 p 5 w, and q 5 p. For example, in some theorems we will assume that the functions gi are twice continuously differentiable to obtain that f is continuously differentiable, or to prove that f is continuous we will assume that the functions gi are continuously differentiable. Of course, more refined C P ? ~ - C Q > Pquestions can be also considered, where CP@ means the class of those CP-functions whose p-th derivative is locally Holder continuous with exponent a. 1.20. Remarks. (1) The Coo-version of the above general regularity problem 1.18 was formulated by the author (JArai [MI) in a somewhat weaker form, for the functional equation

The analytic version of the problem was included by J. Aczd in his survey talk on the XXI. International Symposium on Functional Equations among the most important open problems on functional equations (see Aczd [8], p. 256). The practice of the last fifteen years shows that the restriction X = Y introduced implicitly by the term f (y) does not help. Hence we will investigate here the above stronger form 1.18. (2) Earlier versions of this problem were formulated under the supposition that h is defined on the set D x Zn. Moreover, several theorems were proved for such type of h. Recent methods make it possible to treat the above stronger version, and in some applications the stronger version proved to be useful. Note that for those "local" theorems where all the functions are known to be continuous, the weaker version can be applied locally to obtain the stronger one, hence they are equivalent. (3) If instead of supposing that for each x E X there exists a y E Y such dgi that (x, y) E D and that -(x, y) has rank r for 1 5 i 5 n, we suppose that dy dgi the rank of -, 1 5 i 5 n is r for everywhere on D , and for each x E X

8~

there exists a y E Y such that (x, y) E D, we get an equivalent problem. Indeed, D could be replaced by the open subset of D l where the rank of each 8gi - is r .

8~

§ 1. Introduction

29

1.21. The above problem is well formed. We will prove by simple counterexamples that none of the conditions of the problem above can be omitted without introducing new conditions. This supports our calling this problem "the fundamental regularity problem of non-composite functional equations with several variables". On the other hand, these counterexamples show us the limits of possible changes in the conditions. The implicit equation

shows that no regularity is satisfied in general for implicit equations; every function f : R + R is a solution for which f (x) = 0 if x @ ]1,2[. Indeed, if x, y E ] 1,2[ then x y E ]2,4[, hence the product is zero. Some of these functions are measurable, but non-continuous, continuous but non-differentiable, etc. Another example, connected to the equation characterizing Weierstrass's sigma function can be found in Remark 24.3. Cauchy's power equation

+

shows that if the equation is implicit in the sense that substituting new variables we cannot convert it to the form f (x) = . . . , then also no regularity is satisfied in general. Indeed, here all functions f : R -+ R, f (x) = /xiCand f (x) = lxjCsgnx are solutions; c E R and OC understood to be 0. Some of these functions are measurable but non-continuous, etc. That the domain D of the functional equation in the above problem has to be open seems to be a technicality: otherwise it would be harder to speak about the differentiability of the inner functions gi. That it is not possible to omit this condition, even if the functions gi are very nice, is shown by the example of Cauchy's exponential equation. Substituting a new variable we obtain f b ) = f ( x - y ) f (Y). On any domain D containing the origin (0,O) and the open set D o = { (x, y) : 0 < y < x ) but contained in the closure of D o , the function f (0) = 1, f (x) = 0 for x > 0 is a measurable but non-continuous solution. If the outer function h in the above problem is not in CP, then the solution need not be in CP, 0 5 p 5 w . This is proved by the following simple example: Let H : R + R be a one-to-one Bore1 function, f : R -+ R,and let us consider Cauchy's functional equation for H o f , i. e.,

30

Chapter I. Preliminaries

This equation is equivalent to the equation

where h is defined by h(zl, z2) = H-' (H(z2) - H ( z l ) ) whenever zl, z2 E R Hence f is a measurable [continuous, etc.] solution of (1) if and only if f is a measurable [continuous, etc.] solution of (2). But i f f is measurable, then H 0 f is measurable too, hence with some constant c we have H ( f (x)) = cx, i. e., f (x) = H-'(cx) for all x E R. If H-I is measurable but non-continuous [continuous but non-differentiable, etc.] then we see that (2) has a measurable [continuous, etc.] solution, which is non-continuous [non-differentiable, etc.]. A very similar example shows that if at least one of the inner functions gi in the problem above is not in CP, 0 5 p 5 W , then the solution f need not be in CP. Let G : R -+ R be a one-to-one Bore1 function with the property that if A C R is a set having Lebesgue measure zero, then G-I (A) also has Lebesgue measure zero. Let f : R + R be a function and let us consider Cauchy's functional equation for f o G, i. e.,

Substituting x = G ( X ) , y = G(Y), we obtain that this equation is equivalent to the functional equation

+

where g is defined by g(x, y) = G (G-'(2) G-l(y)) whenever x, y E R. Hence f is a measurable [continuous, etc.] solution of (3) if and only if f is a measurable [continuous, etc.] solution of (4). But if f is measurable, then f o G is measurable too, hence with some constant c we have f ( G ( X ) ) = cX, i. e., f (x) = cG-'(x) for all x E R. If G-I is measurable but non-continuous [continuous but non-differentiable, etc.] then we see that (4) has a measurable [continuous, etc.] solution, which is non-continuous [non-differentiable, etc.]. Let us observe that in the special case G ( X ) = x3 we obtain that f (x) = cx1I3. In this case in equation (4) we have g(x, y) = (x1I3 y1/3)3. The function g is continuous on Rx R and infinitely many times differentiable at each point (x, y), x # 0 # y. Even the function y ++ g(0, y) = y is differentiable with a non-vanishing derivative. But for x = 0 there is no such y E R for which g is differentiable at (x, y). This shows that we cannot omit the differentiability of the inner function g. This example also shows that we cannot omit the condition that for each x there is a y such that (x, y) E D.

+

§ 1. Introduction

31

8% has to be equal to r , is shown by the dY simple example obtained from Cauchy's power equation by introducing new variables: f (4= f (y)f ( x ~ Y ) , x E R, 0 # Y E R. Finally, that the rank of

-

&l Here, for x = 0 there is no y, for which (0, y)

# 0,

where g(x, y) =

xly. Among real-valued solutions are f (x) = /xiCand f (x) = IxlCsgn x, c E R,where OC is understood to be 0. Some of these are measurable but non-continuous, continuous but non-differentiable, etc. The same equation yields a counterexample in higher dimensions, too, if x, y E RT, none of the coordinates of y is zero, and the division x l y is understood coordinatewise. Here among the solutions are the functions f (x) = fi(xi) where each function fi has the form fi(xi) = lxilCi or fi(xi) = Ixiici sgn xi. dgi has to be equal to r is Sincov's Another example that the rank of --89

nl=l

The general solution f : R x R -+ R of this equation is f (xl, x2) = g(x2)g ( x l ) , where g : R + R is any function. Indeed, any such function f is a solution. To see that an arbitrary solution f can be represented in this form, let us define g(x) = -f (x, c), where c E IR is an arbitrary constant. 1.22. Transfer principle: a simple example. There are several tricks that make it possible to use our results in cases where seemingly they cannot be used directly: see the chapter containing applications. There is one very important "trick" we have to mention here explicitly. Seemingly, equation 1.18.(1) is rather special in the following respect: it contains only one unknown function. But this restriction is not so strong as it seems to be. If we have a general non-composite functional equation with several variables and several unknown functions, and we can express each unknown function from it, then (after writing different variables in each equation) we may consider a vector-valued function having the different unknown functions as coordinates and use results concerning equations with only one, but vector-valued unknown function. This clearly shows how important it is to discuss vector-valued unknown functions with vector variables. Before we formulate this "transfer principle" in full generality, we illustrate it on the simple example of Pexider's equation

32

Chapter I. Preliminaries

From this equation we obtain the following three:

where with the notation u = ( u l ,u2, u 3 ) , v = ( v l ,~ 2 , 2 1 3 )the function h : R6 + R3 is defined by h ( u ,v ) = (u2 vs, vl - us, ul - v 2 ) . Let us observe that equation ( 1 ) satisfies all the conditions of problem 1.18. This example seems to be rather special, and it is not clear how to ~ defined by generalize it. So instead we use the function h : ( R ~-+ )R3

+

h ( ~ u27 l , u3,u4, u5r ~ 6 =) (u1,2+ u2,3,u3,l - u4,3,U 5 , l - u6,2) where ui = ( u ~U Q,, ui,3) E R3 for i = 1,2,3,4,5,6 and obtain the functional equation

where g is an arbitrary continuously differentiable function mapping R into R and having a non-vanishing derivative everywhere. Equation ( 2 ) also satisfies all the conditions of problem 1 .l8. We remark that the condition that all unknown functions have to be expressible from the original equation cannot be omitted. The following simple examples show this. By equation the functions f , f o , where f ( x ) = cx and f o is any function which is zero outside ]1,2[gives a solution. By the equation

f (4= f ( x + Y ) - f ( Y ) + f O ( e x ~ 3-) f o ( e X y 2 ) f 0 ( ~ ) , f , f o : R + the functions f , f o , where f ( x )= cx and f o ( x ) = 1x1C0 for some co E $ where O C O understood to be zero, yields a solution. Below we formulate the general transfer principle, which shows that the regularity problem of the general non-composite functional equation 1.7.(1), if it implies an explicit functional equation having the type 1.7.(2) for all the unknown functions, can be reduced to problem 1.18.

§ 1. Introduction

33

1.23. General transfer principle. Let Xi c R r i , Yi C PSSi , Zi C R t i , and Di C Xi x Y , be open sets, let ni be positive integers, f i : Xi -+ Zi be functions (i = 1,2, . . . , m ) . Suppose, that 1 5 ki,j 5 m is an integer and gi,j : Di -+ X k i I jis a function for 1 5 i 5 m and for 1 5 j 5 ni. Let moreover W i be an open subset o f Di x Zki,, x . . x ZkiIniand let hi : W i -+ Zi be a function for 1 5 i 5 m . Suppose, that for 1 5 i 5 m we have

and

whenever ( x i ,y i ) E Di. (In these functional equations the functions fi are considered unknown, other functions are considered known.) Then for the function defined by f (x) = ( f l ( x l ) ., . . , f m ( x m ) ) where x = ( X I ,... , x m ) and mapping the open set X = XI x X p x . . x X , C IWT , r = rl + - + rm into the open set Z = Z1 x Z2 x x Z , there exist integers n and s and there exist open sets Y c Rs, D C X x Y , W C D x Z n and functions g j : D -+ X ( 1 5 j 5 n ) , h : W -+ Z such that

and the functional equation

is satisfied whenever ( x ,y ) E D. The functions h and g j may be defined in a way that the following statements also hold true (Cw understood to be the class of analytic functions) : (1) i f each hi, i = 1 , .. . ,m , is in C P;

CP

for some 0 5 p 5 w, then h is also in

(2) i f each g i , j , i = 1, . . . ,m, j = 1, . . . ,ni is in C P for some 0 5 p 5 W , then g j , j = 1,2,. . . ,m are also in CP. (3) i f the functions gi,j are in C 1 and for each 1 5 i 5 m and for each xi E Xi dg, there exists a yi such that ( x i ,y i ) E Di and rank A ( x i ,yi) = rki,j for 15 j

5 ni, then for each x agj

and rank - ( x , y ) dy

E

= r for 1

X there exists a y

5 j 5 n.

a ~ i E Y such

that ( x ,y ) E D

Chapter I. Preliminaries

34

(4) if there are compact sets Ci C Xi given with nonvoid interior and yi can be chosen so that gi,j(xi,yi)E Cki,j for 1 5 i < m and 1 5 j 5 ni, then y can be chosen so that gj(x, y) E C for 1 < j 5 n , where C = C1 x C2 x . . . x C,.

<

Proof. First of all, let us choose numbers s:, 1 i 5 m such that si s{ 2 r j is satisfied for each 1 j m. Let Y,' = RS;. For each 1 5 i , k 5 m , let us choose an arbitrary analytic submersion g:,k of Yi x Y,' into the interior of Ck. (Remember that a differentiable map from an open set of some Euclidean space into another Euclidean space is called submersion, if the rank of its derivative is equal to the dimension of the range space everywhere.) For each 1 5 i 5 m and 1 5 j ni let us choose a permutation Pi,j mapping {1,2, . . . ,m ) onto itself, such that pi,j (ki,j) = i . Let us define gi,j,k by gi,j,k(xi,~i,y:)= gi,j(xi,yi) if (xi,yi,y;) E Di x Y,' and = If k f ki,j1 then let gi,j,k(x,i,j(k),~ p i . j ( k ) c = g~i,j(k),k(y~i,j(a);y~i,j(k)) whenever (xpi,,( k ) , YPi,j ( k ) E Dpi,j( k ) x Yii,,(k). With this definition, gi,j,k maps Dpi,j( k ) into X k . Finally, let us define gj, , where j' = n l nz . . ni-1 j so that gjt (x, y) is defined on the set D of those pairs (x, y) for which (xk, yk, y i ) E Dk x YL for k = 1 , 2 , .. . , m where x = (xl, x2,. . . , x,), y = (fh,U2, . . . ,Urn), and yk = (yk,yi) for k = 1 , 2 , . . . , m; the function gjt (x, y) is defined so that it has the coordinates gi,i,k(xpi,,( k ) , yPi,,(k), k = 1 , 2 , . . . ,m. Let us observe that by this definition,

+

< <

<

YL~,~(~))

+

+ +

+

Finally, with this notation, let W be the set of those points (x, y, zl , . . . ,z, ) for which (xi, yi, z ; , ~., . . ,x;,,~) E Wi for 1 i 5 m where is equal to the kilj-th coordinate of ~ ~ ~ + , . , + ,E~Z1 - ~x + j x Zm for 1 j ni; let the function h be defined such that its coordinates at the point (x,y, zl, . . . ,2,) are equal to hi(xi, pi, zll1, . . . ,ziIni),I 5 i 5 m.

<

< <

1.24. Summary of results about the main problem. The complete answer to problem 1.18 above is unknown. We may divide the problem into parts corresponding to steps (I)-(VI) in 1.7. The problems corresponding to steps (I), (11), (IV), and (V) are solved, and step (111) is solved under a weak additional compactness condition, which is usually satisfied by applications. Concerning step (VI) we have results only under rather strong conditions. We will treat steps (I), (11), (111), (IV), (V), and (VI) in $5 8, 9, 11, 14, 15, and 16, respectively. Other related regularity results for problem 1.18 can be found in 55 5, 6, 7, 10, 12, 13. Here we sum up some of the consequences of the more general results of these sections for problem 1.18.

§ 1. Introduction

35

1.25. Theorem. With the notation ofproblem 1.18, i f h is continuous and the functions gi are continuously differentiable then every solution f , which is Lebesgue measurable or has Baire property, is continuous.

Proof. This is a consequence o f Corollary 8.7 and Corollary 9.5 if we apply them locally. 1.26. Theorem. With the notation ofproblem 1.18, i f the functions h and gi are p times continuously differentiable, then every almost everywhere differentiablesolution f is p times continuously differentiable ( 1 5 p 5 m ) .

Proof. Follows from Theorems 14.2 and 15.2.

1.27. Theorem. With the notation of Problem 1.18, i f the functions h and gi arep times continuously differentiable,then every solution f which has locally bounded variation is p times continuously differentiable ( 1 5 p 5 CQ). Proof. From Theorem 1.25 it follows that f is continuous. Theorem 13.2 implies that f is locally Lipschitz. From Theorem 1.26 it follows that f is p times differentiable. 1.28. Theorem. With the notation of Problem 1.18, i f the functions h and gi are max(2,p) times continuously differentiable and there exists a compact subset C of X such that for each x E X there exists a y E Y satisfying gi ( x ,y) E C besides other conditions in 1.1 8. (3), then every solution f , which is Lebesgue measurable or has the Baire property, is p times continuously differentiable ( 1 5 p 5 m ) . Let us observe that the additional compactness condition in this theorem is retained by the transfer principle.

Proof. Follows from Theorems 1.25, 1.26, and 11.6.

1.29. Theorem. W i t h the notation o f Problem 1.18, i f t = 1, n = 2, g1 ( x ,y) = y, and the function h is p times and the function gz is max(2,p) times continuously differentiable, then every solution f , which is Lebesgue measurable or has the Baire property, is p times continuously differentiable (1 5 P 5 CQ). Proof. The statement is a consequence of Theorems 1.25, 11.4, and 1.26. 1.30. Theorem. With the notation o f Problem 1.18, i f equation 1.18.(1) has the following special form:

36

Chapter I. Preliminaries

where the functions hi : D x Z -+ Rt are p times, the functions gi are max(2,p) times continuously differentiable, then every solution f , which is Lebesgue measurable or has the Baire property, is p times continuously differentiable ( I 5 p 5 m).

Proof. This theorem summarizes Theorems 1.25, 11.3, and 1.26. 1.31. Theorem.

t

With the notation of Problem 1.18, suppose that

= 1, D = ]a, b[x]a, b[, the equation has the special form

with ci E R (i = 1 , 2 , . . . ,n ) , the functions

are given, and the following conditions are satisfied: (2) gi(x, y) is between x and y whenever x, y €]a, b[; (3) gi is analytic and with some constant 0 < A < 1 we have

whenever x, y €]a, b[ and p = 1 , 2 , . . . ;

-

(4) for all x €]a, b[ and i = 1,2, . . . ,n the mapping y t+ gi (x, y) of ]a, b[ into ]a, b[ is strictly monotonic, and the function gi, for which the mapgi(x, y), is twice ping z ++ gi(x, z) is the inverse of the mapping y continuously differentiable on its domain. Then every solution f which is Lebesgue measurable or has the property of Baire is analytic.

Proof. This theorem summarizes Theorems 1.30 and 16.2. 1.32. Regularity on manifolds. For functional equations arising from geometry or from physics sometimes the natural domain of definition of the functional equation is a manifold. Most of the results of 55 5-15 are general enough to apply to strictly nonlinear equations, vector-vector solutions, and have a local flavor. Hence we may expect that most of the results can be generalized to manifolds. For local results this does not cause any problem but global results need a new proof. In § 17 we will treat the generalization to manifolds. It is useful to reformulate the main regularity problem of noncomposite functional equations with several variables for manifolds:

§ 1. Introduction

37

1.33. Problem. Let X , Y, and Z be analytic manifolds, let D be an open subset of X x Y and let W be an open subset of D x Zn.Let f : X + Z , gi : D + X (i = 1 , 2 , .. . ,n), and h : W -+ Z be functions. Suppose that

(2) h is analytic;

(3) gi is analytic and for each xo E X there exists a yo for which (xo,yo) E D and y c, gi (xo,y) is a submersion at yo for i = 1 , 2 , . . . ,n. Is it true that every f , which is measurable or has the Baire property is analytic? We remark that here as in 1.19 it is possible to formulate the CP problem or the CP-Cq problem, etc., which include that the manifolds X , Y, and Z , too, are supposed only to be CP manifolds. 1.34. Summary of results on manifolds. The complete answer to the problem above is unknown. Just as problem 1.18, this problem can also be divided into parts (I)-(VI). The complete solution of the problems corresponding to (I), (11), (IV), and (V) easily follows from results of 55 8, 9, 14, and 15. We solve (111) for compact X ; in the noncompact case we will obtain results under some additional conditions. Our results can be summarized in the following theorems: 1.35. Theorem.

With the notation of Problem 1.33 if X , Y, and

Z are Cm manifolds, h is continuous and the functions gi are continuously differentiable, then every solution f which is measurable or has the Baire property is continuous.

Proof. It follows from Theorems 17.1 and 17.2 if we apply them locally. 1.36. Theorem. With the notation of Problem 1.33, if X , Y, and Z are Cm manifolds and the functions h and gi are p times continuously differentiable, then every locally Lipschitz solution f is p times continuously differentiable (1 5 p 5 m).

Proof. It follows from Theorems 17.6 and 17.7

38

Chapter I. Preliminaries

1.37. T h e o r e m . With the notation of Problem 1.33, if X , Y, and Z are C" manifolds, the functions h and gi are max(2,p) times continuously differentiable and there exists a compact subset C of X such that for each xo E X there exists a yo E Y, for which in addition to condition (3) of the problem above gi(xo,yo)E C (i = 1 , . . . , n ) then every solution f which is measurable or has the Baire property, is p times continuously differentiable (1 l p l 00).

Proof. It follows from Theorems 1.35, 1.36, and 17.3. 1.38. T h e o r e m . With the notation of the Problem 1.33, if X, Y, and

Z are Cw manifolds, X is compact, moreover h and gi are Cm functions, then every solution f which is measurable or has the Baire property is in C". Proof. It follows from the previous theorem. 1.39. T h e o r e m . With the notation of Problem 1.33, if gl (x, y) = y, n = 2, X , Y, and Z are C" manifolds and dim(Z) = 1, the function h is p, and the function g2 is max(2,p) times continuously differentiable, then every solution f which is Lebesgue measurable or has the Baire property, is p times continuously differentiable (1 p 5 m ) .

<

Proof. This is a consequence of Theorems 1.35, 17.5, and 1.36. 1.40. T h e o r e m . With the notation of Problem 1.33, if X and Y are C" manifolds, Z is an open subset of Rt, and the functional equation has the following special form:

where the functions hi : D x Z + Rt are p times continuously differentiable and the functions gi are max(2, p) times continuously differentiable, then every solution f which is Lebesgue measurable or has the Baire property, is p times continuously differentiable (1 5 p 5 m ) . Proof. This theorem summarizes Theorems 1.35, 17.4, and 1.36. 1.41. Regularity results w i t h fewer variables. Roughly speaking, all the results of § § 5-17 prove regularity of an r place function f which is the solution of a functional equation only if there are at least 2r variables in the functional equation. The reason for this, using notation of Problem d g i is equal to r, the dimension 1.18, is that the condition that the rank of -

8~

§ 1. Introduction

39

of X , implies that s, the dimension of Y cannot be less than the dimension of X . The last two examples of 1.21 show that the condition about the dgi rank of - cannot be simply omitted. Nevertheless, that, at least in some dy cases, the number of variables can be reduced down to r 1 is shown by some regularity results of ~ w i a t a k . (The case of r variables in a functional equation for an r place function would already mean a functional equation in a "single variable" .) ~ w i a t a khas proved general regularity results having the type "continuous solutions are C"". She obtained her results using distributions. The essence of her method is to prove that solutions in the distribution sense satisfy a linear partial differential equation having only infinitely many times differentiable solutions. This idea was used by ~ w i a t a kto prove general regularity results for the functional equation

+

where f is the only unknown function. Roughly speaking, she applies a partial differential operator d; in y to the equation in the distribution sense. Of course, the nonlinear term on the right-hand side disappears. If there exists a yo such that gi(x, yo) x for 1 5 i 5 n (a very strong assumption), then after substituting this fixed yo we obtain a linear partial differential equation P ( d , x) f = H. If the partial differential operator P ( d ,x) is of constant strength and hypoelliptic at an x = xo, then by the regularity theory of partial differential equations all distribution solutions correspond to a Cm function. We will briefly discuss the results of ~ w i a t a kin 5 18. Generalizing our methods we may hope to obtain regularity results for general nonlinear functional equations. This seems to be impossible using the method of ~ w i a t a kbased on Schwartz distributions. We can avoid the very strong assumption that there is a yo such that gi(x, yo) = x for 1 5 i 5 n. The somewhat artificial condition of hypoellipticity also has to disappear. We prove "measurability implies continuity" type results for nonlinear functional equations with "few" variables in § 19. Well-known analogies between measurability and Baire property (see Oxtoby [161]) suggest trying to prove analogous results for Baire property. Differences, such as the lack of the %-technique", the lack of convergence in measure and theorems connected with it (for example Riesz' theorem), the lack of Hausdorff measure, etc., justify a separate discussion. "Baire property implies continuity" type results for general nonlinear functional equations with "few" variables will be

=

40

Chapter I. Preliminaries

proved in 5 20. Finally, in § 21, we will prove "continuity implies continuous differentiability" and "p times differentiability implies p 1 times differentiability" type results for functional equations with "few" variables. In the latter case, our theorems are applicable to general nonlinear equations. In the former case, they presently are applicable only to linear equations, but without several other strong assumptions. We will also give some examples and investigate the connection with the results of ~ w i a t a k .

+

2. NOTATION A N D TERMINOLOGY 2.1. Sets and functions. The cardinal number of a set A will be denoted by card(A). If card(A) 5 No, we will call A countable. The cardinal number continuum, i. e., 2'0 will be denoted by c. We will denote by N,23, Q, R, C, and T the set of nonnegative integers, integers, rational, real, complex, and unit length complex numbers, respectively. I f f is a function, dmn f and rng f denote the domain and the range of f , respectively. The multiplicity function Nf is defined by Nf(y) equal to the number of elements of the set f -'(y) (possibly m). We will also use the notation { f = y) = f -l(y) and analogously {f > y), {f E A), {f = g), etc. All normed spaces are supposed to be real; the norm is denoted by I 1. The sign I/ (1 used only for the operator norm of a linear operator. I f f : D + Y is a function mapping an open subset of a normed space into a normed space, then f ' will denote the derivative of f . If D c X I x Xg x x X,, we will use the partial sets

The partial functions fxi : DXi + Y are defined by

whenever ( x l , . . . ,x,) E D. Also Dxil,... and fxil,... larly. Now, if Xi and Y are normed spaces and

are defined simi-

is an open subset of Xi, we define the partial derivative denoted by

5 2.

Notation and terminology

as the derivative of f,, ,,,,,,i-l,,i+ dex, let

41

(if it exists). If a E W is a multiin-

,,, n

If D C Rn and Y = Rm, then f ' may be identified with the matrix

where f = (fj)y=l and x = If y = f ( x ) , 1 5 i l , . . . ,i, 1 5 jl , . . . , j, 5 m, then we denote the determinant of the matrix

5 n,

a ( ~ j , r . . i. ~ j , ) d(xil1. . - xi,) ' Other notions of calculus are used in the usual way. In case of doubt the reader is referred to Rudin [172]. 2.2. Topology. We will use the terminology and notation of Bourbaki [38] concerning topology and topological groups. Hence, every regular, completely regular, normal, compact or locally compact space is supposed to be Hausdorff. For any class E of subsets of a set X , let E, and Isdenote the class of all countable unions and countable intersections of members of El respectively; the empty union is the empty set and the empty intersection is X. The topology of a topological space, i. e., the class of all open sets, will be denoted by 5;, and the class of closed sets will be denoted by F. So the classes G6, F,,, GsU = (6s)alFU6 = etc. are defined. A path in X is a continuous function mapping some closed interval of R into the topological space X. For a topological space X the weight of X is the least cardinal number of an open basis. The character of a topological space X is the least cardinal number n for which there is an open basis at x with cardinal number not greater than n whenever x E X. If x, y are points of a metric space and a > 0, we say that x and y are a-near if their distance dist(x, y) is less than a. Similarly, if x and y are points of a uniform space and a is a relation from

Chapter I. Preliminaries

42

the uniformity we say that x and y are a-near if (x, y) E a. In a metric space the closed ball having radius r 2 0 and center x will be denoted by I& ( x ) . Concerning topological dimensions dim, ind, and Ind we refer the reader to the classic book of Hurewicz and Wallman [76]. All results concerning topological groups used here can be found in the monograph [66], [67] of Hewitt and Ross.

2.3. Continuity modulus, Holder a n d Lipschitz functions. For a function f mapping a subset of a metric space into a metric space we define the continuity modulus wf of f by

<

whenever 0 5 r m. If the domain of f is not empty then always wf ( 0 ) = 0. It may happen that (even for every positive r ) we have w f ( r )= m. Clearly wf is continuous (from the right) at 0 if and only i f f is uniformly continuous. If for some exponent 0 < a 5 1 and constant 0 5 Ha < m we have w f ( r )5 H , . r f f whenever 0 5 r < oo, then we say that f is Holder continuous with exponent a and Holder constant Ha. If a = 1, the Holder continuous functions and Holder constants are called Lipschitz functions and Lipschitz constants. We will use the local notion of Holder continuity, Holder constants, Lipschitz function, and Lipschitz constant. In connection with Lipschitz and locally Lipschitz functions see the book of Federer [53]. 2.4. M e a s u r e theory. Concerning measure theory, we follow the terminology of Federer [53]. Hence a measure means a countably subadditive extended real-valued nonnegative function defined on all subsets of a set; this is called outer measure in other terminology. The a-algebra of measurable sets is defined by the Carath6odory condition: a set A C X is called p measurable if p(T n A ) ,u(T \ A ) = p ( T ) for every T C X . The measure p is called finite if p ( X ) < m. If p ( X ) = 1, then we say that p is a probability measure. If a set A C X can be represented as a countable union of measurable sets having finite measure, then it is called a-finite. If X itself is a-finite, then the measure p is called a-finite. A measure is called diffuse if any set of only one element has measure 0. If Y C X, the restriction of p to 2Y is a measure on Y denoted by py. Let p [Y defined by ( p[ Y )( A ) = p(Y n A) for a11 A C X . Every function f : X -+ Y induces a map f# of measures over X into the class of measures over Y defined by ( f # p )( B ) = p (f - ' ( B ) ) for B C Y. If a nonnegative set function v is defined on an arbitrary class 3-1 of subsets of a set X , then

+

p ( T ) = inf{CiEI v ( H i ) : I is countable, Hi E 3-1 if i E I}

5 2.

Notation and terminology

43

for each T c X defines a measure p on X . (Empty sum is understood to be zero.) The set function p is an extension of the set function v if and only if v is countably subadditive. It is not hard to prove that a set A C X becomes p measurable if and only if

whenever H E 31. Specially, if 31 is a a-algebra and v is countably additive on 31 then p is an extension of v and all H E 31 is p measurable. If p is a measure on X we say that the measurable set B c X is a p hull ofAifACBCXand

p (A n T) = p ( B n T)

for every p measurable set T.

A measure p over X is called regular if for each set A C X there exists a p measurable set B such that A c B and p (A) = p ( B ) . Let p be a measure on X . A base of the measure p is a class B of p measurable sets such that for any p measurable set A and for any E > 0 there exists a B E B such that p(A A B ) < E. The weight of the measure p is the least cardinal number of a basis of p. The character of p is the least cardinal number n for which p y has a base with cardinal number not greater than n whenever Y is p measurable with finite p measure. We remark that if p is a finite measure, then the weight of p and the Hilbert space dimension of the corresponding L2 space are equal, whenever one of them is infinite. (See Hewitt and Ross [66], (16.12).) If p and v are measures over X we say that the measure p is an expansion of the measure v if any v measurable set A is also p measurable and p(A) = v(A). If p is a measure on a group X , we say that p is left [right] invariant, if for each z E X and A c X we have p(xA) = p(A) [p(Az) = ,u(A)]. An will denote the Lebesgue measure on Rn, and X h i l l denote the kdimensional Hausdorff measure on a metric space. The Lebesgue density of a set A c Rn at a point x E Rn is defined as lim,40 An (I$.(x) n A) / A n (I++ (x)) if the limit exists. Let Y, be a measure on X and let Y be a topological space. A function is called p measurable on a set A c X if its domain contains almost all of A, its range is in a topological space Y, and if A f? f -'(v)is p measurable whenever V is an open subset of Y. Especially, A has to be measurable. We refer the reader to Federer [53] concerning measure theoretical results used here.

Chapter I. Preliminaries

44

2.5. Topological measures. Our terminology concerning topological measures is somewhat different from that of Federer's book [53]. We call a measure p over a topological space X topological measure if all open subsets of X are p measurable. Of course, if p is a topological measure on X, then Borel sets of X - elements of the smallest a-algebra containing all open sets - are measurable. The support of the topological measure p is defined by sptp = x

\ U{V

: V

cX

open, p(V) = 0 ) .

A topological measure is called Borel regular if for each A C X there exists a Borel set B such that A C B C X and p(A) = p ( B ) . By a Radon measure we mean a topological measure p defined on a Hausdorff space X, with the following properties: (1) The p measure of any compact subset K of X is finite;

(2) for every open subset V,

(3) if A is any subset of X , then p (A) = inf { p (v): A

c V,

V open).

Clearly, a Radon measure is uniquely defined by its values on compact sets. Let p be a Radon measure over X and A C X . The set A is p measurable if and only if P(V n A) \ A) 5 P(V)

+

for any open set V. Indeed, if this condition is satisfied, then for any T C X we have

whenever T c V and V is an open set. Taking infimum for all V containing T , we obtain that A is measurable. Similarly, A is p measurable if and only if P ( K n A) + P ( K \ A) i p ( K ) for any compact set K . Indeed, let V be an open set for which p(V) < cc, and let K, be compact subsets of V for which p ( V \ K,) < l l n . Then p ((vn A) \ (K, r l A)) p(V \ K,) < l l n , and hence p(K, r l A) 4 p(V rl A)

<

5 2. as n -+ m. Similarly, p(K, is satisfied, then

Notation and terminology

\ A) -+ p(V \ A).

45

Hence, if the above condition

p(V) = lim p(K,) = lim p(K,nA)+ lim p ( K n \ A ) = p ( V n A ) + p ( V \ A ) . n+m

n+m

n-+m

Finally, A is p measurable if and only if A n K is p measurable for all compact subsets K . Indeed, this condition implies that

for any compact set. It is not hard to prove that if p is a Radon measure, then p(A) = s u p { p ( ~ :) K

c A, K compact)

whenever A is a p measurable set with finite p measure. 2.6. Lusin measurable functions. Let p be a Radon measure on a Hausdorff space X and let Y be a topological space. Let f be a function mapping almost all of a subset E of X into Y. The function f is called a Lusin p measurable function on E, if for each measurable subset A of E having finite measure and for each E > 0 there is a compact subset C of A such that p ( A \ C ) < E and f lC is continuous. In this setting Lusin's theorem reads as follows (see moreover Federer 1531, 2.3.6 for further weakening of the conditions) :

2.7. Lusin's Theorem. Let p be a Radon measure on a Hausdorff space X , let E be a p measurable subset of X and let Y be a topological space. A function f mapping a subset of X into Y which is Lusin p measurable on E , is p measurable on E. Conversely, i f f is p measurable on E and the values o f f are for almost all of E in a subspace of Y whose topology has a countable base, then f is Lusin p measurable on E .

Proof. We have to prove that if V is an open subset of Y, then E n f -'(v) is measurable. It is enough to prove that ~ f lf -'(v) ~ n is measurable whenever K is a compact subset of X . The set K n E can be written as the union of a null set N and a sequence of compact sets C, such that f lC, is continuous. Now Cn n f -'(v) is relatively open in C,, hence a Bore1 set. This implies the statement. In the proof of the converse we follow the proof of Oxtoby [161], 8.2. Let V, be a countable base of the given subspace of Y, let A be a measurable subset of E and E > 0. By approximating A from the inside with a compact

46

Chapter I. Preliminaries

set, we may suppose that A is compact. Let us choose for each n an open set U, and a compact set C, such that Cn C A n f (v,) c Un and p(Un \ C,) < ~ 1 2 , . If C = A \ Up?l(Un \ Cn), then C is compact, p(A \ C ) < E and f-'(v,) n C = U, ri C is relatively open in C . This implies that f lC is continuous.

-'

2.8. Haar measure. On a locally compact group G, there exists a left [right] invariant Radon measure X for which the measure o f any nonvoid open set is nonzero. A n y such measure is called a left [right] Haar measure. For any two such measures A, A' there is a constant 0 < c < cc such that A' = cX. Moreover, there exists a unique continuous homomorphism A o f G into the multiplicative group o f the positive real numbers (the so-called modular function o f G) such that i f X is a left Haar measure, x E G and L is X measurable, then Lx and L-' are also X measurable

and

X(L-') =

A(x-')dX(z).

On a locally compact group left Haar measure is usually denoted by A.

2.9. Baire category. The most important facts concerning Baire category can be found in Bourbaki [38];see chapter IX, § 5, and the corresponding exercises, but here we will use the different (and more usual) terminology of Oxtoby [161]. For clarity we summarize the notions and facts we will use. We will say that a subset A of a topological space X is of first category, if A can be represented as a countable union of nowhere dense sets (i.e., sets having closure with empty interior), otherwise A is of second category. It is easy to see that if X is a topological space, Y is a subset of X , A is a subset of Y, and A is nowhere dense [is of first category] as a subset of the subspace Y, then A is nowhere dense [is of first category] as a subset of X , too. Conversely, if Y is an open subset, and A is nowhere dense [is of first category] as a subset of X, then A is nowhere dense [is of first category] as a subset of Y, too. A topological space X is called Baire space if the intersection of any countable class of dense open sets is dense. The theorem of Baire states that locally compact spaces and complete metric spaces are Baire spaces. We note that the topology of a subspace of a complete metric space can be derived from a metric of the subspace for which it is complete if and only if the subspace is a Gs subset of the space.

5 2.

Notation and terminology

47

Let E be a subset of the topological space X. We say that E is of second category at a point x E X if E n V is of second category for every neighborhood V of x. Let D ( E ) denote the set of all points x E X for which E is of second category at x. Then D ( E ) = 0 if and only if E has first category. Moreover D ( E ) is closed and the set E \ D ( E ) is of first category. We will say that E C X has the Baire property if there exists an open set V such that the symmetric difference E A V is of first category. All subsets of X having Baire property form a a-algebra. Of course this a-algebra contains Borel sets, the members of the smallest a-algebra containing all open sets. With the notation of the previous paragraph, a set E C X has the Baire property if and only if E \ D ( E ) is of first category. The set E has Baire property if and only if each point x of X has an open neighborhood U such that U Ti E has the Baire property in X. Let X and Y be topological spaces and suppose that the topology of one of them has a countable base. A subset E x F of X x Y has the Baire property if and only if one of the sets E, F is of first category or both of them have Baire property. Combining these facts with the proof in Oxtoby [161], chapter 15, we get the following form of a well-known theorem of Kuratowsky and Ulam: 2.10. Kuratowsky-Ulam T h e o r e m . Let X and Y be topological spaces, and suppose that Y has a countable base. Let E be a subset of X x Y having the Baire property. Then except for a set of points x of X which is of first category, the set Ex has the Baire property. Moreover E is of first category if and only if the set Ex is of first category in Y with the exception of a set of x 's of first category.

2.11. B a i r e p r o p e r t y of functions. The function f has the Baire property on E if the domain of f contains E except for a set of first category, the range of f is in a topological space Y and E n f -'(v)has the Baire property in X for every open subset V of Y. We simply say that f has the Baire property, if it has the Baire property on X. This definition is very similar to the definition of a Borel function. A function f mapping some subset of a topological space X into another topological space Y is called a Borel function, if for each open subset V of Y the set f - ' ( ~ ) is a Borel subset of X. The properties of functions having the Baire property are very similar to the properties of measurable functions. We will use the following statements.

If f is any function defined on a subset of X having the Baire property, then all subsets B of Y for which f - ' ( B ) has the Baire property form a

48

Chapter I. Preliminaries

a-algebra. Hence i f f has values in the topological space Y and has the Baire property on a subset of the topological space X , moreover i f g is a Borel function on a subset o f the topological space Y , then g o f has the Baire property on its domain.

ni

Suppose that Y = Y, is a countable product o f spaces each having a countable base o f topology. A function f mapping a subset o f a topological space X into Y has the Baire property i f and only i f all functions pi o f have the Baire property where pi is the natural projection o f Y onto Y i . Suppose that X is a topological space, Y a metric space, and the functions f , f n ( n = 1 , 2 , . . . ) are defined on X except for a set o f first category (depending on the function), have values in Y , and have the Baire property. Then the set E = { X : fn(x) + f ( x ) ) has t h e Baire property. Indeed, the sets

have the Baire property and E differs from the set

only in a set of first category.

Suppose that X is a topological space, Y is a metric space, the functions f , f n ( n = 1 , 2 , . . . ) are mappings from X into Y , the functions f n have the Baire property for all n, and f n ( x ) + f ( x ) for all x E X except for a set o f first category. Then f has the Baire property, too. Indeed, if V is an open subset of Y and V, is the open subset of Y containing those points of Y having distance larger than l / i from Y \ V , then f -'(v)differs from the set

only in a set of first category. We say that a function mapping some subset E of a topological space X into another topological space Y has the Lusin-Baire property on E , if there

5 2.

Notation and terminology

49

exists a set F of first category such that f I E \ F is continuous. The following theorem is analogous to Lusin's theorem: 2.12. Theorem. Suppose that X and Y are topological spaces and E is a subset o f X which has the Baire property. I f a function f mapping some subset o f X into Y has the Lusin-Baire property on E , then f has the Baire property on E. Conversely, i f f has the Baire property on E and the values o f f for all points o f E except a set o f first category are in a subspace o f Y whose topology has a countable base, then f has the Lusin-Baire property on E .

Proof. Suppose that F is a set of first category and let V be any open subset of Y. Since f 1 E \ F is continuous, (E \ F) n f (v) has the form U n E \ F for some open subset U of X, hence it has the Baire property. Since F n f-'(V) is of first category, it also has the Baire property. Hence E Ti f -l(V) also has the Baire property. Conversely, let V, be a countable base of the topology of the given subspace of Y. Since E n f -'(Vn) has the Baire property, it can be written A are of first category. Now let in the form U, \ F, U FA, where F, and F F = Up?=, (F, U FA). Then ( E \ F) f l f -'(Vn) = (E \ F) n U,, hence it is relatively open in E \ F . Since V, is a basis, this means that f lE \ F is continuous. From the Kuratowski-Ulam theorem the following analogue of Fubini's theorem follows directly: 2.13. Theorem. I f X , Y, and Z are topological spaces, Y and Z have countable bases, f : X x Y + Z has the Baire property, then except for a set o f points x which is o f first category, the function f , has the Baire property. 2.14. Bounded variation. Besides the elementary notion of bounded variation function of a real variable we will use the notion of bounded variation for functions having vector variables. Following the book of Giusti [60], if X c Rn is an open set, we will say that the function f : X + R has bounded variation if it is integrable and

f divg:g

E

J C ' ( X ; R ~and ) lgl 5 1

is finite; here K1(X;R n ) denotes the class of all continuously differentiable functions having compact support mapping X into Rn, and / / is the norm on R n . If f is continuously differentiable, then the expression above is equal to

Chapter I. Preliminaries

50

Sx

I f ' l . We will use the same symbol to denote the above supremum for arbitrary functions having bounded variation. An equivalent condition to having bounded variation would be that the gradient of f in the distributional sense is representable by integration with respect to a vector-valued finite Radon measure. Jx 1 f ' 1 is the total variation of this measure. Strictly speaking, this notion is not a generalization of the usual notion of a one variable function having bounded variation on an interval. But rather it is a generalization of the notion of a function having bounded essential variation. This connection and the basic properties of functions having bounded variation can be found in Giusti [60]. We will use results from this book only. Further results can be found in Federer [53], 4.5.9, 4.5.10. We say that a vector-valued function with values in a Euclidean space has bounded variation if all its coordinates have bounded variation. We will say that f has locally bounded variation on X if each point of X has a neighborhood such that the restriction of f to this neighborhood has bounded variation. 2.15. Manifolds. In general we will use the terminology of Dieudonnk [49] about (Cm or analytic) manifolds. Hence, a manifold is always understood to be finite dimensional, separable, and metrizable. We use the notion of a Banach manifold as defined by Zeidler [209], book IV, chapter 73. Concerning Lebesgue measures, Lebesgue measurability, null sets on manifolds we follow the terminology of Dieudonnk [49],see 16.22.2. A Lebesgue measure p on a pure n-dimensional Cm manifold X is characterized by the property that if cp is a chart, then a set A C dmn(p) is p measurable if and only if cp(A) is Lebesgue measurable in Rn; moreover for the chart cp there exists a positive real valued Cm function f , defined on rng(cp) such that p (A) =

J

f , (x) dXn (x),

whenever A C dmn(p) is measurable.

4.4)

Although the notion of a Lipschitz function cannot be transferred to manifolds, this is possible for the notion of a locally Lipschitz function: a function f : X 4 Y between manifolds is called a locally Lipschitz function if for each x E X there is a chart cp on a neighborhood of x and there is a chart $ on a neighborhood of f (x) such that $ o f o p-I is a Lipschitz function on a neighborhood of cp(x). Then for any other charts @ and defined on a neighborhood of x and f (x), respectively, it is also true that of o is a Lipschitz function on a neighborhood of @(x).It is not hard to prove that if X and Y are connected, then choosing arbitrary Riemannian structures on X and Y compatible with the manifold structure, the function f is a locally Lipschitz function by the definition above if and only if it is

4,

4

§ 2. Notation and terminology

51

a locally Lipschitz function with respect to the distance resulting from the Riemannian structures. 2.16. Distributions. We will use the standard notation of the book of Hormander [74] about distributions. The class of test functions and distributions will be denoted by D and Dl (or more precisely D(Rr) and D1(Rr), etc., if necessary), respectively. All results we will use about distributions can be found in Hormander's book [74], chapters I-VI. We remind the reader that if f 6 D'(Rr) and h : Rr + C is a Cm function, then h f 6 Dl is defined by ( hf ) ( p ) = f (hp) whenever p E D. Moreover, if g : Rr + IWT is a diffeomorphism, then the distribution f o g is defined by , ~ ( g - l )denotes the absolute value (f 0 g) ( p ) = f ( ( p o g-I) ~ ( g - I ) ) where of the Jacobian of g-l.

Chapter 11.

STEINHAUS TYPE THEOREMS

The generalizations of Steinhaus's theorem and its Baire category analogy, Piccard's theorem, play a key role in several later sections as well as certain applications. This chapter deals with these generalizations.

3. GENERALIZATIONS OF A THEOREM OF

STEINHAUS A famous theorem of Steinhaus [I891 from 1920 asserts that, for any measurable set A c PS with positive Lebesgue measure the set A - A contains an interval. More generally, if A, B C Rn are measurable sets with positive Lebesgue measure, A B has an interior point; see for example the paper of Kemperman [123]. The proof can be based on Weil's idea [206] that the convolution of the characteristic functions XA and XB (in case A and B has finite measure) is a continuous function, hence the function

+

Chapter 11. Steinhaus type theorems

54

is continuous and as follows from Fubini's theorem, not everywhere zero. This means that A B contains a nonvoid open set. This proof works directly when X is a Haar measure on a locally compact group. This theorem allows various generalizations and modifications. In the generalizations the following problem is treated: if we replace the addition by a binary operation F ( x , y), under what conditions on F can we prove that F ( A , B) contains a nonvoid open set? The first step was done by Erdos and Oxtoby [52] proving in the case x, y E R that, if F is a continuously differentiable function with nonvanishing partial derivatives, then F (A, B) contains a nonvoid open set. Further generalizations detail the case when x and y are from different topological measure spaces and F satisfies certain solvability conditions in x and y. See in this direction Kuczma [I301 and Sander [178]. Sander has pointed out that one of the sets A, B may be nonmeasurable. These results apply to the case when x, y E Rn and F is a continuously differentiable function of which the partial derivatives are nonsingular. In this 5 we will treat a generalization for function F with more then two variables. Of course, if F maps IR x R x R into R we obtain a problem already solved by the theorem of Erdos and Oxtoby. To obtain a really interesting new problem, we have to consider a function with values in R2. The condition about the nonvanishing partial derivatives will be substituted with the condition that the null space of the derivative (as linear mapping) is in general position. Theorem 3.11 and Remark 3.13 are such generalizations of Steinhaus' theorem. As a corollary we obtain the well-known special case F : R n xRn + W . Our proof depends on a very general version of the theorem about the continuity of the convolution formulated in Theorem 3.4. As corollary we obtain a result about the continuity of a mapping

+

where the functions gi,t do not map sets with positive measure into zero sets and depend smoothly on the parameter t . The investigation of this mapping occurs in an implicit way in JArai [go] and in an explicit way in Krausz [125]. As we explained in the introduction, the lower semicontinuity of mapping (1) and several other variants of Steinhaus' theorem have applications in the theory of functional equations. The results of this 5 are used in $5 5, 6, 8, 10, and when discussing applications in 5 23. More detailed references about earlier results may be found in Kuczma [130], Kuczma and Kuczma [131], Sander [178], and Grosse-Erdmann 1611.

5 3.

55

Generalizations of a theorem of Steinhaus

Another kind of refinement of Steinhaus' theorem using one-sided lower densities in R is given by Raikoff [I681 and by Matkowski and ~ w i ~ t k o w s [156]. ki The results of this § are published in the paper JBrai [99]. We start with the investigation of an important condition. 3.1. Condition. In this 5 and in $5 8, 10, and 19 a measure theoretical condition will play an important r61e. In general form this may be formulated as follows.

Let X and Y be sets with measures p and v, respectively, let T be a set, x Y, and g : D + X a function. Our condition is the following: (1) For each E > 0 there exists a 6 > 0 such that if B C Dt, v(B) E, t E T , then p (B)) 2 6.

D

cT

>

Concerning this condition we summarize some simple properties. We treat this condition in 3.10 in the most important case when X and Y are open subsets of Euclidean spaces. Further results can be found in J&rai [80]. 3.2. Remarks. With the notation of the previous point,

if Dl C D2 C T x Y, g2 : D2 3.1.(I), then gl too;

-+ X , g1 = g2/ D l , and g2 satisfies condition

if D = D l U D2, g1 = 9101, 92 = g10 condition 3.1.(1), then g too;

2 ,

moreover gl and g2 satisfies

if for all E > 0 there exists a decomposition D = D l U D2 such that 9102 satisfies condition 3.1.(1) and p (gt ( ~ 1 , ~<) E) for all t E T, then g also satisfies condition 3.1. (1); condition 3.1. (1) is equivalent with the following one: for all E > 0 there exists a 6 > 0, such that if A c X , p(A) < 6, then v ( g , l ( ~ ) ) < E for all t E T; if p is a regular measure, then it is enough if (4) is satisfied for all measurable A; if X and Y are Hausdorff spaces, p and v are Radon measures, all gt is continuous on the Borel set Dt and the condition 3.1. (1) is satisfied, then gt-l(A) is measurable for all a-finite measurable set A C X whenever t E T; under conditions of (6) if B c A and A is a ,LL hull of B , then g,l(A) is a v hull of g , l ( ~ ) whenever t E T;

Proof. (1)-(5) are trivial. To prove (6) let us represent A as the union of a Borel set B and a set C having measure zero. Then g;l (A) = g,l ( B ) U

56

Chapter 11. Steinhaus type theorems

( C ) where g,l(B) is a Bore1 set and 9,' ( C ) has measure zero. To prove (7) suppose indirectly that for some measurable set E C Y we have

Then by the measurability of g r l ( A ) and since v is a Radon measure there exists a compact set C c E n g,l(A \ B ) such that v ( C ) > 0; but by gt ( C )c A \ B we should have p (gt ( C ) )= 0. 3.3. Lemma. Let T be a topological space, X a uniform space, C a compact uniform space, D C T x C , to E T , { t o ) x C C D , and let g : D + X be a continuous function. Then for any relation a from the uniformity of X there exists a relation ,B from the uniformity of C and a neighborhood V of to such that i f t , t' E V, the points y , y' are ,&near in C and ( t ,y ) , (t',y') E D , then the points g ( t , y ) and g(t', y') are a-near in X .

Proof. Let 6 be a symmetric relation from the uniformity of X for which 6 o 6 c a. For all y E C there exists a neighborhood V, of to and a symmetric relation y, such that if t E V, and the point y' is 7,-near to y , then g ( t , y') and g(to, y ) are 6-near in X . Let us choose for each y, a symmetric /, o ,By C y,. The open kernels of the sets ,By ( y ) give relation ,By for which 3 ,By,, an open cover of C . Let us choose a finite subcover and let ,B = V = V,,. If the points y and y' are p-near in C , then there exists a yi such that y and yi, moreover y' and yi are yyi-near, hence g ( t , y ) and g ( t o ,yi) moreover g ( t ' , y') and g ( t o ,y i ) are &near. This means that g ( t , y ) and g ( t l , y') are a-near in X .

nkl

3.4. Theorem. Let T be a topological space, Y a Hausdorffspace, X i (i = 1 , 2 , . . . ,n ) be completely regular spaces, and let Z , Zi(i = 1 , 2 , . . . ,n ) be Banach spaces. Let v and pi be finite Radon measures over Y and X i , respectively. Consider the functions fi : X i + Zi, gi : T x Y Xi, h : Z1 x . x Zn + 2.Suppose that the following conditions hold: (1) h maps bounded subsets into bounded subsets and is continuous; (i = 1 , 2 , . . . ,n ) ; (2) fi is Lusin pi measurable and is in L m ( p i ) (3) gi is continuous and for each E > 0 there exists a 6 > 0 such that p i ( g i , t ( B ) ) 2 6 whenever B c Y , v ( B ) E , t E T , and 1 5 i 5 n. Then the function

>

is continuous on T .

§ 3. Generalizations of a theorem of Steinhaus

57

Proof. First we prove that the integral exists. We may replace each fi by a bounded Borel function defined on all of Xi, which is almost equal to f i , and is separable valued. This switch does not change the integral, because, by (3), the set of points y for which the value of fi (gi (t,y)) are changed, has measure 0. Hence we may assume that the functions fi are separable valued bounded Borel functions. By (1) the function

is a separable valued Borel function whenever t E T is fixed, and its image is a bounded subset of 2.Hence the integral exists whenever t E T . Now let E > 0 and to E T. Let us choose a real number M > 0, for which the image of (4) is contained in the closed ball with center 0 and radius M . By (3), there exists a 6 > 0 such that B C Y, u(B) 2 E' = ~ / ( 1 6 M n ) t, E T , and 1 5 i 5 n implies pi ( g i , t ( ~ )2 ) 6. Let us choose a compact set C c Y for which u(Y \ C ) < ~ l ( 8 M )There . exists a compact set Ci in Xi for which pi(Xi \ Ci) < S and filCi is continuous. Let us choose uniformities on the spaces C , X I , . . . , X, compatible with their topology. By (1) there exists an a! > 0 such that

z E fi(Ci) and lzi - zQ) < a. Because of the uniform continuity whenever zi, Q of fi ICi there exists a reflexive symmetric relation pi in the uniformity of Xi such that Ifi(xi) - fi(x1) < a

1

whenever xi, xi E Xi and xi and xi are pi-near, that is, (xi, xi) E pi. By Lemma 3.3 there exists an open neighborhood V of to in T and a reflexive symmetric relation y in the uniformity of C such that gi ( t o y) , and gi (t, y') are &-near in Xi whenever t E V and y and y' are y-near in C. Now let t be an element of V and let

Then

58

Chapter 11. Steinhaus type theorems

and hence (using (3) and that pi(Xi \ Ci) < S),

Using this, we have with the notation

that

By (5) the first term on the right side is not greater than 2Me/(4M) = ~ / 2 . By the choice of K, a , P I , . . . ,,On, y, and V, the second term on the right ) ~/2. side is not greater than u ( Y ) E / ( ~ u ( Y )=

3.5. Corollary. Let T, Y, Xi, u, pi, and gi be the same as in the previous theorem. Suppose that condition (3) of the previous theorem is satisfied, and let Ai be a subset of Xi. Suppose that Ai is pi measurable if 2 5 i 5 n. Then the function

is continuous on T.

Proof. Condition (3) of the previous theorem by 3.2.(7) implies that the set 9;: (B1)is a u hull of 9;: (A1) whenever B1 is a p1 hull of A1. Hence

where X A ~is the characteristic function of Ai and function of B1.

XB,

is the characteristic

5 3.

Generalizations of a theorem of Steinhaus

59

3.6. Theorem. Let T be a topological space, X and Y be Hausdorfl' spaces with Radon measures p and v, respectively. Let D be an open subset of T x Y, g : D -+ X, to E T , and let K c X be a compact set. Suppose, that (1) the mappings g and (t, x) t, g,l(x) are continuous and gt is a homeomorphism of Dt onto X if t E T; (2) for each E > 0 there exists a 6 > 0 such that p(gt (B)) > 6 whenever B c Y, v(B) E, and t E T. Then . V ( ~ ; ~ ( K ) A ~ ; ~ (- +KO ) ) if t - + t o .

>

Proof. Let E > 0 and let us choose a 6 > 0 for ~ / 2 .Let C = g;l(K), and let W be an open subset of Dt, containing C for which v(W \ C) < ~ / 2 . Similarly, let U be an open subset of X containing K for which p(U \ K ) < 6. Let us choose an open neighborhood V of to such that if t E V, y E C , then gt(y) is defined and gt(y) E U ; moreover if t E V, x E K , then g,l(z) E W. Then by g;l(K) \gtnl(K) C V \ C we have v ( y ; l ( ~ ) \ g , o ' ( ~ ) < ) ~ / 2 Moreover . v (K) \ 9': ( K ) ) < &/2because the mapping gt maps the set gtil(K) \ g ; l ( ~ ) into U \ K , for which p ( U \ K ) < 6. Summarizing, v (,g;'(K) A g , ' ( ~ ) ) < E if t E V. 3.7. Corollary. Let G be a locally compact group and let X be a left Haar measure on G. Let Ai, i = 1,2, . . . ,n be subsets of G with finite measure. Suppose that Ai is X measurable if 2 I i 5 n. Then the mapping

of Gn into R is continuous.

Proof. Since the replacement of Al by a X hull does not change this function, we may suppose that Al is measurable too. Let E > 0 and T = Gn. Let us choose the compact sets Ki such that Ki C Ai and X(Ai \ K i ) < E are satisfied. Let Y = X1 = X2 = . = X, = G and gi(t,y) = t;ly

if

( t l , . . . , t n ) E T and y E Y.

Since for any measurable subsets B, C with finite measure

Ix(B) and

-

X(C)I I X(B A

c)

Chapter 11. Steinhaus type theorems

60

by the previous theorem we obtain

I x ( ~ ~ K n~ t 2 K 2n - n t , ~ , )

-

x ( t ! ~n, t ; ~ n, . . . n t ; ~ , ) (

n

X(tiKi A tf K i ) + 0 whenever ti

5

-+

ti0 ,

i=l that is

t

x ( ~ , Kn , t , ~ n, . - .n t , ~ , )

t)

is continuous on T . But

5

C X(tiAi \ t i K i )< ne. i=l

Hence

( t l , .. . ,t,) e x ( ~ n~. . An t~, ~ , ) is the uniform limit of continuous functions, and so itself is continuous. In the following lemma which will be needed for the proof of our main result, we give sufficient conditions for the validity of condition (3) in Theorem 3.4. 3.8. Lemma. Let Y be an open subset of IRk, let T be a topological space, yo E Y, and to E T. Let g : T x Y -+ RT be a continuous function and

8s is continuous and suppose that 8~

rank

($( t o ,

y o ) ) = r.

Then there exist open neighborhoods Y* and T* of yo and to, respectively, and there exists a constant 0 < C < cc such that Y* C Y, T * C T, and

whenever B C Y* and t E T*. (Here diamB denotes the diameter of the set B -1 Proof. Let q = k-r and let us divide the coordinates of y = ( y l , . . . ,y k ) into two groups y' = ( y i , . . . ,yb) and y" = (y:, . . . ,y f ) so that the condition

5 3.

Generalizations of a theorem of Steinhaus

61

is satisfied. Let us introduce the notation

Using the proof of the inverse function theorem (see Rudin [172], theorem 9.24) we obtain that, if Y'' is an open ball with center y: in Rr , t E T , (y', y") E Y, and

whenever y" E Y", then gt,,~ is a homeomorphic mapping of Y" onto an open subset U(t, y') of Rr . Now let

and

Using the continuity of the expressions in (2) and (3) we can choose an open ball Y" with center y t and open sets Y' and T * such that to E T*,yb E Y', Y* = Y' x Y" C Y, moreover t E T*, y' E Y', y" E Y" implies that

Let a ( q ) denote the Aq measure of the q dimensional unit ball ( ~ ( 0 = ) 1). We are going to prove that a ( q ) (diam B)"' X ~ BF )A' (gt (B))Y

62

Chapter 11. Steinhaus type theorems

whenever B C Y * and t E T*. Let R = diam B. Then there exists a closed ball V with radius R in IWQ such that B c (V n Y ' ) x Y".Suppose to the contrary that there exists a t E T* for which

where C = a(q)/y.Then we can choose an open set U for which gt(B)c U and X ~ B )> Xr(U)CRq. Let B* = g,l Then B

(u)n ( ( vn Y ' ) x Y " ).

c B*, B* is a Bore1 set and gt(B*)C U , that is

We are going to prove that this is impossible. Let

B,*,= { y"

: ( y ' , y") E

B * } if y' E V n Y ' .

Using the theorem concerning transformation of integrals we have that

whenever y' E V n Y'. By Fubini's theorem

which is a contradiction. Hence the proof is complete. 3.9. Lemma. Under the conditions of the previous lemma, if a subset D of WT has density 1 in the point g ( t o , y o ) , then g ~ l ( Dn) Y* has density 1 in the point yo.

89 Proof. By the continuity of -(to, y ) the function gt, satisfies the Lipdy schitz condition on a neighborhood of yo. Hence there exist a y > 0 and an 0 < M < m such that y E Y * and ( g ( t o , y ) - g ( t o , y o )Mly-yol I whenever

<

§ 3. Generalizations of a theorem of Steinhaus

63

ly - yo/ < y. Let a ( k ) and a(r) denote the X h n d AT measures of the Ic and r dimensional unit balls, respectively. Let E > 0, and let

where C is the constant from the previous lemma. Let us choose a ,B > 0 such that, whenever V is a closed ball with center g(to,yo) and radius less than ,f3, then AT (V n D ) 2 (1 - S)Xr (V). We prove that if W is a closed ball in Y* with center yo and radius less than y and P / M , then xk(wn & ( ~ ) ) 2 (I - E ) P ( w ) . Suppose to the contrary that for such a W with radius R,

Then there exists a compact subset B EXYW). Hence by the previous lemma,

c W \ g;'(~)

for which X ~ B>)

But gto(B) is a compact subset of V \ D , where V is the closed ball in RT with center g(to,yo) and radius M R < P. Since XT(V\ D) < SXr(V) we get

which contradicts the choice of S. 3.10. Lemma. Let Y be an open subset of Rk , T a topological space, D an open subset of T x Y, and (to,yo) E D . Suppose that the function g : D + RT is continuous and continuously differentiable with respect to y.

8s yo) is r, then there exist such neighborhoods If the rank of the matrix -(to, 8~

T* and Y* of to and yo, respectively for which (1) for each E > 0 there exists S > 0 such that XT(gt(B)) 2 S whenever t E T*, B c Y* Ak(B)2 E; (2) ifA is a A' measurable subset ofRr , then g,' (A)n Y * is a X b e a s u r a b l e subset of Y for each t E T*.

Proof. Let us use Lemma 3.8. Shrinking the so obtained set Y* we may suppose that it is bounded. Now from the statement of Lemma 3.8 and from 3.2.(4) we obtain (1) and from 3.2.(7) we obtain (2).

Chapter 11. Steinhaus type theorems

64

3.11. Theorem. Let X be an r-dimensional Euclidean space, and let X I , . . . ,X, be orthogonal subspaces of X with dimensions 9-1,.. . ,r,. 1 (1 5 i 5 n ) and CYylri = r. Let U be an open Suppose that ri subset of X and F : U -+Rm be a continuously differentiable function. For each x E U let N, denote the null space of F1(x). Let Ai be a subset of Xi (i = I , . . . , n ) and suppose that Ai is XTi measurable for 2 5 i 5 n. Let a E U, dim N, = r - m. Let pi denote the orthogonal projection of X onto Xi. Suppose that pi(Na) = Xi and Ai has density 1 in the point pi(a) whenever I 5 i 5 n. Then F (A1 x . x A,) is a neighborhood of F ( a ) .

>

Proof. Let k = r - m. Since x ++ rank F' (x) is lower semicontinuous, and r a n k ( ~ I ( a ) )= m, we may suppose that r a n k ( ~ I ( x ) )= m whenever x E U. Similarly, choosing a smaller U if necessary, we may suppose that pi(N,) = Xi whenever x E U and 1 5 i 5 n; to prove this, suppose to the contrary that there exists an i and for each natural number j there exists an 31) E U and there exist orthonormal vectors e p ) , . . . , e,-,,+, (j) in N,, such that x j -+ a and pi(ep)) = 0 whenever j = 1 , 2 , .. . and 1

< s 5 k - ri + 1.

Using the compactness of the unit sphere we can pass to a subsequence and suppose that e!$ + e, if j -+ m. But this proves that the vectors e, are orthonormal in Na and pi(e,) = 0 whenever 1 5 s 5 k - ri 1, which is a contradiction. Now, choosing a smaller U if necessary and using the rank theorem (see Dieudonni! [49], 10.3.1), we have that there exist mappings u, p, and v and an open neighborhood V of b = F ( a ) in Rm with the following properties: u maps U onto the open cube IT,where I = 1-1, 1[, u is invertible and moreover u and u-I are continuously differentiable; v maps Imonto V, v is invertible and moreover v and v-I are continuously differentiable; p is the projection ( X I , . . .1 2 , ) P : ( X I , . . .7 2 , )

+

of ITonto I m ; and finally F = v o p o u. We may write ITas IT= T x Y where T = Imand Y = I k . Let u ( a ) = (to,yo)E T x Y. Now let us use some facts from differential geometry (see Dieudonni! [49], mainly 16.8.8). U ~i F - I (v(t)) is a closed submanifold of U whenever t E T. The tangent space of this submanifold in a point x E U n F - l (v(t)) is equal to the subspace N, of X . Clearly u-I is a diffeomorphism of the closed submanifold {t} x Y

5 3.

Generalizations of a theorem of Steinhaus

65

of T x Y onto U f l F - I ( v ( t ) .) Let gi = pi o u-I if 1 5 i 5 n. By the choice of U , pi is a submersion of U n F-I ( v(t)) into Xi. Hence the mapping gilt : Y + Xi is a submersion too, that is, its derivative has rank ri whenever Y and t E T . Now, by Lemma 3.8, there exist open sets T* and Y * and there exists a 0 < K < cc such that to E T* c T , yo E Y * C Y , and

whenever B C Y * ,t E T * . Let X,* = X i , A: = Ai, and gg the restriction of gi onto T * x Y * . Applying Corollary 3.5 to the sets and functions marked by stars we have that the function

is continuous on T * . By Lemma 3.9, g;;:(~i) has density 1 in the point yo. Since g:,;: ( A i )n Y * is measurable by Lemma 3.10 if 2 5 i 5 n, we have that

has density 1 in the point yo. Hence f ( t o )> 0 and we have that there exists a neighborhood V of to for which f (t) > 0 if t E V . Clearly v ( V ) is a neighborhood of b in Rm. If z E v ( V ) , then t := v-' ( z ) E V and hence the set n

is nonvoid. If y is an element of this set, then ~ ( u - ' ( t y, ) ) = v ( p ( t ,y ) ) = v ( t ) = z and xi = pi (u-'(t, y)) = g:,t ( y ) E Ai if 1 5 i 5 n. This means that F ( x l ,. . . , x n ) = Z . 3.12. Corollary. Let U be an open subset o f W x RT and F : ( x ,y) ++ F ( x ,y) a continuously differentiable mapping of U into RT. Let A , B C RT and suppose that B is AT measurable. If ( a ,b) E U ,

dF det -(a, dx

b) # 0 ,

dF det -(a, dy

b) # 0,

A has density 1 in the point a and B has density 1 in the point b, then F ( A ,B ) is a neighborhood of F ( a , b).

66

Chapter 11. Steinhaus type theorems

Proof. By Theorem 3.11 we have to prove only that pl (Na,b) = RT and p2(Natb)= Rr where Nalbis the null space of F1(a,b). Let (x, y) E Na,b. If pl (x, y) = 0, then x = 0. Hence dF 0 = F 1 ( a ,b) (x, y) = -(a, dy

b) (y).

aF But det -(a, b) # 0, hence y = 0. This proves that pl : Na,b + Rr is a dy = Rr. one-to-one mapping, that is, pl (Na,b)= Rr . Similarly p2 3.13. Remark. The previous theorem may be stated in the following global form: IfN, is in general position, i.e., dim N, = r-m and pi (N,) = Xi for all x E U (i = 1,2, . . . ,n ) , moreover A1 x A2 x . x A, C U, Xri(Ai) > 0 (i = 1,2,. . . ,n ) , and the set Ai is Xri measurable for 2 5 i 5 n , then F(A1 x . . . x A,) contains a nonvoid open set.

Proof. Let us choose a point a E U for which Ai has density 1 at pi(a) whenever 1 5 i 5 n , and let us apply the previous theorem.

4. GENERALIZATIONS OF A THEOREM OF

PICCARD Piccard's result [I671 analogous to the theorem of Steinhaus states that the sum of two Baire sets having second Baire category has an inner point. Very strong generalizations exist; in this case also addition can be replaced by a two variable function with weak solvability conditions. These results are useful in the proof of "Baire property implies continuity" and "Baire property implies boundedness" type regularity theorems for functional equations. We refer the reader to the papers Sander [179], [181], [183], Kominek [124], J h a i [86], Grosse-Erdmann [61], Lindberg and Szymanski [139], and the references cited in them. The purpose of this section is to give a generalization of Piccard's theorem analogous to the results of the previous section. These results were published in J&-ai [103]. The following theorem is an abstract version of our generalization of the theorem of Piccard.

§ 4. Generalizations of a theorem of Piccard

67

4.1. Theorem. Let T , Y , and Xi be topological spaces, gi : T x Y + Xi continuous functions, and suppose that giSt(B)has second Baire category whenever B c Y is a subset of Y with second Baire category. Suppose that Ai C Xiand Ai is a Baire set whenever 1 5 i 5 n. Then the set V ofpoints t E T for which n

is of second category is an open subset of T Proof. The sets Ai can be written in the form Ai = Ei A Mi, where Ei is open and Mi is of first category. Suppose that to E V, and let K = n>,g; ( A i ) ,K t = K n (n:=,g;,fi ( E , ) ) . Since 9;; (Mi) is of first category in Y whenever i = 1 , 2 , .. . ,n , we have that

is of first category, hence K t is of second category in Y . Let yo be a point of K ' for which W f l K t is of second category in Y for each open neighborhood W of yo. Clearly gi(to,y o ) E Ei if i = 1 , 2 , . . . ,n. Since the sets Ei are open and the functions gi are continuous, the sets g Z - l ( ~ , )are open and contain the point (to,y o ) , hence there exist open sets V' and W', such that to E V', yo E W', and V' x W' c n~=lgz:l(~i). We will prove that

is of second category for each t E V'. Were this not true, the sets

would cover - except for a set of first category - the set W'. If we prove that these sets are of first category, then we have a contradiction. But this follows from the inclusion

which is a consequence of W' C 9,:

(E,).

Chapter 11. Steinhaus type theorems

68

4.2. Remark. I f we suppose that Y is a complete separable metric space and X1 is metrizable, then we may omit the condition that the set A1 has the Baire property.

To prove this, let C1 denote the set of all points xl E X1 such that for each neighborhood U1 of xl the set Ul n A1 is of second Baire category. It is known that C1 is a closed set and A1 \ C1 is of first category. Let B1 denote the set of inner points of C1. Then B1 is open and Al \ B1 is also of first category. As in the previous proof we obtain that W' \ g<;(B1) and W' \ g,;,'(Ai), 2 i 5 n are of first category. It is enough to prove that W' f l (A1) is of second category, because then it follows that

gl,i

<

cannot be of first category. Suppose that W' \ gl,; (A1) is of first category. Then, using that W' n g,(B1) i is an open set of second category, we obtain that

is a Baire set of second category. Let G be a Gs subset of second category of the set above. Then gl,t(G) is of second category as a subset of X I . By Bourbaki [38], IX, 56, Exercise 10, gl,t(G) is a Baire set in X I . Clearly ~ I , ~ (C G B1 ) \ Al. Writing gl,t(G) = U A F where U is open and F is of first category, we see that U n B1 is a nonvoid open set for which the intersection with A1 is of first category. This contradicts the definition of B1. In Laczkovich [135], 11.9.9 it is proved that the continuous image of a Polish space in a Hausdorff space is a Baire set. This shows that it is enough to suppose that X1 is Hausdorff. The following lemma allows us to use derivatives to verify that the conditions on the functions gi in the previous theorem are satisfied. 4.3. Lemma. Let Y be an open subset of R\ T a topological space, D an open subset of T x Y, and (to,yo) E D. Suppose that the function g : D -+ RT is continuous and has continuous partial derivative with respect &I to y. If the rank of -(to, yo) is r , then there exist open neighborhoods T*

8~

and Y* of t o and yo, respectively, such that

§ 4. Generalizations of a theorem of Piccard

69

(1) if B has second category in Y *, then gt (B) has second category in RT for each t E T *; (2) if A is a Baire set in RT, then g,l (A) n Y * is a Baire set in Y for each t E T*. Proof. We have proved in the proof of Lemma 3.8, that there exist open sets T* and Y' and an open ball Y" centered at y{ such that t o E T*, yb E Y', T* x Y' x Y" C D, and gt,,, is a homeomorphism of Y" onto an open subset U(t, y') of RT whenever t E T * and y' E Y'. Suppose tha,t there exists a subset B of Y* = Y' x Y" of second category, and a t E T*, such that gt(B) is of first category in RT. Let us choose a Borel set U of first category in RT, for which gt(B) C U C gt(Y*) and let B * = g , l ( ~ ) n Y*. This set B* is a Baire set and is of second category in Y*, but gt(B*) is of first category in IWT. By the Kuratowski-Ulam theorem the set of all points y' E Y' for which B;, is of second category is a set of second category. On the other hand, by the same theorem, the set of all points y' E Y' for which B,", is not a Baire set, is of first category. From this it follows that there exists y' E Y' for which B;, is a Baire set of second category in Y". Since gt,,/ is a homeomorphism of Y" onto U(t, y'), the set gt,,, (B,",)is of second category in RT. This is a contradiction, because gt,,~(By*,)C gt (B*). Hence (1) is proved. To prove (2) suppose that A is a Baire set in RT, and let us choose a Borel set B for which A C B and B \ A is of first category. Then

Using that g,l(B) is a Borel set and g c l ( B \ A) n Y* has first category by ( I ) , we have that g,' (A) f l Y * is a Baire set. Now we are prepared to prove the local version of our generalization of the theorem of Piccard for a function from an open subset of RT into IWm. The condition of the following theorem means, roughly speaking, that the null space of the derivative is large enough and is in general position. 4.4. Theorem. Let X be the r-dimensional Euclidean space, and let X I , . . . ,X , be orthogonal subspaces of X with dimensions r l , . . . ,r,, respectively. Suppose that ri 2 1 whenever 1 5 i 5 n and Cy=lri = r . Let U be an open subset of X and F : U + Rm a continuously differentiable function. For each x E U let N, denote the null space of F1(x). Let Ai be a Baire subset of Xi ( i = 1 , 2 , . . . ,n), and suppose that a E U and

70

Chapter 11. Steinhaus type theorems

dim Nu = r - m. Let pi denote the orthogonal projection of X onto X i . Suppose, that pi ( N u )= Xi and pi ( a ) has a neighborhood Uisuch that Ui\ Ai is offirst category if 1 5 i 5 n. Then F ( A 1 x x A n ) is a neighborhood of F(4.

Proof. Let us define t o ,yo, T , Y , and gi as in the proof of 3.11. By the above lemma there exist sets T* and Y * such that t o E T* c T , yo E Y * c Y , and gi,t(B) is of second category whenever B C Y * has second category and t E T * . Let X t = X i , A f = A i , and g: be the restriction of gi to T * x Y * . Applying the theorem above to the sets marked by a star we have that the set V * of points t for which n

is of second category is open in T * . Since gljtomaps Y * onto an open neighborhood of pi(a) and g* ( t o ,yo) = pi ( a ) , there exists an open neighborhood W of yo in Y * such that W \ g;;: (A,) is of first category if 1 5 i 5 n. This proves that t o E V * . Clearly v ( V * ) is an open neighborhood of b in Rm . If x E v ( V * ) ,then v - ' ( z ) E V * , and hence the set

is nonvoid. If y is an element of this set, then u - l ( t , y ) E F - ' ( x ) and xi = pi (u- ( t ,y ) ) E Aiwhenever 1 5 i 5 n. This implies F ( x l , . . . ,x n ) = z , which is enough since v ( V * )is an open neighborhood of b = F ( a ) .

4.5. Corollary. Let W be an open subset of Rr x IF,and let F : ( x ,y) H F ( x ,y ) be a continuously differentiable mapping of W into IF. Suppose, that A , B C Rr and A, B are Baire sets. If (a, b) E W,

dF det -(a, ax

b) # 0 ,

dF det -(a, 8~

b) # 0,

and there exist neighborhoods U and V of a and b respectively such that U \ A and V \ B is of first category, then F ( A ,B ) contains a neighborhood of F ( a , b).

Proof. Similar to Corollary 3.12 of the previous section. The above theorem can be formulated in the following global form:

§ 4. Generalizations of a theorem of Piccard

71

4.6. Theorem. Let X be the r-dimensional Euclidean space, and let X I , . . . ,X , be orthogonal subspaces of X with dimensions r l , . . . ,r,, respectively. Let pi denote the orthogonal projection of X onto Xi. Suppose that ri 2 1 whenever 1 5 i 5 n and Cy=l ri = r . Let U be an open subset of X and F : U + Rm a continuously differentiable function. For each x E U let N , denote the null space of F ' ( x ) . If dim N x = r - m and p i ( N x ) = Xi whenever x E U and i = 1 , 2 , . . . , n, moreover A1 x . x A , c U and Ai is a Baire set having second category i f 1 i n, then F ( A 1 x x A,) contains a nonvoid open set.

< <

a

Proof. A Baire set Ai can be written in the form Ui A Fi where Ui is a nonvoid open set and Fi is o f first category. For any a E U1 x U2 x . x U, we may apply the previous theorem. 4.7. Remark. Using the previous Remark, we may omit the condition that Al is a Baire set.

Chapter 111.

BOUNDEDNESS AND CONTINUITY OF SOLUTIONS

In this chapter we discuss how we can prove boundedness, local boundedness, or continuity of solutions of functional equations from their measurability or Baire property.

5. MEASURABILITY AND BOUNDEDNESS The classical method treated in the introduction to prove continuity of solutions works only for locally integrable solutions. Hence for measurable solutions we have to prove that they are bounded and hence locally integrable. Therefore we need "measurability implies boundedness" type theorems. For certain special equations such type of results can be found for example in the papers of VajzoviE [204], Dar6czy [40], and Sander [179]. First we prove a general theorem about a functional inequality. This theorem was first published in the paper JBrai 1801. As we have already mentioned in the introduction the results of 5 8 make it possible to directly conclude continuity of solutions from measurability if the measures are Radon measures. Hence "measurability implies boundedness" type results are used today only in cases where there are non-Radon measures involved.

5.1. Theorem.

Let T, Y, and Xi (i = 1 , 2 , ,. . ,n) be sets, u and Xi (i = 1 , 2 , . . . ,n), respectively. Further

pi be finite measures on Y and

Chapter 111. Boundedness and continuity of solutions

74

IetDcTxYandf :T+R, fi:Xi+R, fo:Y+R,gi:D+Xi ( i = 1,2, . . . ,n ) , h : D x IRnf R be functions. Assume that the following conditions hold: ( I ) for every (t,y) E D

> 0 there is a K > 0 such that xi I k ( i = 0,1, . . . , n ) and ( t ,y) E D implies h ( t ,y, xo,2 1 , .. . , xn) I K ;

(2) for every k

(3) the functions f i , i = 1,2,. . . ,n are measurable; (4) for each E > 0 there is a S > 0 such that p i ( g i , t ( ~ )2) S whenever 1 5 i 5 n, t E T , B c Dt, and u ( B ) 2 E ;

(5) there exists an E > 0 such that u ( D t ) 2 E for every t E T . Then f is bounded ii-om above on T . Proof. Since the sequence of sets

is increasing and it's union is Y , for the that with the notation

E

> 0 of (5) there exists

a ko such

. we have u ( C o )2 u ( Y ) - ~ / 2 Hence

By (4) there exists a 6 > 0 such that pi ( g i , t ( ~ ) > ) S whenever 1 2 i I n , t E T , B C Dt, and u ( B ) _> ~ l ( 2 n ) Since . for 1 I i 5 n the increasing sequence

{ x :x

€ x if,i ( x ) 5 k ) ,

k = 1,2,...

of measurable sets has union X i , there exist ki's such that with the notation

we have pi(Xi

\ C i ) < S.

We will prove that the set

5 5.

75

Measurability and boundedness

is nonvoid for all t E T . Indeed, if this set would be void, then the set Dt n Co would be covered by the sets g$ (Xi \ Ci). Since u(Co n D t ) 2 r/2, for some 1 5 i 5 n we would have u ( g $ ( ~ i \ C,)) rl(2n). This would imply that pi(Xi \ Ci) 2 S, contradicting the choice of Ci. Now let k be the maximum of the numbers ko,kl, . . . , k, and let y be any element of set (6). Then gi (t, y) E Ci and hence by (1) and (2) we obtain that

>

f(t)

< h (t, Y

fo (I/), fi (91(t, 9))

1

, f n (9n (tr Y))) 5 K.

5.2. Remark. Applications of the previous theorem are not restricted to real-valued functions. For example if functions f , fi have values in a semimetric space then we may take the distance of the function value from a fixed point of the space and apply the previous theorem to these real-valued functions. Even the condition that the measures v and pi are supposed to be finite is not a strong restriction: choosing appropriate smaller sets T , Y, D , and Xi finiteness can be obtained. We will use these observations in the proof of the following statement.

5.3. Statement. Let G be a locally compact group, H a metric group with left invariant metric d, and let f : G -+ H be a homomorphism. I f f is measurable with respect to some invariant expansion p of left Haar measure, then f is bounded on compact sets.

Proof. Let T be a compact subset of G and let Y be a compact subset of G having positive measure. Let D = T x Y and X = TY. Then

Introducing the notation and

f (x) = d(e, f (x)) the

function

If is p measurable

-

f (t) I f ( t y ) + f ( y ) if t , y E D .

Applying the previous theorem we obtain that f is bounded on T .

76

Chapter 111. Boundedness and continuity of solutions

6. CONTINUITY OF BOUNDED MEASURABLE

SOLUTIONS As we mentioned in the introduction, "bounded measurable solutions are continuous" type results are not so important today as were before because of the new "measurability implies continuity" type results of 5 8. Hence we discuss only two - unpublished - results, 6.6 and 6.8. Our interest in these theorems is that they can be applied to covariant expansions of the Eebesgue measure and to invariant expansions of Haar measure. Note that Theorem 3.4, treated among Steinhaus type theorems is also connected with this topic and can be used to prove continuity of bounded measurable solutions. First we have to survey some results about invariant expansions of the Haar measure and covariant expansions of the Lebesgue measure. The full discussion can be found in JBrai [82]. In 1950, Kakutani and Oxtoby [I211 proved that Haar measure on the circle group T can be expanded to an invariant, countably additive measure for which the corresponding L2-space has Hilbert space dimension 2', where c = card(T) is the cardinal number of the continuum. The same result holds for any compact metric group (see Hewitt and Ross [66], 5 16). On the other hand, using the result of Kakutani and Kodaira [122], in 1959 Hulanicki [75] proved that 2' discontinuous characters of the circle group can be made simultaneously measurable under an invariant expansion of Haar measure. This result was extended by Itzkowitz [77] for connected compact infinite Abelian groups and by Hewitt and Ross [68] for arbitrary locally compact Abelian groups by showing that 2card(G)characters of any infinite locally compact Abelian group G can be made simultaneously measurable under an invariant expansion of Haar measure. In JBrai [82] the author generalized the theorem of Kakutani and Oxtoby for arbitrary locally compact groups. This result is based on an abstract expansion theorem for a given finite measure to a very much larger measure so that the expanded measure has the same transformation law for a certain class of transformations as the original measure. To explain these results we introduce two notions. 6.1. Definition. A measure p is called J covariant measure if and only if a set T of mappings r is given such that each r E T maps a p measurable set into X, and for every r E T an extended real-valued nonnegative p measurable function J ( T ) is defined satisfying the following property:

5 6.

Continuity of bounded measurable solutions

77

(1) the multiplicity function N (r/A,y) is p measurable and

whenever A is a ,u measurable set and T E T = dmn J . If T is one-to-one and J(r)is p almost equal to the characteristic function of r E T, then the measure p is called T invariant. d m n whenever ~ We remark that if every element T of T is one-to-one, then condition (1) is equivalent to (2) r (A) is p measurable and p (T(A)) = JA J(r)(x) dp (x) whenever A is p measurable and r E T . Hence if every element r of T is one-to-one, then p is T invariant if and only if (3) r ( A ) is p measurable and p (T(A)) = p(A) whenever A is p measurable, 7 E T , and A c d m n ~ . 6.2. Examples. (1) Let X = Rn, let p be the Lebesgue measure on Rn, and let T be the group of all translations of Rn. Then p is T invariant. = 1 for every x E Rn whenever r E T . In this case, J(T)(x) (2) Let X = R n , let p be the Lebesgue measure on Rn, and let T be the set of all linear mappings of Rn into itself. If J(r)(x) = 1 det r / whenever x E Rn and T E T , then p is J covariant. (3) Let X = Rn,let p be the Lebesgue measure on Rn, and let T be the set of all continuously differentiable functions r , each mapping an open subset of Rn into Rn. Let J ( r ) ( x ) be equal to the absolute value of the Jacobian of r at x if x E d m n r and 0 if x $ d m n r . Then p is J covariant.

6.3. Remark. If p is a J covariant measure, then we have for every nonnegative p integrable function f the following formula concerning the transformation of integrals: (1) for every r E dmn J the function y ++ C,ET-liy) f (x) is a p measurable function and

This statement is obvious in the case when f is the characteristic function of a p measurable subset of X. Hence, using approximation by linear combination of characteristic functions we obtain the statement for nonnegative integrable functions. Our abstract expansion theorem is as follows:

78

Chapter 111. Boundedness and continuity of solutions

6.4. Theorem. Let p be a J covariant measure on X for which 0 < p ( X ) < oo, and suppose that every element of T = dmn J is one-to-one. Suppose that for some infinite cardinal number n and a subset .F of 2 X the following properties hold: (1) card(X) = 2"; (2) card(T) 5 2";

(3) card(.F) 5 2" and for every p measurable A for which p(A) > 0 there exists an F E .F for which card(F) = 2" and F C A. Then there exists a J covariant expansion v of p with weight 22". The proof is based on the method of Kakutani and Oxtoby. The essential difference is in the elimination of the topological feature of the original version. This makes it possible to use only purely set-theoretic and measuretheoretic devices. The proof uses the axiom of choice, but does not use the continuum hypothesis. Our expansion theorem for Haar measure reads as follows:

6.5. Theorem. Let G be a locally compact nondiscrete group and X a left Haar measure on G. Let A denote the modular function of G, and let n denote the character of the topological space G. Then the measure X (which has character n) can be expanded to a measure K such that K: has character 22n and if z E G and K is K measurable then (1) zK is K measurable and K ( z K )= K ( K ) ; (2) K z is K measurable and K ( K z )= A ( z ) K ( K ) ; (3) K-I is K measurable and K ( K - ' ) = l / A ( z )~ K ( z ) .

&

The proof is based on the abstract expansion theorem above. As another application of the abstract expansion theorem we proved that the Lebesgue measure on the n-dimensional Euclidean space can be expanded to a measure for which the corresponding L2 space has Hilbert space dimension 2' and is transformed by differentiable coordinate transformations as the Lebesgue measure, namely, J covariant with respect to J from example 6.2.(3). This is only a special case of a more general result about covariant expansions of the Lebesgue measure. A similar theorem is true for manifolds: see my thesis [98], kj 5.

6.6. Theorem. Let n , k, m l , . . . ,m,, and s be positive integers. Let Xi be an open subset of IW" let T and Y be open subsets of IWS, let Zi be let D be an open subset of T x Y, and let Z be a an open subset of Rmi,

$ 6. Continuity of bounded measurable solutions

79

Euclidean space. Let us consider the functions f : T -+ 2,gi : D -+ Xi, fi : X i + Z i , h i : D x Z i + Z ( i = 1 , 2 , . . . , n ) , a n d h o : D + Z . Let T J(T)be the mapping given in example 6.2.(3). Let p be a J covariant expansion of the Lebesgue measure on Rk.Suppose that (to,yo) E D and (1) for each (t, y) E D we have

(2) hi is continuous (i = 0 , 1 , . . . , n ) ; (3) the function fi is p measurable and bounded on Xi; (4) gi is twice continuously differentiable and

Then f is continuous on some neighborhood of to. Proof. Using coordinates, we may suppose that Z = R Let us choose a compact set K which is a neighborhood of yo and an open neighborhood V of to having compact closure such that V x K C D. If t E V, then integrating over K we obtain

Since p ( K ) # 0, it is enough to prove that the integrals on the right-hand side are continuous functions of t. This is clear for the first integral because ho is uniformly continuous on V x K . Leaving out indices, we have to prove

is continuous on V if V and K are small enough. By substituting y = g c l ( x ) and using that the measure p is J-covariant, we obtain that

whenever t E 7. Since g is continuous on the compact set 7 x K , the set C = g(V x K) is compact, too. Since g t 1 ( x ) and (~9;') (x) are continuous on

80

Chapter 111. Boundedness and continuity of solutions

the compact set 7x C, they are bounded and uniformly continuous there. Since the range of f is contained in a compact set L and h is uniformly continuous on the set V x K x L, we obtain that for each E > 0 there exists a S > 0 such that with the notation

whenever t , t l E V, It - tll exists an M > 0 such that

IH(~,x)I~M

<

S, and x E g t ( K ) n gt/(K). Moreover, there

if

t t V

and

x€gt(K).

Hence

By 3.6 the multiplier of M goes to zero if t

-+

t', hence the statement follows.

6.7. Remark. In the previous theorem, the sum of functions h l , . . . , hn cannot be replaced by a function having n variables. The addition cannot be replaced even with complex multiplication. Indeed, by the expansion theorem I ' of Hewitt and Ross 1681 a lot of non-continuous homomorphisms f : G 4 ' of a locally compact nondiscrete Abelian group G (for example, G = R) can be made measurable by an appropriate invariant expansion of Haar measure. This is not true for homomorphisms having values in Rn, as shown by the following statement. 6.8. Statement. Let G be a locally compact group and let f : G + Rn be a homomorphism which is measurable for a left invariant expansion p o f left Haar measure. Then f is continuous.

Proof. By the results of the previous section f is locally bounded. Similarly, as in the proof of the previous theorem, if we integrate both sides of the

§ 7. On a problem of Mazur

81

equation f (t) = f (ty) - f (y) over a compact set K having positive measure, we obtain K

The mapping

r

is continuous, because

where X is Haar measure. The expression on the right-hand side continuously depends on t and is zero if t = to, hence f is continuous.

7. ON A PROBLEM OF MAZUR In about 1935 S. Mazur asked (Problem 24 of "The Scottish Book" [157]) the following question: An additive functional f in a (real) Banach space X is given with the property

(P)

for each path g in X , the function f

o

g is Lebesgue measurable.

Is it true that f is continuous? This long-standing question was answered in 1984: Labuda and Mauldin [I341 proved, that the answer is yes, more generally, i f f is an additive operator mapping X into a Hausdorff topological vector space, and f has property (P), then it is continuous. The result was extended by Lipecki [I401 to the case when X and Y are Abelian Hausdorff topological groups with X metrizable, connected, locally arcwise connected, and complete. More general functional equations have also been considered. R. Ger [58] has obtained similar results concerning property (P) for Jensen convex functions, and L. Szkkelyhidi [I911 for exponential polynomials. The aim of this section is to prove that, on certain topological spaces, property (P) is equivalent to a more usual measurability condition, namely,

Chapter 111. Boundedness and continuity of solutions

82

universal measurability (J8rai [89]). This result makes it possible to use "measurability implies continuity" type results, that are known for more general functional equations, to get "property (P) implies continuity" type results. It is clear that several questions are open in this field, but the results below show that the two questions are strongly connected and may be starting points for further research. In particular, using a result from the classic book of Schwartz [187], we obtain the result of Labuda and Mauldin (see 7.5).

7.1. Definitions. For a measure p let M, denote the a-algebra of all p measurable sets. For a topological space X let Ux = n{M, : p is a Borel regular measure on X ) , the set of universally measurable subsets of X . Similarly, if X is a Hausdorff space, let Rx = n{M, : p is a Radon measure on X). In these definitions the expression "Borel regular measure" and "Radon measure" may be replaced with the expression "diffuse probability Borel regular measure" and "diffuse probability Radon measure", respectively. In the case when X is a complete separable metric space, Rx = Ux. More general results can be found in Schwartz [187], pp. 117-130. See also Federer [53], 2.2.16. Let

Px

=

{A : A C X , g-l(A) is Lebesgue measurable for each path g in x).

Trivially, Px is a a-algebra, and if f : X -+ Y is a function mapping X into a topological space Y, then f -l(V) E Px is satisfied for any open subset V of Y if and only if for each path g : [a, b] + X , the function f o g is Xra,b1 measurable. So we have to investigate only the connection of Px with other a-algebras. Clearly, Bx c Px, where Bx denotes the class of all Borel subsets of X. More generally, we have 7.2. Theorem. With the notation of the definition above, if X is a Hausdorff topological space, then Rx c Px.

4

Proof. Suppose that A is a subset of X for which A P x . Then there exists a path g : [a, b] + X for which g-'(A) is not Lebesgue measurable. Let p = g#(X[,,b1). Using results 2.2.17, 2.1.5(4) from Federer [53], we get that p is a Radon measure, and A cannot be p measurable.

5 7 . On a problem of Mazur

83

The converse is not true in all Hausdorff spaces. Clearly, if X is totally disconnected, than Px = 2 X . Hence some connectedness properties are needed. To prove the converse for a wide class of spaces, we need a lemma, which is more or less equivalent to a theorem of Marczewski published in Polish* (see [140]).

7.3. Lemma. Let p be a diffuse probability Radon measure on a complete metric space X. Then for each E > 0 there exists a compact subset C of [ O , l ] and a homeomorphism g : C + g(C) c X such that g is measure preserving between C and g(C), that is, g#(Xc) = pLg(C), moreover 1 > p(g(C)) > 1 - E . Proof. Let G(x, r) denote the open ball, B ( x , r) denote the closed ball with center x and radius r, respectively. Let t, be a strictly decreasing sequence tending to 1. Let us choose a compact set K C X for which 1 > p ( K ) > I - ~ 1 2 .We may suppose that spt(p1K) = K . For a fixed x E K , the function r H p ( K n B(x, r ) ) is positive for each positive r, monotone increasing, and tends to 0, as r tends to 0. Hence, we can choose a number 0 < r, < 112, for which the function is continuous in the point TX and has values less than i p ( ~ ) Choosing . a finite subcovering from the open covering G(x, r,), x E K , we get points xi and radii ri = r,,, for which

Moreover, we can choose radii r,! > ri, i = 1 , 2 , .. . , n , for which

Let K1 = K f l B ( x l , r l ) , and by induction let

If one of the sets Ki has measure zero, let us drop it, decreasing n. Finally, let n

Clearly, the sets Ki are disjoint compact sets with diameter less than 1, for each i the inequality 0 < p(Ki) < 112 holds, K(') C K , and p ( K \ ~ ( ' 1 ) < *Recently collected works of Marczewski have been published in English.

84

Chapter 111. Boundedness and continuity of solutions

~ / 2 Let ~ .us select disjoint closed intervals Ci, i = 1,2,.. . , n in [O,1],such that p(Ki) < X(Ci) < t l p ( K i ) ,whenever i = 1 , 2 , . . . , n and let c(')denote their union. In the second step, let us repeat this process for each Ki instead of K . We can get the disjoint compact subsets

of Ki with diameter less than 1/2, with measure positive but less than 1/4, such that with the notation

K ( ~C) K(') and p(K(')\ ~ ( ~ <1 ~ ) / 2 We ~ . choose the corresponding closed subintervals

Ci,l,Ci,2,.. . , Ci,ni

is satisfied. Let

n

nil

Continuing this process, by induction we get the systems

and the sets K(" and c @ ) . Let

To each point x E C there corresponds a unique sequence

Similarly, to each y E K , there corresponds a unique such sequence. Let g(x) = y if the corresponding sequences are equal.

85

§ 7 . On a problem of Mazur

To prove that g is continuous, let q > 0. If 1/2"' < 7 , and 6 is the minimal distance between sets Ci, ,i, ,.. ,ik, then 1x - x'l < 6 implies that g(x) and g(xl) are in the same set Kil,i2 ,,,. ,ik, hence have distance less than q. If q > 0 and tk/2k < q, let S be the minimal distance between sets Kil,i2,,., ,ik. If the distance between g(x) and g(xl) less than 6, then x and x' are in the same set Cil, i 2 , . .. , i k , hence lx - x11 < q and g-' is continuous. To prove that g# (Ac) = p [g(C), it is enough to prove that ,

because every open subset of C and g(C) is a disjoint countable union of open subsets of C and g(C) of this form, respectively, and the measures are Radon measures. But

7.4. Theorem. Let X be a complete, connected, and locally arcwise connected metric space. Using the notation of the previous definition, Px c

Ex. Proof. Suppose, that A is a subset of X for which A f! Rx. We will prove that A f! Px.Let us choose a diffuse probability Radon measure, for which A f! M,. Let A' denote the complement of A. Then p(A) p(A1) > p ( X ) = 1. Let us choose an E > 0 for which p(A) p(A') > 1 2 ~ .By the previous lemma, there exist a compact subset C of [O,1] with X measure greater than 1-E and a measure preserving homeomorphism g between C and K = g(C) C X. By theorem 5, $50 in Kuratowski, [132],g can be extended to a path h mapping [ O , 1 ] into X. Clearly h#(XLC) = g#(Xc) = p [ K . By the measurability of K, p(A) = p ( A n K ) + p ( A \ K ) , hence p ( A n K ) > p(A) -E. Similarly, p(A' n K ) > p(A1)-e. This proves, that A is not p LK measurable. By 2.1.2 of Federer [53], h-'(A) is X measurable if and only if A is h#(XLB) measurable for each subset B of I, which is not the case if B = C. Hence

+

A$

Px.

+ +

86

Chapter 111. Boundedness and continuity of solutions

7.5. Corollary. The answer to the problem of Mazur is affirmative, i. e., if for an additive functional f of a (real) Banach space X the function f 0 g is Lebesgue measurable for all continuous functions g : [a,b] + X, then f is continuous. Proof. Using the notation of the previous definition, we obtain from the previous theorem that f is universally Radon measurable. Since for all x E X the mapping t H f (tx) is Lebesgue measurable and satisfies Cauchy's equation, it is continuous. Hence we obtain that f is homogeneous. By a theorem from the book of Schwartz [I871 (p. 157), a linear and universally Radon measurable functional on a (real) Banach space is continuous. It follows that f is continuous. 7.6. Remark. The previous corollary can be extended to additive operators mapping a (real) ultrabornological topological vector space into a (real) locally convex Hausdorff topological vector space, if we use the ideas of the proof on p. 159 in the book of Schwartz [187].

8. CONTINUITY OF MEASURABLE SOLUTIONS Several papers dealt with the following question. Does, for a given type of functional equations, measurability of a solution imply its continuity? Among several papers dealing with measurable solutions of a special functional equation, the following papers treat general types of functional equations: Baker, [29], McKiernan [I581, Itzkowitz [78], Paganoni [162], and Sander [180]. Szkkelyhidi's results concerning regularity properties of (generalized) polynomials and exponential polynomials on locally compact Abelian groups can also be considered as general "measurability implies continuity" type results. See his book [193]. In this section first we will treat results first published in the papers JBrai [80], [all, [86]. The presentation here is somewhat more general than there, and several earlier results can be obtained as special cases. These theorems yield several new results; some of these will be treated in the last chapter among applications. Then we will discuss a result of K.-G. Grosse-Erdmann [61] and a new result of ours connected to it. We will refer to a quite recent "measurability implies continuity" type result of Guzik among applications, in 22.4. The reader can find a large number of further references in the papers mentioned above.

§ 8. Continuity of measurable solutions

87

8.1. Theorem. Let Z and Zi ( i = 1,2,. . . , n ) be completely regular spaces and Zo a a-compact completely regular space. Let Xi ( i = 1,2, . . . ,n ) and Y be locally compact spaces and T be arbitrary topological space. Let v be a Radon measure on Y and let pi ( i = 1,2,. . . , n ) be Radon measures on X i . Suppose that v ( Y ) < m, p,(Xi) < cx (i = 1,2,. . . , n ) , and let D be subset of T x Y . Consider the functions f : T -+ Z , f o : Y 4 Zo, f i : X i + Z i , h : D x Z o x Z 1 x . . . x Z n + Z , g i : D + X i ( i = 1 , 2, . . . , n ) . Let to be a fixed element of T, and suppose that the following conditions hold: (1) for each ( t ,y) E D,

h is continuous; (3) the function f i is Lusin pi measurable on Xi (i = 1,2,. . . ,n ) ; (4) gi iscontinuouson D (i = 1,2, . . . , n ) ; (5) Dt is measurable and there exists an 77 > 0 such that v(Dt n Dto) 2 7 for each t E T; B)) (6) for each E > 0 there exists a S > 0 for which P ~ ( ~ ~ ,2~S (whenever B c Dt, v ( B ) E , t E T , and I < i 5 n . Then f is continuous at the point to. (2)

>

>

Proof. By ( 5 ) , there exists an 17 > 0 for which v ( D t n D t o ) 77 whenever t E T. Let C be a compact subset of Dto for which v ( C ) > v ( D t o )- 7714. Now let E = 77/(8n). Using (6), we get a 6 > 0 for which 1 5 i n, t E T , B C Dt, and v ( B ) 2 17/(8n) imply p i ( g i , t ( ~ )2) 6. There exists a compact subset Ci of Xi for which pi(Xi \ C i ) < S and filCi is continuous. Hence the v measure of the open sets g ; ; , ' ( ~ i \ Gi) is less than 7 / ( 8 n ) whenever 1 i n and t E T. Consider now the set

<

< <

This is a measurable subset of Y, and it has a measure greater than 712. Let B 1 ,B q ,B 3 , . . . be an increasing sequence of compact sets in the space Zol for which U E , Bi = Zo. Then lim v ( f o 1 ( ~ i = ) )v ( Y ) < m,

i+m

88

Chapter 111. Boundedness and continuity of solutions

and, for a suitable i , the compact set Co = Bi satisfies the condition

Comparing this with what we have established before, we see that the set

is not empty for any t E T. Consider on the spaces Z , Zo, Zi, Xi, and Y uniformities compatible with the topology. Let a be a relation from the uniformity of Z and consider the compact set c x Co x fl(C1) x . . . x fn(Cn). By (2) and Lemma 3.3 there exists a neighborhood V' of to and there exist relations ,&, . . . ,,Bn from the uniformities of Z1,. . . , Zn, respectively, such that the points

are a-near whenever t E V', (t, y) E D , y E C, fo(y) E Co, zi, 2: E fi(Ci), and the points zi, zl are ,&-near if i = 1 , 2 , . . . ,n. Now, fi is uniformly continuous on Ci because Ci is compact. Hence there exists a relation yi from the uniformity of Xi such that the points fi(xi) and fi(x',) are &-near whenever xi,xi E Ci are yi-near. By (4) and Lemma 3.3 there exists a neighborhood V C V' of to for which the points gi(t, y) and gi(t0,y) are yi-near whenever t E V, (t, y) E D and y E C. Let t E V. Let us choose an arbitrary element y from the set (7). Then we have that (t,y) E D , (to,y) E D, fo(y) E Co, gi(t,y) E Ci, and gi(to,y) E Ci (i = 1,2, . . . , n ) . Hence the points fi(gi(t,y)) and fi(gi(to,y)) are ,&near, and so

are a-near. This shows that f (t) and f (to) are a-near whenever t E V, that is, f is continuous at to.

§ 8. Continuity of measurable solutions

89

8.2. Theorem. Let Z, Zi (i = 1 , 2 , . . . , n ) be completely regular spaces, Zo a a-compact completely regular space, Y and Xi (i = 1 , 2 , . . . , n ) locally compact spaces, T an arbitrary topological space. Suppose that D c T x Y . Let f : T -+ 2,f o : Y -+ Z o , g i : D - + X i (i = 1 , 2, . . . , n ) , fi : X i -+ Zi (i = 1,2, . . . , n ) , a n d h : D x Z o x Z l x . . . x Z n -+ Z befunctions, v a Radon measure on Y and pi a Radon measure on Xi (i = 1,2, . . . , n ) . Suppose that to E T and the following conditions are satisfied:

(1) for each (t,y) E D ,

(2) h is continuous;

(3) fi is Lusin pi measurable on the subset Ai of Xi ( i = 1,2, . . . ,n ) ; (4) gi is continuous on D (i = 1 , 2 , . . . , n ) ;

(5) there exist sets V and K such that V is open, K is compact, V x K to E V, v ( K ) > 0, and K c n ~ = ~ ~ ~ ; ~ ~ ~ ( ~ i ) ~

c D,

(6) for each E > 0 there exists a S > 0 such that B c K and v(B) implies pi ( g i , t ( ~ )2) 6 whenever 1 5 i 5 n and t E V.

2

E

Then f is continuous on a neighborhood of to.

Proof. We will reduce this theorem to the previous theorem by replacing D by a suitable smaller set D*. Let us choose a S > 0 for v ( K )l(4n) and let Xf = gi,t, ( K ) and let Wi be an open set with compact closure containing Xf for which pi(Wi \ X f ) < S. By Lemma 3.3 we may choose an open neighborhood T * c V of to for which gi,t(K) C Wi whenever i = 1 , 2 , . . . ,n and t E T*. Let Y* = K , v* = v(K)/2, and t t be an arbitrary element of T*, D * = {(t*,y*) : t* E T * and y* E n~=lgz;$ (x:) n Y*}, Zz*= Zi

( = O , .

n),

Z* = Z,

f * = flT*,

fo* = folY*,

and let v* and p: be the restrictions of v and pi to Y* and Xf (i = i , 2 , . . . ,n ) , respectively. If we prove that the conditions of the previous theorem are satisfied for the sets, points, functions, and measures marked by stars, then the proof is complete. Clearly, the conditions 8.1. ( I ) , 8.1.(2), and 8.1. (4) are satisfied, and 8.1. (3) follows by the choice of X,?;. 8.1. (6) also follows from the definition

90

Chapter 111. Boundedness and continuity of solutions

of T * and Y * . Hence we only have to prove that 8.1.(5) is satisfied. It is enough to prove that

u ( g ~ ( ~ : ) n Y2*2 7) ' - E

if

l
and

~ E T * ,

because from this

((n n

v ( D ~n * D,:) = u

g;;*

n(n n

(xt)n Y*)

i=l

gL;:

(x:)n Y * ) )

i=l

Suppose to the contrary that there exists an i (1 5 i 5 n ) and t E T * for which V (Y*n g,;t(x:)) < 24' - E l

On the other hand, by the choice of T * if y E Y * \ g , ~ , ' ( X r )then , gi,t(y) E Wi but g i , t ( ~ $) X:, i.e.1

This is a contradiction because pi(Wi \ X t )

<S

8.3. Theorem. Let Z , Zi (i = 1,2,. . . ,n ) be completely regular spaces, Zo a a-compact completely regular space, and T an arbitrary topological space. Let Y be an open subset of IW'C , Xi an open subset of F i ( i = 1,2,. . . ,n ) , and D an open subset of T x Y . Consider the functions f : T + 2, f o : Y + Z O , g i :D + X i , fi : X i + Zi (i = 1 , 2 , . . . , n ) , and h : D x Zo x Z1 x . . . x Zn 4 2 . Suppose that to E T and the following conditions are satisfied: (1) for each ( t ,y) E D ,

(2) h is continuous; (3) fi is Lusin XTi measurable on the subset Ai of Xi ( i = 1,2,. . . , n ) ;

5 8.

Continuity of measurable solutions

91

(4) gi is continuous on D (i = 1 , 2 , . . . , n ) ; g;;o~~i)) > 0; agi (6) the partial derivative - is continuous and has rank ri for each (t, y) E

(51 X~

8~

D (i = 1 , 2, . . . , n ) . Then f is continuous on a neighborhood of to. Proof. We will reduce this theorem to the previous theorem by replacing D by a suitable smaller set D*. Let yo be a density point of the set

By Lemma 3.10 there exist open neighborhoods V and W of to and yo, respectively, with the following properties:

(7) for each E > 0 there exists a 6 > 0 such that Xri (gilt(B)) > S whenever 1 Ii I n , t E V , B C W, and Xk(B) > E ; (Ai) n W is a X h e a s u r a b l e subset of ~ h h e n e v e r1 5 i I n and (8) t E V; (9) V and W are compact sets, V x W c D and

Let D* = V x W, h* = hlD* x Zo x Z1 x x Zn, gd = gilD* (i = 1 , 2 , .. . ,n), and let us apply the previous theorem on the set D* instead of D.

8.4. Remark. Some conditions of the theorems above may be weakened at the expense of other conditions. In this way we obtain other versions of the above theorems. On such versions see the papers Jhrai [81] and [86]. In his paper [61] K.-G. Grosse-Erdmann proved an interesting "measurability implies continuity" type theorem. Although his result can be applied only for Cauchy-type equations

(where f and f l are the unknown functions), his conditions are remarkably weak and generalize several earlier results about Cauchy-type equations.

92

Chapter 111. Boundedness and continuity of solutions

Roughly speaking, he removes absolutely all conditions about the dependence of h on variable y. As a special case, h may depend on fo(y),where f o and its range are arbitrary. His conditions on the inner function G are also somewhat weaker than ours. (To compare the two conditions, substitute t = G(x, y) locally.) Here we formulate one of his results. The proof will not be given: this is Theorem 5.A4 in the paper [61] of K.-G. Grosse-Erdmann. 8.5. Theorem [K.-G. Grosse-Erdmann]. Assume that X and Y are Hausdorff topological spaces with Radon measures p and v, respectively. Let T be a first countable topological space, D an open subset of X x Y, and G : D + T a continuous mapping. Let Z and Z1 be topological spaces, let Z1, iand h : Y x Z1 + Z bemappings and suppose f : T i Z, f l : XI that: (1) f ( G ( x , y ) ) = h ( y , f 1 ( 4 ) for all (x,Y)E D ; (2) for each fixed y E Y the mapping h is continuous in the other variable; (3) f l is Lusin p measurable on a measurable subset Al of Xl of positive measure; (4) for every to E T thereisan open subset U x V o f D with0 < p ( A l n U ) < cc and v(V) < cc such that, for every compact subset K of Al n U with p ( K ) > 0, there is a S > 0 and a neighborhood W of t o such that for each t E W the set { y : (z,y) E

D and G(x,y) = t forsomez E K )

contains a v measurable set having measure not less than 6. Then f is continuous. Unfortunately, the proof of K.-G. Grosse-Erdmann does not work if there are several unknown functions fi on the right-hand side. But if we combine his technique with the strong results of 5 19, infra, then under the usual condition 3.1 on the inner functions we can remove any condition concerning the dependence of h on y also in the more general case with several unknown functions. 8.6. Theorem. Let T , Z , and Zi (i = 1 , 2 , .. . , n ) be topological spaces, Y and Xi (i = 1,2, . . . , n ) Hausdorff spaces. Suppose that D c T x Y a n d W C D x Z 1 x . . . x Z n. Let f : T + Z , g i : D + X i ( i = 1 , 2 , . . . , n ) , fi : Xi -+ Zi (i = 1 , 2,... , n ) , and h : W + Z befunctions, v a Radon measure on Y and pi a Radon measure on Xi (i = 1 , 2 , .. . ,n). Suppose that to E T has a countable base of neighborhoods and the following conditions are satisfied:

5 8.

Continuity of measurable solutions

93

(2) for each fixed y in Y , the function h is continuous in the other variables; (3) f i is Lusin pi measurable on the measurable subset Ai of Xi (i = 1,2, ... , n ) ; (4) gi is continuous on D (i = 1 , 2 , . . . ,n); (5) there exist sets V and K such that V is open, K is compact, V x K c D , to E V , u ( K )> 0, and K c ~ Y = ~ ~ ; ( A ~ ) ; (6) for each E > 0 there exists a S > 0 such that B c K and u ( B ) implies pi (gi,t ( B ) )2 6 whenever 1 5 i 5 n and t E V . Then f is continuous at to.

>

E

Proof. Let t , be a sequence in V converging to to. Applying Theorem 19.5 infra for cp defined by cp(y, t ) = gl ( t ,y ) whenever ( t ,y ) E V x K and for the restriction of the measure u to the subsets of K we obtain ~ y ) ) for almost all a subsequence t,, for which f 1 ( g l (tm,, y ) ) + f l ( g (to, y E K. By applying the theorem again for fi, g z , and the subsequence t,,, we obtain a sub-subsequence, etc. Finally, we obtain a subsequence t,, of the original sequence t , such that for almost all y E K we have f i (gi (t,, , y ) ) --+ fi (gi (to,y ) ) for 1 5 i 5 n. Fixing any such y E K, by the functional equation and the properties of h this means that each sequence t , -+ to in T has a subsequence t,, for which f (t,,) + f ( t o ) .Suppose, that f is not continuous at to. Then there exists a neighborhood Wo of f ( t o )for is not a neighborhood of to. If Urn,m = 1 , 2 , . . . is a countable which f neighborhood base of t o ,then let us choose a sequence t, E Urn \ f -'(Wo). No subsequence of the sequence f (t,) can converge to f ( t o ) .

(wo)

8.7. Corollary. Let T , 2, and Zi (i = 1 , 2 , .. . ,n ) be topological spaces. Let Y be an open subset o f R k , Xi an open subset of Rri (i = 1 , 2 , .. . ,n ) , D an open subset of T x Y , and W c D x Zl x x 2,. Let f : T + 2 , g i : D -+ Xi (i = 1,2,... , n ) , fi : X i + Zi (i = 1,2, . . . , n ) , and h : W -+ Z be functions. Suppose that to E T has a countable base of neighborhoods and the following conditions are satisfied:

(2) for each fixed y in Y , the function h is continuous in the other variables;

94

Chapter 111. Boundedness and continuity of solutions

(3) fi is Lusin X T i measurable on the subset Ai of Xi (i = 1 , 2 , . . . ,n); (4) gi is continuous on D (i = 1,2, . . . , n ) ;

(5) X k ( f l ~ = l ~ ~ t l o ( > A i0;) ) (6) the partial derivative

D (i = 1 , 2,... ,n).

dgi dy

- is

continuous and has rank ri for each (t,y) E

Then f is continuous at to.

Proof. This corollary follows similarly from Theorem 8.6, as Theorem 8.3 follows from Theorem 8.2.

9. CONTINUITY OF SOLUTIONS HAVING

BAIRE PROPERTY Results similar to the theorems in 5 8 can be proved for solutions having the property of Baire. Results of this kind can be found in the papers of Haupt [64], Sander [179], and Grosse-Erdmann [61]. Here, again, we start with the investigation of the most general problem, and specializing it we will obtain a less general but easy-to-use result. Theorems 9.1, 9.2 were published in the paper J&rai [81]. Finally, generalizations of the results of Grosse-Erdmann will be proved (unpublished). Further references can be found in the papers cited above. 9.1. Theorem. Let T , Y, Xi, Zi (i = 1 , 2 , . . . , n ) be topological spaces, Zo a a-compact space, and Z a completely regular space. Let D c T x Y, Ai C Xi (i = 1,2, . . . , n ) , to E T, and consider the functions f : T -+ Z , f o : Y + Z o , g i : D -+ Xi, f i : Xi + Zi (i = 1 , 2, . . . ,n ) , h : D x Zo x Z1 x . . . x Z, -+ Z. Suppose that the following conditions hold:

(1) for each (t, y) E D we have

(2) h is continuous; (3) fi has the Lusin-Baire property on the subset Ai of Xi (i = 1,2, . . . , n);

(4) gi is continuous (i = 1 , 2 , . . . , n ) ;

§ 9. Continuity of solutions having Baire property

95

(5) t h e r e e x i s t s e t s V a n d K s u c h t h a t V x K C D , V i s o p e n , t o ~ V , K i s of second category and has the Baire property, and

(6) if B is of second category subset of Y and B c K , then gi,t(B) is of second category subset of Xi whenever 1 5 i 5 n and t E V. Then f is continuous on a neighborhood of to. Proof. Since Ai has the Baire property, there exist sets Ei,M,!, M!', for which Ei is open in Xi, M!', c Ei, the subsets Ml, M,!' of Xi are of first category in Xi, M,! n Ei = 8 , and Ai = (Ei \ Mi') U M,!. First we prove that there exist sets V' and K' for which V' is open in T, K ' is of the second category in Y and has the Baire property, to E V' C V, V' x K ' c D , and K ' C K , and further, n

K' i

n

g$

(Ei)

whenever t E V'.

i=l

Let K" = K n (fl,",lgz~~(Ei)).Since, by condition (6), K n gt;i0(M,!) is of first category in Y whenever 1 i n, we have by condition ( 5 ) that

< <

is of first category in Y. Hence Kt' has the Baire property and is of second category in Y, and n

Let yo be a point of K " such that for every open set W containing yo the set W n K " is of the second category in Y. Since the functions gi are continuous on D , the sets gz:l(~i) are open, and contain the point (to,yo), there exist open sets V' and W' for which to E V' C V, yo E W' c Y , and

It is clear that the set K ' = Kt' n W' satisfies the required conditions.

96

Chapter 111. Boundedness and continuity of solutions

The proof will be complete if we prove that f is continuous on V'. Let t E V' and let U be a neighborhood of f (t) in Z. Since the function fi has the Lusin-Baire property on the subset Ai, there exists a subset Mi" C Ai which is of first category in Xi such that the restriction of fi to Ai \ M,!" is continuous. Let Mi = Mi' U M y . Then the restriction of fi to Ai \ Mi" is continuous (i = 1,2, . . . ,n). Since by condition (6) the sets K ' n :9; (Mi) are of first category in Y, the set

has the Baire property and is of second category in Y. Let B1,B 2 , .. . be an increasing sequence of compact sets of Zo for which U,OO,,Bi = Zo. Since

there exists a compact subset Co = B j of Zo for which f c l ( C o ) f l K"' is of the second category in Y. Let y be a point of this set with the property that W n f;l(Co) n K"' is of second category in Y whenever W is a neighborhood of y. With the notation x; = gi (t, y) and zi = fi(x;), i = 1 , 2 , .. . ,n we have that xi E E i \ M i , fo(y) E CO,and

Let us choose a uniformity on Z for which the uniform topology is the original topology of the space. We can find a reflexive symmetric relation a from the uniformity of Z with the property that z E U whenever z and f (t) are a-near in Z. By Lemma 3.3 there exist neighborhoods V", W", and Ui of t, y, and xi, respectively, such that h(t, y', zo,x i , . . . ,z;) and h(tl, y', zo,zy, . . . ,z:) are a-near for all zo E Co whenever t' E V", y E W", and zj, zl' E Ui. Using the fact that the restriction of fi to Ei \ Mi is continuous at xi we can choose an open neighborhood V , of xi in Ei for which fi (xi)E Ui (i = 1,2, . . . ,n ) , whenever xa E V, \ Mi. Since gi is continuous at (t,y), there exist open sets V"' and W'" for which t E V'" c V" n V', y E W"' c W" n W', and gi(V1" x W"') C V (i = 1,2, . . . ,n). This proves that

n n

w"' n f c 1 ( c o )n K"' c

9%;

i=l

(v,),

5 9.

Continuity of solutions having Baire property

whenever t' E V"'. By condition (6), the sets K"' fl g,;! category in Y if t' E V"', and hence the set

97

( M i ) are of first

is nonvoid whenever t' E V"'. Let y' be an arbitrary element of this set. Then ( t ,y') E D , (t',y') E D, ~ o ( Y ' = ) zo E CO,and gi ( t ,Y ' ) E V , \ Mi, gi (t',Y ' ) E V , \ Mi for i = 1 , 2 , . . . ,n. Hence fi(gi(t,y'))= Z( E Ui, fi(gi(t',y'))= z,!' E Ui for i = 1,2,.. . ,n , and by the choice of V"', W'", Co, and Ui (i = 1,2,.. . , n ) we have that

are a-near in 2, that is, f (t') E U. This proves that f is continuous on V'.

9.2. Theorem. Let T and Zi ( i = 1 , 2 , . . . ,n ) be topological spaces, Y an open subset o f R k ,Xi open subset o f F i (i = 1,2,. . . , n ) , Z a completely regular space, and let Zo a a-compact space. Let D be an open subset of T x Y . Takethefunctionsf : T + Z , f o : Y -+Zo,gi : D - + X i , fi : X i + Z i ( i = 1 , 2 ,. . . , n ) , a n d h : D x Z o x Z 1 x . . - x Z , + Z . Suppose, t h a t t o E T and that the following conditions hold: ( I ) for each (t ,y ) E D ,

(2) h is continuous;

(3)

fi

has the Lusin-Bairepropertyon thesubset Ai o f X i (i = 1,2,.. . , n ) ;

(4) gi is continuous (i = 1,2,.. . , n ) ; (5) the set

g2;:0

( A i )is of second category in Y ;

agi (6) the partial derivative - is continuous and has rank ri whenever ( t ,y ) E 8~ D ( i = l , 2,... , n ) .

98

Chapter 111. Boundedness and continuity of solutions

Then f is continuous on a neighborhood of to.

Proof. We will reduce this theorem to the previous theorem using Lemma 4.3. It is enough to prove that conditions (5) and (6) of the previous theorem are satisfied if K and V are suitably chosen. Because of condition (5) of the present theorem, the set

is of second category in Y, so there exists a yo for which (to,yo) E D and for each open neighborhood W of yo the set

is of second category. By Lemma 4.3 we can choose open sets V and W such that the following conditions hold:

(7) gi,t(B) is of second category as a subset of IFi whenever i = 1 , 2 , . . . , n, t E V, B C W, and B is of second category; 9 , ~ ;(Ai)) ~ has the Baire property and is of second category (8) W n as a subset of Y;

(9) V x W Now let

c D, t o E V, and yo

E W.

Then the conditions of the previous theorem are satisfied and our proof is complete.

9.3. Remark. Some conditions of the theorems above can be weakened if other conditions are strengthened. See the papers J b a i [81] and [86] in connection with such variations. In his paper [61] K.-G. Grosse-Erdmann proved a "Baire property implies continuity" type theorem for the equation f (G(x,y)) = h (y, f 1 (x)), where f and f 1 are the unknown functions. (Theorem 5.B1 in [61]). The virtue of his result is that he completely removes any condition about the dependence of h on the variable y. Apart from this, as the substitution t = G(x, y) shows, his conditions are not very different from ours. Therefore, we do not repeat his results here, but prove a version with several unknown functions on the right-hand side. The proof is based on a result of 5 20 infra.

§ 9. Continuity of solutions having Baire property

99

9.4. Theorem. Let T , Z , Y , Zi, and Xi ( i = 1,2,. . . , n ) be topological spaces. Let D c T x Y , W C D x Z1 x . . x Z,, and consider the functions f : T - + Z , g i : D + X i , f i : X i + Z i ( i = 1 , 2 , . . . ,n ) , h : W + Z . Suppose that t o E T has a countable base of neighborhoods and the following conditions hold:

(2) for each fixed y in Y , the function h is continuous in the other variables;

(3) fi has the Lusin-Baire property on the subset Ai of Xi ( i = 1,2, . . . , n ) ; (4) gi is continuous on D ( i = 1,2,. . . , n ) ;

(5) there exist sets V and K such that V x K C D , V is open, to E V , K is of second category, has the Baire property, and

(6) i f B isasecond categorysubset ofY and B c K , thengi,t(B) isasecond category subset of Xi whenever 1 5 i 5 n and t E V. Then f is continuous at t o .

Proof. Let t, be a sequence in V convergent to t o . Applying Theorem 20.4 infra to cp defined by cp(y,t ) = gl ( t ,y) whenever (t,y ) E V x K, we have f 1 ( g l (t,, y ) ) + f l ( g l ( t o y , ) ) for all y E K except for a set of first category. By applying the theorem again for f z , 92, etc., we have for all y E K except for a set of first category, fi (gi (t, , y )) + f i (gi ( t o ,y)), whenever 1 5 i 5 n . Fixing any such y E K , the functional equation and the properties of h imply that for each sequence t , + to in T we have f (t,) -+ f ( t o ) Since to has a countable base of neighborhoods, this means that f is continuous at t o . 9.5. Corollary. Let T , Z , and Zi ( i = 1,2,. . . , n ) be topological spaces. Let Y be an open subset of Rk,Xi an open subset of F i( i = 1 , 2 , . . . ,n ) , D an open subset of T x Y, and W C D x Z1 x . . . x Z,, and consider the functions f : T + Z , gi : D i X i , f i : Xi -+ Zi ( i = 1,2,. . . , n ) , h : W -+ Z . Suppose that to E T has a countable base of neighborhoods and the following conditions hold:

Chapter 111. Boundedness and continuity of solutions

100

(2) for each fixed y in Y, the function h is continuous in the other variables;

(3) fi has the Lusin-Baire property on the subset Ai of Xi(i = 1 , 2 , . . . ,n); (4) gi is continuous on D (i = 1 , 2 , .. . ,n ) ;

(5) the set

n:=, g;jo (Ai) is of second category in Y;

agi (6) the partial derivative - is continuous and has rank ri whenever (b, y) E

D (i = 1 , 2,... , n ) .

aY

Then f is continuous at t o . Proof. This corollary follows similarly from Theorem 9.4, as Theorem 9.2 follows from Theorem 9.1.

10. ALMOST SOLUTIONS In 1960 P. Erdos raised the following problem: Suppose that a function

f : IR + IR satisfies the relation

for almost all (x, y) E R2 in the sense of the planar Lebesgue measure. Does there exist an additive function g : R R such that f (x) = g(x) almost everywhere in the sense of the linear Lebesgue measure? A positive answer was given to this question by N. G. de Bruijn, and independently, by W. B. Jurkat. For further references, see the book [127] of Kuczma, pp. 443-467. Such "almost solutions" , i. e., functional equations satisfied almost everywhere, arise naturally if we differentiate a functional equation having monotonic or locally Lipschitz solutions. In this case the solutions are known to be measurable. Another case where measurable almost solutions arise naturally are characterization problems of probability distributions. A prototype of such results is the well-known theorem stating that if J and q are independent random variables and J q and J - 17 are independent, too, then all have normal

+

§ 10. Almost solutions

101

+

distributions. Supposing that (, q, ( q, and ( - q have density functions f t , fi7,fCfl),and ft-,, respectively, we obtain that

= 2/B

+

f6,i7(~

U,U

-

u) dudu,

for any Bore1 subset B of R2, where

for almost all (u, v ) E R2. This is a functional equation which is a special case of the equation treated in § 24. A much more complicated characterization problem of probability distributions will be considered in § 23. A further example from harmonic analysis where measurable almost solutions arise naturally is shown in Aczd [7]. If we apply the distribution method to a locally integrable solution of a functional equation (see 5 18) we obtain only that the solution is almost equal to a smooth function. If we can conclude that the solution is continuous, then it is equal to this smooth function. Hence we arrive again at a regularity problem of measurable almost solutions. As the above examples show, the following question concerning measurable almost solutions of functional equations is of great importance in practice: Suppose that the general functional equation considered in 1.7

with unknown measurable functions f , f l , f 2 , . . . , f n is satisfied for all (x, y) from a subset E' of some open subset E of RT x IRk, for which E \ E' has Lebesgue measure zero in RT x Rk. Here all functions are supposed to be defined on some open subset of some Euclidean space, and taking values in some Euclidean space. The known functions H and G, G I , . . . , G, are supposed to be smooth. Is it possible to find functions such that f" = f = fi almost everywhere for i = I , . . . , n and such that replacing and f , f l , . . . , f n with f", f;,. . . , fn, respectively, the functional equation (1) is satisfied everywhere on E?

f"i

102

Chapter 111. Boundedness and continuity of solutions

We will prove that under reasonable conditions the answer is yes and the functions f", i = 1 , . . . , n are continuous. Of course, this question is related to results of 5 8. The tools there seem to be strong enough to answer this problem too, but technically it will be simpler to use the tools of $19, infra. Most of the results of this section was published in JBrai [105]. First, we discuss the connection between almost solutions of explicit and implicit equations. Then we prove two theorems: the first one is a general version, and the second one considers the special case of several real variables. We will also deal with the analogous Baire category question. Some related results of ~ w i a t a kare referred to in l8.5.(2).

f"i,

10.1. Connection between almost solutions of explicit a n d implicit equations. Under certain condition equation

just as in 1.7 - can be reduced to a simpler explicit equation. Namely, suppose, that the term f (G(x,y)) can be expressed from equation ( I ) , and it is possible to introduce a new variable t = G(x, y) instead of x to obtain the functional equation

-

Here D' is the image of E' by the mapping (x, y) (G(x,y), y) . We suppose that this mapping is a diffeomorphism mapping the open set E onto the open set D , hence this mapping and its inverse carry sets having Lebesgue measure zero into sets having Lebesgue measure zero. Now, if we are able to prove that there is a subset T' of the domain T of f such that T \ T' has measure zero and f IT' can be (uniquely) extended to a continuous function f" defined on T , then the functional equation (2) is satisfied with f" instead of f almost everywhere on D, namely, on the set Dh = Di n (Ti x Ktk). This means that if we replace f by f", then equation (1) is still satisfied, at least on some subset Eh of Eo for which E \ Eh has measure zero. Now, repeating this process with f l , we may obtain a subset Ei of Eh and a function f1continuous and equal almost everywhere to f l such that (1) is satisfied i f f and f l are replaced by f" and f;,respectively. Finally, we obtain a subset E" = EL such that E \ Eil has measure zero and (1) is still satisfied if we replace f , f l , . . . , f, by the continuous functions f", f;,. . . , fn equal almost everywhere to them. Since the left-hand side of (1) is continuous on E and equal to zero on a dense

5 10. Almost

solutions

103

subset E" C E, we obtain that equation (1) is satisfied on E if f , f l , . . . ,f, are replaced by f", f;,. . . , f,, respectively. Summarizing, our problem concerning equation (1) can be reduced to the following problem: Suppose that equation (2) is satisfied on a subset D' of the open set D for which D \ Dl has measure zero. Give reasonable conditions under which there is an appropriate subset T' of the domain T of f such that T \ T' has measure zero and the function f IT' has a (unique) continuous extension f" to T . We will prove that the set T' can be chosen to be the set of all points t E T for which the set {y : (t, y) E D \ D') has measure zero. By Fubini's theorem, T \ T' has measure zero. More generally, f" is determined by any subset of this T' which is still dense in T . This will be proved in a much more general setting in the next theorem. The proof uses Theorem 19.5. infra. Note that we use the terminology of Bourbaki concerning topology. The limit of a function at a point is defined by Bourbaki without excluding the point. 10.2. Theorem. Let T and Zi (i = 1,2, . . . , n ) be topological spaces, Z , Y, and Xi (i = 1,2, . . . ,n ) Hausdorff spaces. Suppose that D C T x Y , T' c T , X,' C Xi, W c D x Z1 x . . . x 2,. Let f : T' -+Z, gi : D + X i (i = 1 , 2 , . . . ,n ) , fi : X,! + Zi (i = 1 , 2 , .. . ,n ) , and h : W + Z be functions, v a Radon measure on Y, pi a Radon measure on Xi, and pi (Xi \ X,!) = 0 (i = 1 , 2 , .. . ,n ) . Suppose that to E T has a countable base of neighborhoods and the following conditions are satisfied:

(1) for each fixed y in Y, the function h is continuous in the other variables; (2)

is Lusin 1,2, . . . , n ) ;

fi

measurable on the measurable subset Ai of Xi (i =

(3) gi is continuous on D (i = 1 , 2 , .. . , n ) ; (4) there exist sets V and K such that V is open, K is compact, V x K to E V , v ( K ) > 0, and K c n:,lgz$J~i);

c D,

(5) for each E > 0 there exists a 6 > 0 such that B C K and v(B) implies pi ( g i , t ( ~ )2) S whenever 1 5 i 5 n and t E V ;

2

E

(6) there exists a subset V' of TI n V such that to is contained in the closure o f V 1and for each t E V' for almost ally E K we have

104

Chapter 111. Boundedness and continuity of solutions

Then limtEvl,t-ttof (t) exists.

Proof. Let t, be a sequence in V' convergent to to. Applying Theorem 19.5 for cp defined by p(y, t) = gl (t, y) whenever (t, y) E V x K and for the measure VK we obtain a subsequence t,, for which f l (gl(tmk,y)) + f l (gl(to,Y)) for almost all y E K. Now repeating this with f n , g2, and the subsequence t,,, we obtain a sub-subsequence, etc. Finally, we obtain a subsequence t,, of the original sequence t , such that for almost all y E K we have fi (gi (trn,,Y)) -+ fi (gi (to, Y)) for 1 I i I n and

Fixing any such y E K , by the functional equation and the properties of h this means that each sequence t, -+ to in T has a subsequence t,, for which f (t,,) is convergent. Of course, its limit zo E Z does not depend on y. Hence, for all y from a set K ' C K for which u ( K \ K t ) = 0 we have zo = h (to, 3, h (a(to, V ) ) , . . .

, f n (gn(to, y )) )

Suppose that it is not true that limtEv~,t+t,f (t) = xo. Then there exists a neighborhood Wo of zo such that there is no neighborhood U of to for which the set U n V' is mapped by f into Wo. If Urn, m = 1,2,. . . is a countable neighborhood base of to, then let us choose a sequence t& E U,nV' for which f (t,) @ Wo. Repeating the above process with t k instead of t,, we obtain a subsequence t L j such that for almost all y E K we have ) 1 I i I n and fi(gi(tL,,y)) -+ f i ( g i ( t 0 7 ~ )for

Now, if we choose a y to be in K', then using the functional equation and (7) we obtain a contradiction. 10.3. Theorem. Let Z be a regular space, Zi (i = 1 , 2 , .. . ,n ) topological spaces, and T a first countable topological space. Let Y be an open subset of EXk, Xi an open subset of Rri (i = 1 , 2 , . . . , n ) , D an open subset o f T x Y, and W C D x Z1 x x 2,. Let T' c T be a dense subset, and h : W + 2 . Suppose that the function fi is f : T' + 2, gi : D + Xi, almost everywhere defined on Xi with values in Zi (i = 1,2, . . . ,n ) and the following conditions are satisfied:

105

§ 10. Almost solutions

(I) for all t E T' for almost all y E Dt we have

(2) for each fixed y in Y, the function h is continuous in the other variables; (3) fi is Lusin Xr2 measurable on its domain (i = 1 , 2 , . . . , n); dgi (4) gi and the partial derivative - is continuous on D (i = 1 , 2 , .. . , n); dy (5) for each t E T there exists a y such that (t, y) E D and the partial dgi derivative - has rank ri at (t, y) E D (i = 1 , 2 , .. . ,n). dy Then f has a unique continuous extension to T

Proof. We will reduce this theorem to the previous theorem. We will prove that the limit limtET,,t,t, f (t) exists for each to E T. If this is proved, then defining the extension of f to T by this limit, the theorem is proved. Let us choose for a given to a yo such that (to,yo) E D and the partial dgi derivative - has rank ri at (to,yo) (i = 1 , 2 , . . . ,n). To prove that the

8~

limit exists, we will replace D by a suitable smaller set D*. By Lemma 3.10 there exist open neighborhoods V and Wo of to and yo, respectively, with the following properties: (6) for each E > 0 there exists a S > 0 such that Xri (gi,t(B)) S whenever 15 i < n , t E V, B c Wo, and XYB) 2 E ;

>

(7) g,2(dmn f,) r l Wo is a X"easurab1e subset of Ti@ whenever 1 < i and t E V; (8) Wo is a compact set, V x Wo c D and


Let D * = V x Wo, h* = hlW n (D* x Zl x . . . x Zn), g: = gijD* (i = 1,2, . . . , n ) , and let us apply the previous theorem on the set D* instead of D . Now we consider the problem of Baire property solutions of a functional equation satisfied except for a first category set. The arguments of 10.1 can

106

Chapter 111. Boundedness and continuity of solutions

be repeated in this situation, and therefore here again it is enough to consider the explicit equation 10.1.(2). The following theorem goes back to the work of Haupt 1641.

10.4. Theorem. Let T , Y, Zi, and Xi (i = 1 , 2 , .. . , n ) be topological spaces. Let Z be a Hausdorff space. Let D c T x Y, T' c T , X,/ c Xi, W C D x Z1 x x Z,, and consider the functions f : T' i Z , gi : D i Xi, fi : XI i Zi (i = 1 , 2 , .. . ,n),h : W + 2. Suppose, that Xi \ X,' is of first category and that to E T has a countable base of neighborhoods and the following conditions are satisfied: (1) (2) (3) (4)

for each fixed y in Y, the function h is continuous in the other variables; fi has the Lusin-Baire property on the subset Ai of Xi (i = 1 , 2 , . . . ,n ) ; gi is continuous on D (i = 1,2, . . . , n ) ; there exist sets V and K such that V x K c D, V is open, to E V, K is of second category and has the Baire property, and

ng;t'o(~i); n

KC

i=l

(5) i f B isasecond categorysubset ofY and B C K , thengi,t(B) isasecond category subset of Xi whenever 1 5 i 5 n and t E V; ( 6 there exists a subset V' of T' n V such that to is contained in the closure of V' and for each t E V' except a set y E K of first category of we have

Then limtEv~,t+to f (t) exists.

Proof. Let t, be any sequence in V' convergent to t o . Applying Theorem 20.4 infra for p defined by p(y, t) = gl (t, y) whenever (t, y) E V x K, we have f 1 (gl (t,, y)) i f 1 (gl (to,y)) for all y E K except a set of first category. By applying the theorem again for f 2 , g2? etc., for all y E K except a set of first category we have fi (gi (t,, y)) i fi (gi (to,y)) for 1 5 i 5 n and

Fixing any such y E K , by the functional equation and the properties of h we get that for each sequence t, i to in V' the sequence f (t,) is convergent. Of course, its limit zo E Z does not depend on y. Since to has a countable f (t) = ZO. base of neighborhoods, we have that limtEV/,t+to

§ 10. Almost solutions

107

10.5. Remarks. (1) If the above theorem can be applied for each to E T , and Z is a regular space, then we obtain that f has a unique continuous extension to T . For example, using the method of the proof of Theorem 9.2, we may obtain a result analogous to 10.3. (2) The above theorem can be proved using the simpler methods of 5 9 without referring to Theorem 20.4, although the proof is somewhat longer. (3) As we have already mentioned, the above theorem goes back to the paper [64] of Haupt. He formulated his theorem for the functional equation

which is not a significant difference, but makes the conditions somewhat more complicated. The main difference between his and the above theorem (and the results of § 9) is, roughly speaking, that he assumed that W = D x Z1 x . . . x Zn, h is continuous on W, V = T , and K = Y. (4) In his paper Haupt considers the case when there are countable infinite functions f l , f 2 , . . . on the right-hand side. Theorems 10.4, 10.2, 9.4, 9.5, 8.6, and 8.7 remain true for this more general case. (5) In his paper Haupt also considers more general a-ideals than the a-ideal of the sets of first category. Roughly speaking, he supposes only that the complement of any set from the a-ideal is dense, and the restrictions of the functions fi is continuous on the complement of an appropriate set from the a-ideal. Hence his results can be applied for sets of first category and functions having the property of Baire, but cannot be applied in the case of measure zero sets and measurable functions. (6) In his paper Haupt also proves that even if we do not suppose any continuity or Lusin-Baire property about the functions h, f l , f 2 , . . . , the extension of f (if any) is unique. We can formulate this as the following statement. 10.6. Statement. Let T , Y, Z, Zi, and Xi (i = 1 , 2 , . . . , n ) be sets and y and Xi be proper a-ideals of Y and Xi, respectively. Let D c T x Y, W c D x Z1 x - - .x Zn, and consider the functions gi : D -+ Xi, f i : Xi -+ Zi (i = 1 , 2 , .. . ,n ) , h : W -+ Z . Suppose that the following conditions are satisfied: (1) for each t E T we have Dt $ y; (2) for each t E T if B C Dt, B $ y, then gi,t(B) $ Xi whenever 1 5 i

5 n.

Then there exists a t most one function f" : T -+ Z for which there exist functions fi : Xi + Zi such that fi = fi on Xi except a set in Xi such that

108

Chapter 111. Boundedness and continuity of solutions

for each (t,y ) E D we have

Proof. We have to prove that for any t E T, f(t) is uniquely defined. Let us observe that if the map

is constant on the set Dtl then this constant is uniquely determined even by the functions f i , because the map

takes this value everywhere except a set in

Y

Chapter IV.

DIFFERENTIABILITY AND ANALYTICITY

This chapter deals with the problem of a chain of deductions about solutions of functional equations. Namely, starting from continuity, deducing local Lipschitz property and hence almost everywhere differentiability, then using that for deducing continuous differentiability, then from there multiple differentiability and finally concluding with analyticity. We also examine Holder continuous solutions and solutions having locally bounded variation.

11. LOCAL LIPSCHITZ PROPERTY OF

CONTINUOUS SOLUTIONS The results of this section - and also several results of the subsequent sections - crucially depend on a theorem on differentiability of the parametric integral

F(t)=

Of course, the differentiability of such parametric integrals for h and g smooth enough is classic. For example it plays a r61e in the derivation of the equations of hydrodynamics, see Zeidler [209], IV., pp. 441-445. For us the most

110

Chapter IV. Differentiability and analyticity

important fact is that it is not necessary to suppose that h is smooth. A theorem of this kind was proved in JBrai [86] under the condition that h and dh are continuous and g is twice continuously differentiable. Mikl6s Laczkovich suggested a formula for the derivative of the above parametric integral in which no second partials of g appear. This version was proved with the use of smoothing in the paper JBrai [91]. Although using the homotopy formula of currents and differential forms (see Federer [53], p. 363) it would be possible to give a shorter proof, the essence of the argument, however, would be the same. Therefore we will present here the elementary proof from [91]. For most of the results in this section we make restrictions on the form of the functional equation. First we will prove that for a "sum form" functional equation, continuity implies continuous differentiability. This result will be very useful in subsequent sections. We remark that the same method can be applied to somewhat more general equations; we refer the reader to the applications in chapter VII. Then we treat the special "Cauchy-type" case where the unknown functions appear only three times in the functional equation. These results were published in JBrai [86]. (Formulation of Theorem 5.3 therein contains a misprint.) Finally, we will prove a theorem for functional equations of general type mentioned in Problem 1.18, using an additional compactness condition. The first version of this theorem has been proved for real-valued solutions with real variables (in this case the estimates are simpler) and was published in the paper JBrai [96]. The more general variant is from JBrai [94]. Although this theorem does not solve Problem 1.18 completely, it can be applied to several concrete equations: see for example the "equation of the duplication of the cube" 22.15 and the characterization of Weierstrass's sigma function in section 24. We remark that very recently A. Lundberg found a way to derive differentiability from continuity using sequential derivatives. His method works for real-valued solutions defined on open intervals of the functional equation

See his forthcoming paper [150]. 11.1. Theorem. Let S be a simplex with nonvoid interior in R k , and let U , V, and W be open subsets of R k , RS, and R k , respectively. Let g : (t,y) t, x be a function mapping V x W into U , and let h : (t, x) t, x be a function mapping V x U into R. Suppose, that S c W and

§ 11. Local Lipschitz property of continuous solutions

111

(1) g is continuously differentiable, ( J g t ) y )# 0 if t E V and y E W, and gt is invertible for all t E V; a h are continuous. (2) h and at Then the function r

F(t)=

h(t, x ) dXk ( x ) whenever t E V

is continuously differentiable on V and its partial derivatives are bounded by

where H is the bound of h and H' is the bound of the partial derivatives iJg - on the sets g t ( S ) , t E V ; moreover GI is the bound of the absolute value

at;

ofthe Jacobian Jgt and G is the bound of the norm of the gradient of g on

V x S. Proof. We may suppose without loss of generality that V is connected, and W is convex, so the Jacobian Jgt does not change sign if y runs in W and t runs in V. Let i be an index between 1 and s. In the first step we will a F exists and we will give a formula for it supposing that g and prove that ati h are smooth enough. In the second step using this formula we prove that aF is continuous. In the third step we start to investigate the general case with the help of smoothing of g and h. In the fourth step we prove that the formula we had in the smooth case is valid in the general case as well. In the fifth step the proof will be finished. I. First suppose that h is continuously differentiable and g is twice continuously differentiable. With a simple substitution we get that

and hence denoting the coordinates of g by upper indices,

112

Chapter IV. Differentiability and analyticity

Changing back the integrals to integrals over gt (S), let us investigate the second and the third integral. If H ( t , x) denotes the vector-valued function having h(t, x)

a ' (t,g;' $!ti

(a))as its j-th coordinate, then

hence, from the second and third integral using the Gauss-Green theorem and denoting by n the outer normal we have

where

We will prove that G is zero. Let g, and [ e l , . . . , ek] denote the inverse of gt and the determinant of the vectors e l , . . . , ek, respectively. Moreover, let the upper indices stand for the coordinates. Then

then we have

§ 11. Local Lipschitz property of continuous solutions

113

On the other hand,

that is G(t, x) is equal to zero. 11. As a second step we observe that, under the conditions of the first aF K may be written as the sum of step> d F is continuous on V. Indeed, 7 integrals

Let E o ,El, . . . ,El, be the sides of S . The last integral can be written as

Let us investigate one of the terms of this sum. For simplicity let us omit the index of E. Let us choose an orthonormal base

such that e k is orthogonal to E. Omitting a sign which does not depend on t , we have

where .Jt(x) is the square root of the Gram determinant of the directional

8

&

( t ,g;l(x)), . . . , (t, CJ;'(X)). derivatives ek-1 integral above, we get the integral

By substitution, from the

114

Chapter IV. Differentiability and analyticity

So we have to prove that the integrals (4) and (5) depend continuously on the parameter t. This easily follows from the fact that the functions are continuous, hence uniformly continuous on compact sets. 111. We want to use smoothing to prove that the same representation is valid for dF if we only suppose that g is continuously differentiable and that dh exists and is continuous. Let t' be an arbitrary but fixed point of V. Let us choose an open ball V* with center t' which has a compact closure contained in V and a convex open set W* containing S with compact closure - contained in W. The image of the compact set V* x W* under the mapping g is compact as well. Let us choose an open set U* which contains this compact image and which has a compact closure contained in U. We will investigate the function F on the set V*. The values of F on this set depend only on the values of g on V* x W* and on the values of h on V* x U*; hence multiplying these functions with an infinitely many times differentiable function with value 1 on V* x W* and V* x U*, respectively, F does not change, so we may suppose that g and h have compact supports. Let us extend g and h to the complement of their domain as zero, and for simplicity let us denote the extended functions also by g and h. Let w be an infinitely many times differentiable function with support contained in the unit ball of IWS x Rk w dXS x Xk = 1. Let wE(u) = W ( U / E ) / E ~ + ~ and with the property if E > 0 and let g' = g * w' and h' = h * w' be defined by convolution over RS x IRk. It is well known that g' and h' are infinitely many times a h * we, moreover differentiable, = * w', = * we, ah' -

haXIWk

$

a ,&at-"-+ -+ 3

g, "

ge -+ g, h' i h, and + if&$O, and the convergences are uniform on compact sets. We want to use the results of I and I1 for the smoothed functions on the sets marked by *. To do this we have to prove that gf maps W* into U*, it is invertible on W*, and has a Jacobian different from zero if E is small enough. If K is a compact neighborhood of - gt and Jg; -+ Jgt uniformly on K , hence only V* x W* in V x W , then gf the invertability of g: is nontrivial. Let us observe that the matrix of the inverse of gi may be expressed with the determinant and subdeterminants of the matrix of g:, hence there exits an EO > 0 such that if (t, y) E K and 0 < E < EO, then the inverse of gf'(y) is defined and g:'(y)-l i gi(y)-l uniformly if EJO. Using the proof of the inverse function theorem we get that if Y is an open ball in Rk with center y' E W*,(t: y) E K , and

5 11. Local Lipschitz

property of continuous solutions

115

for each y E Y, then gZ is a one-to-one function on Y. We prove that the radius of the ball Y may be chosen to be greater than a number r > 0 which does not depend on t and y'. Indeed, choosing a positive constant smaller than the infimum of the numbers

and using that

7J

is uniformly continuous on K , we may choose the number

r so that if ly - y'l

< r, then

I ~ ( Y ) - s:(d)/I< c. Now, decreasing

EO

if necessary, we have

on K if 0 < E < E O . So we have proved that g,E cannot map points with distance less than r into the same point. For points having distance not less than r this is impossible as well. To prove it observe that the infimum of the continuous ( 3 )- gt (9') is positive on the compact set function

lgt

/

hence choosing a positive number d less than this infimum, decreasing E O if necessary, we have lgf ( y ) - (y') 2 d on this set if 0 < E < E O . IV. We will prove that in the general case F is partially differentiable and its partial derivative has the same representation as a linear combination of integrals of type (4) and (5) as in the case of smooth g and h. Let

1

if t E V* and E is sufficiently small. Representing F as the last integral on the right side and using the fact that the functions are uniformly continuous on compact sets, and the convergences are uniform on compact sets if ~ $ 0 it, follows that F c ( t )-+ F ( t ) if t 6 V*. Let f l , . . . , f, denote the usual base of RS.Clearly there exists a 6 > 0 for which t' 7f, E V* if I T / 6. Let

+

<

Chapter IV. Differentiability and analyticity

116

By what we have already proved the function p E ( r )can be represented as the linear combination of integrals of type (4) and (5) if we replace t by t' r f i , h by h E , and g by g'. Let us consider the same integrals for g and h and let p ( r ) denote the same linear combination of them. In the same way as in , on the interval the case of functions F E we get that p' -+ p if ~ $ 0 uniformly [-S, 61. Hence the functions

+

converge uniformly to the function

On the other hand,

1-

p E ( e )de = F E(t'

+ r f i )- F E(t'

-

6

6fi)4 F(tl

+ r f i )- F ( t l

-

6f i )

if ~ $ 0 Hence . we have

By step I1 the functions pE are continuous, hence p is continuous on the a F exists and equals p(O), that is the interval [-S, S]. This implies that -

at;

partial derivative

x is the linear combination of integrals of type (4) and

(5). V. To finish the proof we observe that in step I1 we have used only the continuity of functions in the integrals (4) and ( 5 ) . Hence repeating a F is continuous. Since the partial the argument used in step I1 we get that ati derivatives of F are continuous, we have that F is continuously differentiable. Finally, using the representations (4) and ( 5 ) we obtain the required estimate. 11.2. Lemma. Let T , Y , and U be open subsets of 1WS , IW" and RT, respectively. Let D be an open subset o f T x Y . Let h : D x U + R be a ah ah continuous function for which - and - are continuous and g : D 4 U be at av a twice continuously differentiable function, ( t o , y o )E D , and suppose that

§ 11. Local Lipschitz property of continuous solutions

117

@ ( t o ,yo) has rank r . Let S be a simplex in the unit ball o f Rk with nonvoid 8~ interior. Then there exists a convex neighborhood Vo of t o , an Ro > 0 real number, and a constant C such that for each R satisfying 0 < R < Ro we have R S yo C Dt and the function

+

is continuously differentiableon Vo and its gradient is bounded by C R ~ - '

Proof. The proof depends on the previous theorem about differentiation of parametric integrals. . , x r ) . Without loss of generality we may Let g t ( y ) = x = ( x 1 , x 2 ,.. suppose that the Jacobian

is positive at the point ( t o ,y o ) Let us choose open balls Vo and W o with centers t o and yo, respectively, such that Vo x W o c D is satisfied and the Jacobian above is positive if t E Vo, y E W o . Let Ro be the radius of W o . Let x* = ( x ; ,. . . , x,*,x,*+~, . . . , x i ) be defined in the following way: xd = xi, if 1 5 i r and x: = yi if r < i 5 k. We will denote the mapping y F-+ x* by gf . Let p(x*) = x . It is clear that the Jacobian

<

is equal to the Jacobian above and hence nonzero if t E Vo and y E W o . Let L ( t ) = g;'(yo). Suppose that

if y E W o Then by the proof of the inverse function theorem (see Rudin [172],Theorem 9.24), g; is a homeomorphism of W onto an open subset of Rk.Let 1

Taking a smaller Ro and V if necessary we may suppose that

Chapter IV. Differentiability and analyticity

118

whenever t E V and y E W . Now let Uo be an open subset whose compact closure is contained in g,"o ( W ) . By taking a smaller V , we may suppose that Uo c gF ( W ) for all t E V. Now shrinking V and decreasing Ro we may suppose that g,* ( W ) C U for each t E V. Taking x* = g,*(y) we have that the parametric integral

for all t E V satisfies all conditions of the previous theorem. Applying the previous theorem we have that F is continuously differentiable over V and for some constant C . its gradient is bounded by CR"' 11.3. Theorem. Let n, k , r l , . . . ,r,, m l , . . . ,m,, and s be positive integers. Let Xi be an open subset of RTi ( i = 1,2,. . . , n ) , T be an open subset of R S , Y be an open subset o f IRk, Zi be an open subset of Rmi ( i = 1,2, . . . , n ) , D an open subset o f T x Y , and let Z be a Euclidean space. We consider the functions f : T + 2,gi : D + X i , f i : Xi + Zi, , E D and hi : D x Zi -+ Z , ( i = 1 , 2 , . . . ,n ) , and ho : D + Z . Let ( t o yo) suppose that for each ( t ,y) E D ,

ho and

ah0 ahi ahi are continuous and hi, and are continuous (1 5 at dt &I

-

i 5 n); f i is continuous on X i ;

agi gi is twice continuously differentiable and -(to, y o ) has rank ri, 1 5 8~ i 5 n. Then f is continuously differentiable on a neighborhood of to.

Proof. Without loss of generality we may suppose that Z = R. Let us choose a simplex S small enough and an open set V for which S is a neighborhood of yo, V is a neighborhood of to, and V x S c D. Let us integrate equation (1) over S with respect to y. We get that

§ 11. Local Lipschitz property of continuous solutions

119

Since Js dy is a nonzero constant, we must prove that the integrals on the right side of (5) are continuously differentiable over V with respect to t whenever S and V are sufficiently small. This follows from the previous lemma. 11.4. Theorem. Let T , Y, X I , Zo, and Z1 be open subsets of Euclidean spaces of dimensions s, k , r l , mo, and 1, respectively. Let D be an open subset of T x Y. We consider the functions gl : D + X I , f : T + P, f o : Y + Zo, f l : X1 + Z1, and h : D x Z o x Z 1 + R m . Supposethat to E T and the following conditions hold: (1) for each (t, y) E D ,

(2) h is locally Lipschitz;

(3) gl is twice continuously differentiable; (4) f o and f l are continuous;

(5) there exists a yo, for which (to,yo) E D and

&Jl

- has

dy

rank r l .

Then f is Lipschitz on a neighborhood of to.

Proof. We must prove that there exists an open neighborhood V of to and a number M for which

Let zo = fo(yo),zl = f l (gl(to,yo)). Let us choose balls V c T , W C Y , Uo C Zo, and Ul C R with centers to, yo, zo, and zl, respectively, such that fo(y) E Uo, f1(91(t,y)) E Ul, and

whenever t , t' E V and y E W. Now, if we prove that (for a smaller V if necessary) there exists a number M3 such that, for every t , t' E V there exists a y E W such that

then the proof is complete.

120

Chapter IV. Differentiability and analyticity

Let S be a simplex which is a neighborhood of yo and let

By the previous lemma F is continuously differentiable with bounded derivatives if S and V are sufficiently small, that is, there exists a number M4 for which IF(t) - F ( t 1 ) [5 M 4 / t- t'l if t, t' E V. On the other hand, since f l is a real-valued continuous function and

there exists a y E S for which, denoting by XIC the Lebesgue measure on R k , F ( t ) - W') = AYS) . (fl (g1(t, Y))

-

fl(91 (it,Y ) ) )

With this y and with the notation M3 = M~/X'(S)we have

Hence the proof is complete. Under certain additional "compactness" conditions on the given functions gi, we are able to prove that the continuous solutions of a functional equation containing one unknown function are locally Lipschitz functions. The case when the unknown function is real-valued with one real variable was first examined in JBrai [96]. The proof of the general case of a vectorvalued function with a vector variable depends on the differentiability of parametric integrals with respect to the parameter and on the explicit estimate of the derivative which we will use in the form of the previous lemma. This more general theorem - in a somewhat different form - was published in the paper JBrai [91]. A part of the proof will be contained in the following lemma.

11.5. Lemma. Let Z be an open subset o f a Euclidean space, let Z1, Z z , .. . , Zn be convex open subsets of Euclidean spaces. Let T and Y be open subsets o f Rs and IRk, respectively and let Xi be an open subset o f IFi ( i = 1 , 2 , . . . ,n). Let D an open subset o f T x Y , (to,yo) E D. Consider the functions f : T i Z , fi : Xi + Zi, gi : D iXi(i = 1,.. . ,n ) , h : D x Z1 x Z2 x . . . x Zn + Z . Suppose, that

§ 11. Local Lipschitz property of continuous solutions

121

(1) for each ( t ,y ) E D ,

z

(2) h is continuously differentiable,the functions ah are Lipschitz continu-

ous with Lipschitz constant Li, the functions by BI, and Bi, respectively (i = 1 , 2 , . . . ,n ) ;

1 $$/ 1 I and

are bounded

(3) gi is a twice continuously differentiableLipschitz function with Lipschitz constant Li and % ( t o , y o ) has rank ri for i = 1 , . . . ,n; dY (4) the functions fi are continuous with continuity modulus wi Then there exist constants Ro > 0 and co > 0 and an open ball V centered at to such that i f t , t' E V , 0 < R < Ro, then

Proof. Let us fix a simplex S contained in the unit ball of IRk and containing the origin in its interior. Applying Lemma 11.2 for the coordinates of fi we obtain 6 > 0 , Ro > 0 , and C such that for the open ball V centered at t o ,having radius 6 and the closed ball W centered at yo, having radius Ro the set V x W is contained in D and the mappings

are continuously differentiable on V and the norm of the gradient of each of these mappings is bounded by CRk-' whenever 0 < R < Ro. Let t , t l E V . Let R be an arbitrary real number for which 0 < R < Ro. Let us integrate both sides of the functional equation over the simplex RS yo with respect to y . We have

+

Chapter IV. Differentiability and analyticity

122

where IS1 > 0 is the measure of the simplex S. Hence

To get a good upper estimate for the left-hand side we need an upper estimate for the difference

We may apply Taylor's theorem for the function h with points

z = ( t ,y , z l , . . . , z n ) and

2'

=

(t',y, x i , . . . ,z k ) ,

where t', t E V , y E W , zi = fi ( g i ( t ,y ) ) , and 2,' = fi (9i(t1,y ) ) if i = 1 , . . . ,n. The points z and z', and hence the segment with these endpoints are contained in V x W x Z l x - x Z n . Hence

+

" i=l

1 dr, ldh

(rz

+ ( 1 - r ) z i )(zi - zZ))

dr.

Using this and omitting the variables we have

+

By the triangle inequality we obtain n 1 terms on the right-hand side. The trivial upper bound of the first term is 1 SI~ " 6 It' - tl, where B&is an upper bound of Let us use the notation zp = fi (gi ( t , (i = 1 , 2 , . . . ,n ) ,

/ I.

§ 11. Local Lipschitz property of continuous solutions

123

and z0 = (t, yo, z!, . . . ,z:). If hi denotes the value of the partial derivative at zO,then the other terms can be written as the norm of (rz

+ (1 - r ) z i )

-

hi)

First we give an estimate for the norm of the first term of this sum. lzi - zi I has an upper estimate wi ( ~ i J-ttll), because Li is a Lipschitz constant for gi on V x W. Hence

We need an estimate of the difference

+

This is not greater than Li times the norm of r z (1- r ) z l - zO,i.e., Li times the maximal distance between the vectors z1 and zO = (t,yo, zy, . . . , z:), where L: is a Lipschitz constant for The maximal distance between z' and z0 can be estimated by

z.

Hence we obtain the upper bound

for the first term. To obtain an upper bound for the second term, we need an upper bound for the absolute value of

124

Chapter IV. Differentiability and analyticity

because h : is trivially bounded by the upper bound Bi of

1g1.From the

previous lemma we obtain the upper bound It - t'lCRk-I for this integral. Summing up all these estimates, we get the estimate

which gives the statement of our lemma with the notation co = C/lSI. 11.6. Theorem. Let Z be an open subset of a Euclidean space. Let T and Y be open subsets of RS and R k , respectively. Let D be an open subset of T x Y, C a compact subset of T , and W C D x Z n . Consider the functions f : T -+ Z , gi : D + T (i = 1 , .. . ,n ) , h : W + 2.Suppose, that (1) for each ( t ,y) E D we have

(2) h is twice continuously differentiable; (3) gi is twice continuously differentiable on D and for each t E T there d exists a y such that (t, y) t D, gi(t,y) E C, and $(t, y) has rank s for Y z = 1, . . . ,n; (4) the function f is continuous. Then f is a locally Lipschitz function on T .

Proof. The key idea in the proof is to show that the continuity modulus of f satisfies a functional inequality. Since the continuity modulus is not subadditive in general, first we will define a modification of it. For an E > 0 let C, = {x : dist(x, C ) f- E ) denote the (closed) E neighborhood of C. If E f- 0 let C, = C. Let us fix an E > 0 such that C, c T . For each 0 5 r 5 E let

§ 11. Local Lipschitz property of continuous solutions

125

Clearly w is increasing, w(0) = 0, w is continuous at 0 because f is uniformly continuous on the compact set C,, and w (rl r 2 ) 5 w (rl) w (r2) whenever 0 5 rl, r 2 ,rl + r 2 5 E. TOprove the last assertion, suppose that this inequality does not hold. Then there exist x, y such that Ix - yl 5 rl 7-2, x E C,, and y t Ce-lx-vl, but (x) - f (y)l > w ( r 1 ) w ( r 2 ) Suppose that 7-2 5 r l . The point y is contained in the E - lx - yl neighborhood of some point of C , hence x is contained in the E neighborhood of the same point. Choosing x on the segment connecting x and y such that lx - zl 5 TI and Ix - yl 5 r2, we have z E C,-I,-,I, hence (2) - f (z)I 5 w ( ~ I )and (2) - f 5 ~ ( r 2 ) which , is a contradiction. For an arbitrary to E C,, let us choose a yo (depending on t o ) according = f (x:) if i = 1 , 2 , .. . , n. to (3). Let xo = f (to), x: = gi(to,yo), and Let Z* denote the open ball centered at zo, having radius St,. Similarly, let T*, Y*, and Z; denote open balls centered at to, yo, and z t , respectively. Decreasing the radii, if necessary, we may suppose that

+ +

+

If

+

If

If

(Y)I

and that the range of h* = hlT* x Y* x 2; x x Z,* is a subset of Z*. Decreasing further the radii, if necessary, we may suppose that the partial derivatives of h* are bounded Lipschitz functions. Let us choose for the points xf E T open balls X; with radii at most ~ / 3 .Decreasing the radii, if necessary, we may suppose that the range of f: = filX; is contained in 2;. Finally, decreasing the radii of the balls T * and Y* around the points to and yo, respectively, we may suppose that the range of g: = gi IT* x Y* is contained in X;. Let f * = f IT*. Let w," denote the (usual) continuity modulus of f:. We will prove that w,"(r) 5 W ( T ) for a11 r 2 0. If xi, xi E X;, then Ixi - xpl < ~ / 3 and lxi - xsi < ~ / 3 ,hence using that x; E C , we obtain that xi E C, and Ixi - xi I < 2 ~ 1 3 ,which implies that xi E CE-lxi-,:l. Suppose that

Taking the supremum on the left-hand side we obtain the inequality w," (r) 5 ~ ( 4 . Let us apply Lemma 11.5 with the sets and functions marked by *. We get that there exists an open ball KO centered at t o , having radius > 0, and there are constants Rto > 0 and cto such that if t and t' are in this open

Chapter IV. Differentiability and analyticity

126

ball and 0

< R < Rto, then

Of course here the constants Bh, BQ, Li, and Li depend on to, which is not expressed in the notation. Using the inequality between w: and w, from the inequality above we get

For varying to the open balls KOcentered at to and having radii give an open covering of the compact set C,. Hence there exists a finite set To c C, such that the open balls corresponding to the points to E To give an open covering of C,. Let L denote a positive integer, which is greater than or equal to all of the constants Li (i = 1 , 2 , . . . ,n ) corresponding to all points to E To. Similarly, let B', L', and co denote a constant which is greater than Q (i = 0 , 1 , . . . ,n ) , Li (i = 1 , 2 , . . . ,n ) , and cto or equal to all constants B corresponding to all to E To,respectively. We have that if t and t' are in the same ball Vto and 0 < R < Rt,, then

Let S > 0 denote a Lebesgue number corresponding to the compact set C, and open covering K O ,to E To,i.e., let us choose a S such that if t; t' E C,, It - t'l < S, then there exists a to E To such that t , t l E K O . Let 0 < Ro 5 inf{Rto : to E To}. Decreasing Ro and 6, if necessary, we may suppose that L(S Ro) 5 E . Let t be an arbitrary element of C, and let t' be an element of C,-jt-t,i for which It - t'l < S. There exists a to E To for which t , t' E K O . Using the previous estimate and the fact that L is an integer, moreover the

+

5 11. Local Lipschitz

property of continuous solutions

127

subadditivity of w, we obtain that for an arbitrary real number R for which O
I f (t) - f (t')l5 B'lt

If It

- t'l

-

t'l

+ nB'lt - t'l-coR

I R , this estimate can be written

in the form

where cl, c2, c3, and c4 do not depend on t, ti, and R. Taking supremum first on the right-hand and then on the left-hand side for all t E C, and t' E C,-lt-t,l for which It - t'l 5 r , we obtain the inequality

whenever 0 5 r I 6 I R < Ro. If we choose an R which satisfies the condition c2R c3w(R) _< 112 - which can always be done by decreasing 6, if necessary, - we have

+

whenever 0 5 r 5 6. This proves that f is a locally Lipschitz function on C. For an arbitrary t E T, without loss of generality we may suppose that t is an interior point of C , since we may replace C with the union of C and a compact neighborhood of t. Hence the proof is complete.

11.7. Example. Usually the above theorem can be used together with some additional arguments. We illustrate this with a simple example. More complicated cases can be treated similarly. (See the "equation of the duplication of the cube" in 22.15 and the characterization of Weierstrass's sigma function in 3 24.) Suppose, that f satisfies the functional equation

Here f : Rm + is the unknown function, h is a given C" function, Ai, Bi are matrices with m rows and m columns, Bi is nonsingular, and llAil\ < 1.

128

Chapter IV. Differentiability and analyticity

Then every continuous solution is locally Lipschitz (hence all measurable solutions are Cw ) . To prove this, let us fix a natural number N > 0, let T be the open ball with radius N centered at the origin, let us choose an 0 < r < 1 such that liAill +rllBill < 1 (i = 1 , 2 , . . . , n ) , and let D = T x Y , where Y is the open ball with radius r N centered at the origin. Using the previous theorem, we have that f is locally Lipschitz on T , hence it is locally Lipschitz everywhere, because N was arbitrary.

12. HOLDER CONTINUITY OF SOLUTIONS In this section we only prove one theorem. This may turn out to be an important step in the general proof that "continuity implies local Lipschitz property". In any case, it proves that, under the conditions of Problem 1.18 if a solution is locally Holder continuous with some exponent 0 < a < 1, then it is locally Holder continuous with all exponents 0 < a < 1. The proof of the first result of this kind for real-valued functions in the paper JBrai [93] uses the fundamental lemma of the theory of Campanato spaces, which is a generalization of the famous classical Morrey lemma from the regularity theory of partial differential equations. (For further references about this lemma see the book of Zeidler [209], II/A, pp. 90-93.) Later a simpler proof and a generalization for several variables have been found. (JBrai 1921.) This simpler proof is the basis of the proof of Theorem 11.6. 12.1. Theorem. Let 0 < a < 1, and let Z , Zi be open subsets of Euclidean spaces (i = 1 , 2 , . . . ,n). Let T, Y, and Xi be open subsets of JP, Rk,and IFi, respectively. Let D be an open subset of T x Y. Consider the functions f : T + Z, fi : X i + Zi (i = I , . . . , n ) , g i : D -+Xi (i = I , . . . ,n), h : D x Z1 x Z2 x . . . x Z, -+ Z. Suppose, that

(1) for each ( t ,y) E D ,

(2) h is twice continuously differentiable; (3) gi is twice continuously differentiable on D and for each t E T there exists a y such that (t, y) E D and h ( t , y) has rank ri for i = 1 , . . . , n;

8~

(4) the functions fi,i = 1 , . . . ,n are locally Holder continuous with exponent a.

5 12. Holder

continuity of solutions

Then f is locally Holder continuous with exponent 2 a / ( a

129

+ 1).

Proof. We have to prove that for each point to E T the function f is Holder continuous on a neighborhood of to with exponent 2a/(l a ) . Replacing fi with the coordinates we may suppose without restricting generality that Zi c R for i = I , . . . , n. Let us choose yo by (3) for to. By Lemma 11.2 there exist a convex open neighborhood Vo of t o , a simplex with nonvoid interior contained in the unit ball, and constants Ro > 0 and C such that for the closed ball Wo with center yo and radius Ro we have Vo x Wo C D, and the mappings

+

t - 1

fi(gi(t,~))dg RSSyo

are continuously differentiable on Vo with gradient bounded by CR"'. Decreasing Vo and Ro if necessary we may suppose that Ro 5 1 and gi is a Lipschitz function with Lipschitz constant L on Vo x Wo. Similarly, we may suppose that fi is Holder continuous with exponent a and Holder constant H and 1 fi 1 bounded by B on gi (Vo x Wo); moreover on a convex closed set containing

z

the functions ah are Lipschitz continuous with Lipschitz constant L:, and

1% 1 I I

the functions and are bounded by Bb and Bl, respectively (i = 1 , 2 , .. . ,n). Let us fix Ro, Vo, Wo, and yo. We will prove that f is locally Holder continuous on Vo with exponent 2 a / ( l + a ) . Let t, t' denote arbitrary elements of Vo and let 0 < R < Ro. Let us integrate the two sides of the functional equation over the simplex R S yo. We have

+

where c

> 0 is the measure of the simplex S. Hence

130

Chapter IV. Differentiability and analyticity

To get a good upper estimate for the left-hand side we need an upper estimate for the difference h ( t , 3, f l ( g l ( t ,3 ) )7 . . .

, f n (gn ( t ,9 ) ) )

We may apply Taylor's theorem for the function h with points

z = (t,y , z l , . . . , x n ) and

Z'

= (t',y, x i , . . . , z;)

where t ' , t E V , y E W , zi = f i ( g i ( t y, ) ) , and I , . . . ,n. We have

Z:

= fi (gi(t',y ) ) for

i

=

Using this and omitting variables we have

+

Using the triangle inequality, we get n 1 terms on the right-hand side. ' Bb For the first term we get the trivial upper bound c ~ ~ ~ b- ltl,t where is an upper bound of . Let z! = fi ( g i ( t ,y o ) ) (i = 1,2,. . . ,n ) , and let dh z0 = ( t ,yo, Z! , . . . ,z:). If hl denotes the value of the partial derivative dzi at the point xO,than the other terms can be rewritten in the form

1%1

S

(zi-~,!)drdy+h;

(zi

-XI)

dy.

RS+yo

First we give an upper estimate for the absolute value of the first term of - t ' ~ ) where ~ , H is a this sum. An upper estimate of z i - Z : is H ( L I ~ Holder-constant for f i and L is a Lipschitz-constant for gi. Hence

dr dy.

131

fj 12. Holder continuity of solutions

Furthermore we need to estimate the difference

+

This is not greater than LQmultiplied by the norm of r z (1 - r)z' - x O , that is, Li times the maximal distance between the vectors z' and z0 = ( t ,yo , zy , . . . , z:) , where LQis a Lipschitz-constant for ah The maximal

2g.

distance between x' and zOcan be estimated by

Hence we get the upper bound

for the first term. To get an upper bound for the second term, we need an upper bound for the absolute value of

because ihil is trivially bounded by the upper bound Bl of

1

221

. From the

lemma we get the upper bound It - t'ICRbl for this integral. Summing up all these estimates, we get

If It

-

t'l 5 R this can be rewritten as If(t) - f(t')I 5 Colt -t'l

+ Cllt-

It t'l"Rff + Cz-

-

R

t'I

'

where Co, C1, and C2 do not depend on t , t', and R. If we choose R such " ) , we have that it satisfies the condition R = It - t l ~ ( l - " ) / ( l f then

1

+ + C 2 ) ( t t'(201(1Cn)

I f ( t )- f (t') 5 (Co C1

-

and t , t' E Vo. This proves that f is locally whenever It - t' 1 < R0(1+")1(1-") Holder continuous on Vo which implies the theorem.

132

Chapter IV. Differentiability and analyticity

12.2. Corollary. Under the conditions of Problem 1.18, if we only suppose that the functions h and gi are twice continuously differentiable, then every solution f which is locally Holder continuous with some exponent 0 < a < 1, is locally Holder continuous with any exponent 0 < a < 1.

Proof. If f is locally Holder continuous with some exponent a 0 = a , 0 < a 0 < 1, then by the previous theorem it is locally Holder continuous with exponents a, = 2an-l/(l a,-l), n = 1 , 2 , . . . , as well. Since an -f 1, we obtain the statement.

+

13. SOLUTIONS OF BOUNDED VARIATION As we mentioned in 2.14 we follow the terminology of the book of Giusti [60] regarding functions of bounded variation with several variables. Hence, strictly speaking, we are working with a generalization of the notion of bounded essential variation for functions of a real variable (not the generalization of the usual notion of bounded variation). We will prove only one theorem, which says that solutions of locally bounded variation satisfy the local Lipschitz condition. This result appeared in [108]. The following lemma is the key to our main result. 13.1. Lemma. Let V, W, and U be open subsets of RS, R k , and Rr , respectively, f : U -+ R a function of bounded variation, g : V x W + U a continuously differentiable function, to E V, yo E W, and suppose that

%J yo) has rank r. Then there exists a ball Vo c V centered in t o , a ball -(to, 8~

B

C

W centered in yo, and a constant C, such that

whenever t', t E Vo.

Proof. Taking smaller U, V, W if necessary, without loss of generality we may suppose that U, V, and W are bounded convex open sets, W = W' x W" where W' c Rr and W" C lRk-', and g is a Lipschitz function on V x W with Lipschitz constant M, moreover, that the determinant of

5 13. Solutions of bounded

133

variation

is not zero at the point yo, where y' E W ' , y" E W". Let p denote the projection ( y ' , y") +-+ 3' mapping IRk onto Rr. Let g,* denote the mapping ( y ' , y") +-+ (gt ( y ' , y"), Y") of W = W' x W" into U x W". The linear operator L o = gt*,'(yo) has determinant equal to the determinant of

hence it differs from zero. Let

Choose an open convex neighborhood X of Lo such that each operator L in X is invertible and satisfies

Taking an appropriate ball Vo C V centered at to and a smaller W if necessary, we may suppose that g f l ( y o ) E X and

whenever t E Vo and y E W . The last two inequalities imply that

whenever t E Vo and y E W. By the proof of the inverse function theorem (see Rudin [172],Theorem 9.24), from this it follows that gf is a homeomorphism of W onto an open subset of U x W". We need a somewhat stronger result. Let t , t' E Vo and 0 5 T 5 1. Let Gt,tl,T(y) = r g ; ( y ) ( 1 - r ) g ; ( y ) . We need that this function is a homeomorphism. Since

+

and X is convex, this linear operator is in X. On the other hand,

Chapter IV. Differentiability and analyticity

134

and hence

This proves that we may apply the proof of the inverse function theorem for Gt,t~,,and we get that GtItt,, is a homeomorphism of W into U x W". Moreover, from

we obtain the estimate

11

which is valid whenever [IT- SII < l / l I ~ - l / .This implies Ilgt'(y)-l 5 l l c and 5 l / c whenever t , t l E Vo, 0 5 r 5 1, and y E W. Hence the absolute values of the Jacobians of gf-I and G E ~ ,are , bounded by l/ck on g,* (W) and Gt,t~,,(W), respectively. Now we are ready to prove the lemma - first for a continuously differentiable function f . Let B be an arbitrary closed ball in W with center y o Using Taylor's theorem and the substitution x* = Gt,tl,t(y),we have the following estimate:

G:,~.,,(~)-'I

$13. Solutions of bounded variation

135

where ( W UisJ the Lebesgue measure of W" and J denotes the absolute value of the Jacobian. Finally, to get the statement, we use theorem 1.17 from the book of Giusti 1601. By this theorem, there exists a sequence of C" functions on U such that r

if j

+ m, whenever t, t'

E

V. Hence the lemma is proved.

13.2. Theorem. Let T , Y, X I , . . . ,X, be open subsets of PS", FS'", and R r l , . . . , Rrn,respectively, and Z , Zo,21,. . . , Z, be open subsets o f Euclidean spaces. Let D be an open subset of T x Y and W be an open subset of D x Zo x Z1 x x 2,. Consider the functions f : T + Z , f o : Y + Zo, fi : X i + Z i (i = I , . . . , n ) , g i : D + X i ( i = I , . . . ,n ) , h : W -+ 2.Suppose that (1) for each (t,y) 6 D we have

(2) h is continuously differentiable;

136

Chapter IV. Differentiability and analyticity

(3) gi is continuously differentiable on D and for each t E T there exists a y such that (t, y) E D and the rank of (t, y) is ri for i = 1 , . . . , n;

%

(4) f o is continuous and the functions fi, i = I , . . . , n are continuous and have locally bounded variation.

Then f is a locally Lipschitz function.

Proof. Replacing fi (i = 1,2, . . . ,n ) by its coordinates we may suppose that each fi is real valued. Let to E T and let us choose yo by (3). Let us choose open neighborhoods Ui of gi (to,yo) on which fi has bounded variation and neighborhoods V, Wo of to,yo, respectively, such that gi maps V x Wo into Ui. If V and Wo are small enough, we may apply Taylor's theorem for the function h with points z = (t, y, zo, z l , . . . , zn) and z' = (t', y, zo, zi, . . . , z n ) where t t , t E V, y E W, 20 = fo(y), zi = f z ( g i ( t , y ) ) and , = fi(gi(t'7y)) for i = 1 , . . . , n . We have f (t')- f (t) = h(z')

-

h(z) =

[:

-

(rz'

+ (1- r ) z ) (t'

-

t) d r

If V and Wo are small enough, then the partial derivatives of h are bounded by a constant K. Using the previous lemma, we can choose Vo C V and B c Wo such that the statement of the lemma is true for i = 1 , 2 , . . . ,n. Integrating both sides over B we have

where IBI is the Lebesgue measure of B. Using the previous lemma it follows that f satisfies a Lipschitz condition on Vo.

§ 14. Differentiability

137

14. DIFFERENTIABILITY In section 11 we have proved for special "sum form" functional equations that continuity of solutions implies their continuous differentiability. In two other cases we have shown continuity implies the local Lipschitz property. By the well-known theorem of Rademacher (see Federer [53], 3.1.6) a Lipschitz function f : Rm + Rn is Am almost everywhere differentiable. A similar result holds for any locally Lipschitz function mapping some open subset of a Euclidean space into some other Euclidean space. (See the more general theorem of Stepanoff in the book of Federer [53], 3.1.9.) In this section we will prove that differentiability almost everywhere implies continuous differentiability. We will use the results about continuity of measurable solutions. Combining "almost everywhere differentiability implies continuous differentiability" type theorems with the above mentioned classical results, we see that "local Lipschitz property implies continuous differentiability". By Lebesgue's theorem, monotonic functions of a real variable, or vector-valued functions of a real variable that are of bounded variation are almost everywhere differentiable. This result also can be combined with the theorems of this section to prove that "real variable solutions of bounded variation are continuously differentiable". (A theorem of this kind was proved first directly by another method in the paper [94].) In this section we need the fact that the derivative of an almost everywhere differentiable function is Lebesgue measurable. We will prove much more. We will prove that the derivative of any function mapping an open subset of some Euclidean space into a separable Banach space is defined on a Borel set and is a Borel function. This result was published in the paper J&rai [84]and generalizes the analogous result for continuous functions from the book of Federer ([53], 3.1.1.). I believe this result is interesting in itself. 14.1. Theorem. Let U be an open subset of the finite dimensional normed space X , let Y be a separable Banach space, and let f : U + Y be an for any closed subset arbitrary function. Then (f')-'(F) is in the class .Tos F of the normed space L(X, Y) of continuous linear operators mapping X into Y.

Proof. Without loss of generality we may suppose that U = X . It is well-known (see Hewitt and Stromberg [69], p. 78 for instance), that the set C of all continuity points of f is a Gs set of X . It is clear that L ( X , Y) is a separable Banach space. Let us choose a dense subset L1, L 2 ,. . . of F . Whenever i, j, and k are positive integers, let

138

Chapter IV. Differentiability and analyticity

Every Ci,j,k is a relative closed subset of C. To prove this, let x E C \ Cilj,k. Then there exists a n h E X for which lhl < l / j but

+

Let h' be defined by h' = h x - x' whenever x' E X. We observe that h' = x + h, so f (x' h') = f ( x h ) . B y the continuity of f at x, there exists a 6 > 0, such that Ih'l < l / j and

x'

+

+

+

whenever / x - X I / < 6. Now we will prove that

Suppose that f 1 ( x ) = L E F . Then x E C , and for each i there exists an Lk(i)and a n j ( i ) for which

whenever 1 hl

< j ( i ) . This proves

that

whenever lhl < l / j ( i ) ,i. e., x E Cz,j(i),k(i). On the other hand, suppose that x E C and for each i there exists a j(i) and a k ( i ) for which

whenever Ihl < l / j ( i ) . Let il and iq be two positive integers, and let us estimate the distance IILkcil)- Lkci2)11.If Ihi < l / j ( i l ) and jhl < l / j ( i z ) , then we get

5 14. Differentiability

139

This proves that Lk(i),i = 1,2, . . . is a Cauchy sequence in C(X, Y). Let L denote the limit of this sequence. Clearly, L E F. Choosing an arbitrary i, we can choose an i* 2 i for which. IILJ-(~*) - LII < l l i . Hence

whenever I hl < llj (i*), and this proves that L is the differential of f at x. Having proved ( I ) , and from the fact that C is a Gs set, it follows that there exist closed subsets F,,t of X such that

and

i. e., (f')-I (F)E FO6. 14.2. Theorem. Let Z be a normed linear space, Zi open subsets of separable Banach spaces (i = 1,.. . ,n),Y an open subset of R\ T an open subset of IW" , and the Xiopen subsets of Rri . Let D be an open subset of T x Y. Consider the functions f : T -+ Z , fi : Xi -+ Zi, gi : D + Xi (i = 1 , 2 , .. . ,n),h : D x Z1x - .. x Z, -+ Z. Suppose that to E T and the following conditions hold:

(1) for each (t, y) E D ,

dh dh (2) for any fixed y E Y the partial derivatives - and - (i = 1 , 2 , . . . ,n) at dzi are continuous in the other variables;

Chapter IV. Differentiability and analyticity

140

(3) fi is X r i almost everywhere differentiable (i = 1 , 2 , . . . , n ) ;

(4) gi has a continuous derivative, and there exists a yo for which (to,y o ) E D dgi and the matrix -(to, y o ) has rank ri (i = 1,2,.. . , n ) .

d~

Then f has a continuous derivative on a neighborhood o f t o .

Proof. Let us choose open sets T' and Y' with compact closure for which ( t oy, o ) E T' x Y' c T'x Y' c Dl and for every E > 0 there exists a 6 > 0 such that X r i ( g i , t ( ~ ) ) S whenever t E F , B C F. This is possible by Lemma 3.10. Let now D' be the set of all pairs (t',y') for which t' E T ' , y' E Y ' , and f i is differentiable at the point gi(tl,y') for 1 5 i 5 n. Let us observe that D' is a Bore1 set and

>

{y' : y' E Y ' , f,l does not exist at the point gi (t',y ' ) )

has measure 0, whenever t' E TI, because its image by gi,t/ has measure 0 in - Xi. This proves that T' x Y'\ D' has A% X k measure 0. Let X( = gi (T' x Y ' ) (i = 1 , 2 , . . . ,n). Suppose, that (t',y') E Dl and fix y'. We study the function

This function is defined on a neighborhood of t' and has a derivative at t'. Hence f is differentiable at t' and

"

+

C i=l

dh

agi

(t',Y ' , f l

st)), . . . , f. (s. (t',9 ' ) ) )I,'(%(t',9 ' ) ) at (t',y').

(s1(t',

This means that f ' , f l , f i , . . . ,f,, f ; satisfy an equation of the form

for all (t',y') E Dl, i. e., almost everywhere on T' x Y'. We apply Theorem 10.2 to this equation to prove that f' is continuous on TI. Let tb be an arbitrary element of TI. The measurability of the functions fi easily follows from their differentiability almost everywhere. By Lusin's theorem we see that all conditions of Theorem 10.2 are satisfied at tb.

5 15.

141

Higher order differentiability

15. HIGHER ORDER DIFFERENTIABILITY In this section we study the following question: If a solution is p times differentiable, is it also p + 1 times differentiable? Two theorems will be proved. The only difference between them is in their conditions about functions fi and h.

15.1. Theorem. Let Z be a normed linear space, let Zi be open subsets of some separable Banach spaces (i = 1,. . . ,n), let Y be an open subset of some Euclidean space, and let T and Xi, i = 1 , 2 , .. . ,n be open subsets of RS and IWTi, respectively. Let D be an open subset of T x Y. Consider the functions f : T + Z , gi : D + Xi (i = 1 , 2 , . . . ,n ) , fi : Xi Zi (i = 1,2, . . . ,n ) , h : D x Z1 x . . x Z, i Z . Suppose that p > 0 is an integer, to E T, and the following conditions hold: ( I ) for each ( t ,y) E D ,

(2) for any fixed y E Y all partial derivatives 8;' 8; in the other variables whenever 0 5 Icy 5 p;

. . .a,","

h are continuous

(3) the function fi is XT2 almost everywherep times differentiable (1 5 i 5 4 ; (4) gi is p times continuously differentiable and there exists a yo for which &?i (to,yo) E D and -(to, yo) has rank ri (i = 1,2,. . . ,n ) . ay Then f is p times continuously differentiable on a neighborhood of to.

Proof. For p = 1 we obtain the last theorem of the previous section. The proof will be similar to the proof of that theorem. Let us choose open sets T' and Y' with compact closure for which (to,yo) E T' x Y' C T'x Y' C D , and for every E > 0 there exists a 6 > 0 such that XTi (gilt(B)) 2 S whenever B C Y'. This is possible by Lemma 3.10. t E T', Let now D' be the set of all pairs (tt,y') for which t' E TI, y' E Y', exists at the point gi(tr,y') for 1 5 j 5 p and 1 5 i n. Let and X,' = gi(T' x Y') (i = 1 , 2 , . . . ,n). As in the proof of the previous theorem, we obtain that f ', f l , f i , . . . , f,, f; satisfy an equation of type

fp)

<

142

Chapter IV. Differentiability and analyticity

whenever (t',y') E D'. Here hl satisfies the same condition as h but for p - 1 instead of p. Suppose, that (t',9') E D' and let us fix y'. The function

is defined on a neighborhood of t' and is differentiable at t'. Hence f' is differentiable at t'. By differentiating we obtain that

f",fl,f:,f:l,... ;fn,fL,fl satisfy an equation

for all (t',y') E Dl. By induction we obtain that

satisfy an equation

for all (t',y') E D' where h, is continuous. Now as in the previous proof, Theorem 10.2 can be applied for this equation and we obtain that f ( p ) is continuous on TI. 15.2. Theorem. Let Z be a Euclidean space, Y and Zi open subsets o f some Euclidean spaces (1 i 5 n ) , and let T and Xi (i = 1,2,. . . , n ) be open subsets of RS and R T i , respectively. Let D be an open subset of T x Y . Consider the functions f : T + Z , gi : D + Xi ( i = 1,2,. . . ,n), fi : X i :+ Zi (i = 1,2,... , n ) , h : D x Z1x . . . x Zn 4 Z . Suppose that p > 0 is an integer, t o E T , and the following conditions hold:

<

( I ) for each ( t ,y ) E D ,

(2) all partial derivatives 8;' 8F11 .. ever 0 < 1ct.l < p;

h are continuously differentiablewhen-

(3) the functions f i , i = 1 , 2 , ... , n arep times continuously differentiable;

§ 15. Higher order differentiability

(4) gi is p

143

+ 1 times continuously differentiable and

there exists a yo for agi which (to,yo) E D and --(to, yo) has rank ri (i = 1 , 2 , . . . ,n). ay Then f is p 1 times continuously differentiableon a neighborhood of to.

+

Proof. Let 1 5 q 5 s and differentiate equation (1) partially with respect to t,. We have, omitting the variables,

fi = ( f i , j ) , and gi = (gi,k). This equation shows Here xi = (xilj),X i = that, whenever a E Ns , Ial = I , 8" f satisfies the functional equation

for ( t ,y) E D. Here, if the q-th coordinate of a equals 1, all others are 0,

fa,p = -

for some i , j , k ,

8Xi.k

and

for some i, j , k. It is clear that h,,p : D + Z is continuous and its p - 1-th partial derivative with respect to t is continuously differentiable, moreover f,,p : Xi + R for some 1 5 i 5 n and is p - 1 times continuously differentiable if 0 5 p 5 n,. Repeating this process, we have, by induction with respect to la1 that if a E Ns and 1 5 la1 5 p, then 6'" f satisfies the functional equation

144

Chapter IV. Differentiability and analyticity

whenever ( t ,y ) E D. Here : D + Z is continuous and all of its at most p - /al-th partial derivatives with respect to t are continuously differentiable; moreover f,,p : Xi + R for some 1 5 i 5 n is p - Ial times continuously differentiable, and, finally, g,,p = gi for the same i for which dmn f,,p = Xi. Now let a E NS,la/= p, and use Theorem 11.3. We have that every p 1-st partial derivative of f is continuous on a neighborhood of to and hence f has a continuous p 1-st derivative on a neighborhood of t o .

+

+

16. ANALYTICITY In this section a theorem will be proved which can be considered as a first step towards proving analyticity of solutions and was published in the paper JBrai [95]. Another result concerning "Cm implies analyticity" have been found by PBles [163]. He has a general method which is applicable to reduce linear functional equations with two real variables to differential equations. He uses this method to prove that Cm solutions are analytic except for isolated singular points. PQles's method will be discussed, too. We will use some basic properties of analytic functions. All these results can be found in Federer [53]. The following lemma can also be found there (p. 239), but formulated differently. 16.1. Lemma. I f the real functions f and g are p times differentiable at the point t and x = g ( t ) , respectively, then

where the sum is taken for the set S ( p ) o f all sequences a havingp nonnegative integer terms such that P

moreover the coefficients e, are positive integers depending only on a. I f A and B are nonnegative real numbers, then

§ 16. Analyticity

145

Proof. ( 1 ) can be proved easily by induction with respect to p. To prove ( 2 ) let us consider the functions

and

Then

and the derivatives are easy to calculate. Applying (1) for these functions in point t = 0 we get the statement. 16.2. Theorem. Suppose, that the unknown function f : ] a ,b[ + R satisfies the functional equation

where the ci E R (i = 1 , 2 , . . . ,n ) , the functions

are given, and the following conditions are satisfied: (2) g i ( t , y ) is between t and y whenever t , y E ] a ,b [ ;

whenever t , y E ] a ,b[ and p = 1 , 2 , . . . , with some constant 0

< A < 1;

(4) for all t E ] a ,b[ and i = 1 , 2 , . . . ,n the mapping y e gi(t,y ) o f ] a ,b[ into ] a ,b[ is strictly monotonic, and the function gi, for which the mapping x ++ g(t,x ) is the inverse o f the mapping y ++ gi (t,y) , is twice continuously differentiable on its domain. Then every infinitely many times differentiable solution f o f (1) is analytic on ] a ,b[. Proof. We will prove that if [c,d] is a compact subinterval of ] a ,b [ , then there exists a real constant 0 < B < oo,such that

146

Chapter IV. Differentiability and analyticity

for all t E [c,dl. Hence f is analytic on ]c,d[, and the same is true for ] a ,b[. To get the estimation (5) we will use induction with respect to p. For df - the existence of such a B is trivial, because it is continuous. Let us dt dPf (t) is on the left differentiate p times both sides of equation (1). Then dtP side, while the right side is the linear combination of partial derivatives of terms f (gi(t, y)) with respect to t. Now we are going to estimate such a term, for simplicity omitting the subscript i. By the preceding lemma such a partial derivative has the form

where S ( p ) is the set of all sequences a having p nonnegative integer terms such that P

and the coefficients e, are positive integers. Let us integrate both sides of the equation (6) with respect to y from c to d. Then the left side has the form

and the right side is a linear combination of terms of type

In each of these terms let us introduce a new variable with the substitution x = g(t, y). Then we get

By the differentiability of integrals with respect to the limits, and by the theorem concerning differentiability of parametric integrals with respect to the

§ 16. Analyticity

147

parameter, expression (8) is differentiable with respect to t , and its derivative is

We will get an upper bound to the absolute value of this expression in the following way: Since gi is twice continuously differentiable on its domain, there exists a constant 1 < E < m, such that on the compact set

we have

Hence we get the following upper estimate for the second and the third terms in (9):

The estimation of the integral is more difficult. If we take the derivative of the last term in the parenthesis with respect to t, then the upper estimate is

Chapter IV. Differentiability and analyticity

148

and if we take the derivative of the j-th term, then the upper estimate is

Hence the total upper estimate for the integral is

4 (d - c

(A's!)~".

) l ~ ~ p B (Ca)! ' ~ s=l

Summing up for a E S(p), and using the lemma above, the upper estimate for all terms originated from a term f (gi(t, y)) is

Finally, summing up for i, we have that

for all t E [c,

4,if

B>

(

n

2E 4 E 2 +d-- )c ~ / c i ~

and

i=l

Hence it is proved that f is analytic on ] c , d [ .

A 1-A'

B>-

5 16. Analyticity

149

16.3. Pgles's method. Up to the end of this 5 we will treat PAles's general method to find C" solutions of functional equations and to prove their analyticity. Although our treatment will be different from that of Pdes, everything here has to be attributed to him. A large part of the material is in his paper [163]; other parts he explained during our conversations. Let us see first the basic idea. Consider the functional equation

with unknown real functions f o ,f l , . . . , f n and with two real variables x, y. Applying differential operators

we obtain from (1) a system of fiinct ional-differential equations

If we consider all the differential operators having order 5 m, then we obtain (m I ) (m 2)/2 equations. Let us consider the (m 1 ) n unknown quantities f j(0, 1 - 0 , . . . , m , j = 1,.. . , n. If m is large enough we have more equations than unknowns. Solving some of the equations for some of the unknowns and substituting the result back into the other equations we may hope to arrive at an equation having the form

+

+

+

Now let us fix any yo and let us introduce the variable t = go(x,yo). Solving this equation we may express x as a function of t , say x = ljo(t,yo). Hence we obtain the differential equation

Solving this equation, we obtain fo. Similar procedures work for further unknown functions. Let us observe that we may get more than one equation similar to (2). This may be used to eliminate some terms, for example some higher order terms from (3). Another possibility is to apply the procedure to (3) and to use the resulting system to eliminate some higher order terms.

150

Chapter IV. Differentiability and analyticity

More generally, we may consider functional equations in several variables. Let D c RN and D j c RNj , j = 0 , 1 , . . . , n be open subsets, where Nj < N for j = 0 , 1 , . . . , n. Let gj : D -+ D j be given, "sufficiently smooth" functions and h also a "sufficiently smooth" function mapping an open subset of some Euclidean space into R. Let us suppose that the unknown, but "sufficiently smooth" functions f j : D j -+ R satisfy the functional equation

Applying all differential operators id, dl, 6'2,. . . , d ~d:, , d l & , . . . , d;, . . . , d F up to order m we arrive at a system of functional-differential equations

where the symbol (dmf j ) (gj (x)) has the meaning

Now we may eliminate from this system all terms except (dmf o )(go(x)) to obtain a partial differential equation for fo. To include the vector-valued case, too, even more generally, we may start with a system of functional-differential equations such as (6) with k > 0 and m 2 0. For a fixed M 2 m let us apply all partial differential operators d", a E NN, jal 5 M - m to all equations of system (6). We arrive at another system

Here for the number of equations

5 16.Analyticity

151

and for the number of terms coming from (dMf j ) ( g j (x)), j = 1 , 2 , . . . ,n (which we would like to eliminate) we have

To see that for sufficiently large M we have more equations than unknowns we observe that

because the numerator and denominator are polynomials in M having degree m - 1 and m, respectively. Let us observe that this general method works even for composite equations, if we know that the solutions are "sufficiently smooth". After eliminating the outer unknown functions fi, i = 0 , 1 , . . . ,n, we arrive at a functionaldifferential equation for the unknown functions gi. If the equation is still composite we may continue this process. 16.4. PAles's m e t h o d for linear equations. We do not have good tools to solve general nonlinear equation systems. This suggest that we restrict ourselves to the linear case. Let us suppose, as above, that D c IRN and D j C I R N 3 , j = 0, 1 , . . . , n are open sets, where N j < N for j = O , l , . . . , n. For simplicity, let us suppose, that the given functions g j : D + Dj and h, h j : D + R, j = 0,1, . . . ,n are analytic, and D is connected. Let us suppose that the unknown functions f j : Dj + R satisfy the linear equation n

(1)

Chj(~)fj(~j= ( ~h(x) ) ) whenever x E D. = jO

Chapter IV. Differentiability and analyticity

152

If the unknown functions are at least m times continuously differentiable, we may obtain from this equation a system of linear equations

where the functions hi, hi,, : D -+ IF2 are analytic. More generally, starting with any system (2), supposing that the functions f j , j = 0,1, . . . ,n are M times continuously differentiable, and applying any set of linear differential operators

with analytic coefficients a, : D + R, we arrive at a similar system as (2) but with some K instead of k and M instead of m. The advantage of supposing that the functions hi, hh,j are analytic is, that we may consider the quotient field JF of the ring of all real-valued analytic functions over D (or the quotient field of any subring of this ring, if all of the functions hi, hi,, are in this subring). The notions of ordinary linear algebra, such as linear dependence and independence, rank, determinant, etc. can be applied to investigate the solvability of system (2). We will use two matrices H and H*. In both the elements are hj,,, in both the line index i runs from 1 to k. In H the column index runs on all pairs (j,a),1 5 j 5 n , la1 5 m. n , la/ 5 m. In H* the column index runs on all pairs (j,a),0 5 j

<

In this setting, the main result is: 16.5. PBles's t h e o r e m . With the above conditions and notation, suppose that rank(H) < k. Let us suppose that for some

c

{

2.

k}

and

J c { ( a , j ) : la1 5 m , l 5 j l n )

with card(1) = card(J) = rank(H) the determinant of the corresponding submatrix of H is nonzero. Then system 16.4.(2), i E I can be uniquely solved for the unknowns (8" f j ) o g j , (a,j) E J . Putting these solutions back into equations 16.4.(2), the resulting equations do not contain any terms (dof j ) o g j , 1 5 j 5 n, la/ 5 m. If, moreover, rank(H*) > rank(H), then at least one of the resulting equations depends on the term (dmf o ) 0 go. In this setting, the statement is obvious. We remark, that PBles formulated the statement differently, namely, that starting with equation 16.4.(1)

§ 16. Analyticity

153

there exists a non-trivial linear differential operator "killing" all the terms f j (gj (x)), j = 1 , 2 , .. . ,n , and if rank(H) < rank(H*), then this operator does not "kill" the term f o (go(x)). Moreover, he does not suppose that the functions gj, h j are analytic. See his paper [I631 for details.

16.6. PBles's algorithm. The above theorem gives us an algorithm to solve linear functional equations or even linear functional-differential equation systems. Starting with such a system, and getting newer and newer equations by applying differential operators, we may always check the rank of the corresponding H and H * . This can be done most efficiently using Gaussian elimination. It is also possible to add the new equations in blocks, say, for example, all equations corresponding to differential operators with a given order. This procedure will not result in loss of information: If rank(H) < rank(H*) for a given system of equations, then this inequality will stay true if we add new equations. If the equations are not homogeneous, then it is better to calculate the rank of a new matrix, too, which we get by adding a new column hZ, i = 1 , 2 , .. . , k to H*. If this increases the rank, a nontrivial condition has been found which should have been satisfied by functions hZ. e @ to carry We remark that recently A. HAzy [65] wrote a ~ a ~ l program out these calculations automatically. We will give examples in 22.11, how to use this algorithm. Computations will be given as transcripts of computer interaction to avoid programming technicalities that make the essence of the procedure hard to grasp. Another related idea proposed by Baker is to solve the linear functional equation system 16.4.(2) in the distribution sense. This is based on the possibility to define the pullback of a distribution: if D and E are open subsets of Euclidean spaces and g : D + E is a submersion, then there exists a unique continuous linear map g# : D1(E) + V1(D) such that g # ( f ) = f o g when f : E -+ C is a continuous function having compact support. We call g# (f ) the pullback of f . See the details and proofs in Hormander [74], chapter VI. Baker suggested using the pullback of distributions and regularity results for distribution solutions of differential equations to find distribution solutions of functional equations. Two examples are given in his paper [34]. Using the conditions of 16.4, 16.7. PBles's regularity theorem. suppose that N = 2, Nj = 1 for j = 0 , 1 , . . . , n , rank(H) < rank(H*), and that go maps D onto Do. Then f o is analytic on an open subset of Do whose relative complement to Do is discrete. @ ~ a ~is la eregistered trademark of Waterloo Maple Inc

154

Chapter IV. Differentiability and analyticity

Proof. By the above theorem we obtain a differential equation having the form

where a , a j are analytic functions on D , r 5 m, and a, is not identically zero on D . Now we introduce a concept to be used in this proof only. A point t on Do is called a singular point with respect to a real analytic function h defined on D if go (x, y) = t always implies h(x, y) = 0. We show that the set of singular points is a countable set without limit points in D , provided that h is not identically zero. Assume, on the contrary, that there exists a non-trivial convergent set, = to E Do. Without quence t, of singular points such that limn,, loss of generality, we can also assume that this sequence is strictly monotonic, say increasing. Choose a point (xi,yi) € g,l(ti) arbitrarily. Let (x*,y*) E D be fixed. Since D is connected, there exists a continuous function (u, v ) : [O, 11 -+ D such that

Using the Weierstrass approximation theorem and Lagrange interpolation, one can find polynomials arbitrarily close to u and v satisfying (2), too. Thus we may assume that u and v are polynomials themselves. Hence the functions cp, $ defined by

are analytic on [ O , l ] , further we have $(O) = t o , $(I) = tl. By Bolzano's theorem, for each n there exists 7, E [ O , l ] such that $(r,) = t,. Since t, is a singular value this yields cp(r,) = 0. Being analytic, cp must be identically ) h(x*,y*) = 0, and this shows that h is identically zero zero. Then ~ ( 1 1 2 = on D . The contradiction obtained proves our statement concerning singular values. The derivative of go cannot be identically zero on D since then go would be constant, but go(D) = Do is open, so this is impossible. Thus we may assume that &go is not identically zero on D. Using the above statement, we see that the set of singular points with respect to aTdlgois countable and has

155

$16. Analyticity

no limit points in Do. Let to E Do be a regular (i.e. non-singular) point. We show that f o is analytic at to. The non-singularity of to implies the existence of a point (xo,yo) E D such that a, (xo,yo) # 0 and &go(xo,yo) # 0. The latter property says that the function x H go(x, yo) is strictly monotonic in a neighborhood of xo. Thus there exists a strictly monotonic analytic function p such that go (p(t),yo) = t in this neighborhood and p(to) = xo. In a (possibly smaller) neighborhood U of to we also have a, (p(t),yo) # 0. Substituting now x = p(t) (t E U) and y = yo into ( I ) , we arrive at the ordinary linear differential equation

fi') +

c

a; (t)f f ) = a*(t),

t t U7

where a; (t) = a j (,u(t),yo) /a, (p(t),yo) , j = 0, 1,. . . ,r - 1, and a* (t) = a ( p ( t ) ,yo) / a , (p(t), Since all coefficients of (3) are analytic functions, hence f o must be analytic on U.

Chapter V.

REGULARITY THEOREMS ON MANIFOLDS

In this chapter there is only one section. Here we generalize the most important results to manifolds.

17. LOCAL AND GLOBAL RESULTS ON

MANIFOLDS As mentioned in the introduction, most of the regularity results can be generalized to manifolds. Most of the theorems are of a local nature, and thus generalization causes no problem in their case. Theorem 11.6 is of a global nature hence we have to give a new proof for manifolds, which differs in several technical details from the original one. In what follows we only formulate the generalization of the most important theorem for manifolds. In the case of "local" theorems the more general version can be easily obtained from the special one. In the case of the only "global" theorem we give an independent proof. These results appeared in [log].

17.1. Theorem: Measurability implies continuity. Let T and Z be topological spaces and let Zi be topological spaces with countable base. Let Y and Xibe differentiable manifolds (i = 1 , 2 , .. . , n), D an open subset o f T x Y , and W c D X Z1x . - . x 2,. Considerthefunctions f : T + 2,

158

Chapter V. Regularity theorems on manifolds

gi : D + Xi, fi : Xi + Zi (i = 1 , 2 , . . . , n ) , and h : W + Z. Suppose, that to E T has a countable base of neighborhoods and the following conditions are fulfilled: (1) for each (t, y) E D we have

(2) for each fixed y in Y, the function h is continuous in the other variables;

(3) the function fi is Lebesgue measurable on the subset Ai of Xi (i = 1 , 2 , .. . ,n);

(4) n:=,

g,2(Ai) is not a null set;

(5) for each t E T, gi,t is a C1 submersion and considering the open subset of T times the tangent bundle of Y of all triples (t, y, v), where (t,y) E D and v is in the tangent space of Y at y, the mapping

is continuous into the tangent bundle of Xi (i = 1 , 2 , .. . ,n). Then the function f is continuous at to. Proof. The theorem will be reduced to Corollary 8.7 using local coordinates. Since Y can be covered by the domains of countably many charts, there exists a chart cp such that the intersection of its domain Wo with the set n:=,g;;io (Ai) is not a null set. Consider the image of this set by ip and let yo* be a density point of it. On sufficiently small neighborhoods of the points gi(to,yo) there are charts ti defined. Let us choose a small enough neighborhood V of to and let us decrease Wo, if necessary, so that V x Wo c D and gi (V x Wo) c dmnti (i = 1,2,. . . ,n) are satisfied. Let W," = rng(cp), D* = V x W;, X; = rng((i), A: = &(Ai), gZ(t,y*) =
if (t, y*) E D * and (t, ip-'(y*), zl, . . . ,zn) E W. Corollary 8.7 can be applied to the sets and functions denoted by * and hence the proof is complete.

5 17. Local and global results

on manifolds

159

17.2. Theorem: Baire property implies continuity. Let T , Z , and Zi ( i = 1,2,. . . ,n ) be topological spaces. Let Y and Xi be differentiable manifolds ( i = 1,2,. . . , n ) ,D an open subset of T x Y and W C D x Z1 x x 2,. Consider the functions f : T -+ 2, gi : D -+ X i , fi : Xi -+ Zi (i = 1,2,. . . , n ) , and h : W + Z . Suppose that to E T has a countable base o f neighborhoods and the following conditions are fulfilled: (1) for each (t,y) E D we have

(2) for each fixed y in Y , the function h is continuous in the other variables; (3) the function fi has the Lusin-Baire property on the subset Ai o f Xi ( i = 1,2,.. . , n ) ;

(4)

nEl

( A i ) is of second category as a subset of Y ;

g Z ~ ~ O

(5) for each t E T , gi,t i s a C 1 submersion and considering the open subset of T times the tangent bundle of Y of all triples ( t , y , v ) ,where ( t , y ) E D and v is in the tangent space o f Y at y, the mapping

is continuous into the tangent bundle of Xi ( i = 1 , 2 , . . . ,n) Then the function f is continuous at t o . Proof. The theorem will be reduced ordinates. Since f l ~ = l g (zA~i~) is of second for which ( t o ,yo) E D and for each open is of second category -I ( A ) ) WOf l (n?=1gi,to this result reduces to Corollary 9.5.

to Corollary 9.5 using local cocategory in Y , there exists a yo neighborhood W o of yo the set in Y. Now in local coordinates

The next theorem is the most important result of this section; it states that continuous solutions are locally Lipschitz. The proof depends on Lemma 11.5.

17.3. Theorem: Local Lipschitz property of continuous solutions. Let 2, T , and Y be manifolds, D an open subset o f T x Y , C a compact subset of T , and W an open subset o f D x Zn. Consider the functions f : T -+ 2, gi : D i T ( i = 1,. . . , n ) ,h : W 4 Z . Suppose, that

160

Chapter V. Regularity theorems on manifolds

(1) for each (t, y) E D we have

(2) h is twice continuously differentiable; (3) gi is twice continuously differentiable on D and for each to E T there exists a yo such that (to,yo) E D , gi(tO,yo) E C, and y H gi(to,y) is a submersion at yo whenever i = 1, . . . ,n; (4) f is continuous. Then f is a locally Lipschitz function on T.

Proof. Let us choose and fix a Riemannian structure on 2,Y , and T (see Dieudonnk [49], 20.7.13). We will use a distance on all three manifolds resulting from the fixed Riemannian structure, and for simplicity we will denote this with Q in all three cases. This distance is defined in the following way: on all components of the given manifold there exists a natural distance Q' ([49], 20.16.3). Let ~ ( xy), = min{l, ~ ' ( x , y ) )if x and y are in the same component of the manifold, otherwise let Q ( X , y) = 1. With this definition, Q is a distance generating the topology of the manifold, and if ~ ( xy), < 1 then x and y are in the same component, ~ ' ( xy) , is also defined, and equal to d x , Y). The proof uses the continuity modulus of f . Since this is not a subadditive function in general, we will use a modification. For this modification, we will prove that it satisfies a functional inequality, which will imply that f is a locally Lipschitz function. For an E > 0 let C, = {x : Q(X, C ) 2 E ) denote the (closed) E neighborhood of C . Let C, = C if E 0. Since T is locally compact, there exists an open neighborhood of C with compact closure. The distance of the complement of this neighborhood from C is positive. If E is less than this distance, then C, is a compact set. Let us choose for each point t of C an open neighborhood & in the same component, and a number 0 < ~t < 112 such that each open ball with center in & and radius ~t is strictly geodesically convex (see [49], 20.17.5). Let us choose from the open covering &, t E C a finite covering of C , and let E > 0 be less than the corresponding values ~t so that C, is compact. Let

<

§ 17. Local and global results on manifolds

161

Clearly w is increasing, w(0) = 0, and w is continuous at zero, because f is uniformly continuous on the compact set C,. We will prove that

We may suppose that 7-2 I r l . Suppose that w(r1 +r2) > w(r1) +w(rn). Then there exist x and y such that e(x, y) I r l r2, x E C,, and y E CE-e(z,y), but ~ ( (x), f f (y)) > w ( r l ) ~ ( r p )For . some c E C we have

+

+

hence ~ ( xc), 5 E. This implies that for some ~t > E the open ball centered at c and having radius Q is strictly geodesically convex, hence there exists exactly one geodetical arc connecting x with y and having length e(x,y). , Choosing the For an arbitrary point z on this arc, ~ ( xy), = ~ ( xz), ~ ( zy). , ~ ( xz), I r l , and e(z, y) I 7-2, point z in such a way that ~ ( yz), I ~ ( xz), we obtain

+

<

hence z E C,-,(,,,) C C,_,(,,,j. This implies ~ ((x), f f ( 2 ) ) w(r1) and Q (f (z), f (y)) Iw (r2),a contradiction. Let us fix an integer 7 > 1. For an arbitrary to E C, let us choose a yo by (3) (depending on to). Let zo = f (to),20 = gi(to,yo), and z: = f (x;) if i = 1 , 2 , . . . , n. Applying 20.17.1, 20.17.5 from [49] to the component of the manifold Z containing 20, we have that there exists a 0 < St, < 112, such that for all 0 < r I St, the open ball centered at zo and having radius r is strictly geodesically convex. Decreasing dt,, if necessary, we may suppose that the normal coordinates II, are defined on the open ball centered at zo and having radius St,, moreover, that if z and z' are in this ball, then

1 -

7

( z )-z

Ie

,I7

-

II,(z1)I,

i.e., II,and its inverse are Lipschitz functions between the open ball-neighborhood of zo with radius St, and its image in the coordinate space (which is also an open ball having radius St,, centered at z i = II,(zO)).(See 1491, 20.16.4, 20.16.5.) Let Z* denote the open ball centered at zz and having radius St, . Similarly, we may choose strictly geodesically convex open balls to the points to, yo, and z: of the manifolds T, Y, and Z , respectively, such that and their inverses are defined and are the normal coordinates 7 , cp, and

Chapter V. Regularity theorems on manifolds

162

Lipschitz functions with Lipschitz constant q. Let T*, Y*, and Z$ denote the image of these balls in the corresponding coordinate spaces. Decreasing the radii, if necessary, we may suppose that

and that the function

x Z; into Z*. This mapping will be denoted maps the set T* x Y* x Zf x by h*. Decreasing further the radii, if necessary, we may suppose that the partial derivatives of h* are bounded Lipschitz functions. Continuing similarly, let us choose for the points x: of the manifold T strictly geodesically 3 which the normal coordinates convex open balls Xi with radii at most ~ / on ti are defined and are Lipschitz functions together with their inverses with Lipschitz constant at most 17. Let X$ denote the image of these balls in the corresponding coordinate space. Decreasing the radii, if necessary, we may suppose that the mapping

fc.

Finally, decreasing maps X,* into 2;.We will denote this mapping by the radii of the balls T* and Y* around the points t; = r ( t o ) and y; = $(yo), respectively, we may suppose that the mapping

is a Lipschitz map of T * x Y* into X; ; it will be denoted by 9:

. The mapping

t* i $(f (rv'(t*))) of T * into Z* will be denoted by f * . Let w$ denote the (usual) continuity modulus of We will prove that w:(r) 5 qw(qr) for all r 2 0. If xi, xi E Xi, then Q(X~,X:) < &/3 , using that 2; E C , we obtain that xi E C, and and Q(X{,X;) < ~ / 3 hence @(xi,xi)< 2 ~ / 3which implies that xi E C,-,(,,,,;). Suppose that Iti(xi) ti(xi) 5 r . Then @(xi,xi) q r , which implies that ~ ( (xi), f f (xi)) w(qr). Hence

ft.

I

<

<

§ 17. Local and global results on manifolds

163

Taking the supremum on the left-hand side we obtain the inequality wd (r) I nw(rlr). Let us apply Lemma 11.5 with the sets and functions marked by *. We have that there exists a strictly geodesically convex open ball centered at to and having radius 0 < < 112, on which the normal coordinates r are defined, and there are constants Rt, > 0 and ct, such that if t, t' are in the open ball mentioned, and 0 < R < Rt,, then

5 B;ne(t, t')

+

n

B;ne(t, t') i=l

2

Of course here the constants Bh, Bl, Li, and Li depend on to, which is not expressed in the notation. Using the inequality between wd and w, from the inequality above we have

+ C rlw (Liq2@(t,t')) L: ( v Q ( ~t'), + R + i=l

c n

i j w ( n ~ (oe(t, j t')

+~ 1 ) ) .

j=1

give an For varying to the open balls Vt, centered at to and having radii open covering of the compact set C,. Hence there exists a finite set To c C, such that the open balls corresponding to the points to E To give an open covering of C,. Let L denote a positive integer, which is greater than or equal to all of the constants Li (i = 1,2, . . . ,n ) corresponding to all points to E To. Similarly, let B', L', and co denote a constant which is greater than or equal to all constants Bl (i = 0 , 1 , . . . ,n ) , Li (i = 1,2,. . . ,n ) , and ct,

164

Chapter V. Regularity theorems on manifolds

corresponding to all to E To, respectively. We have that if t and t' are in the same ball KOand 0 < R < Rto, then

Let S > 0 denote the Lebesgue number corresponding to the compact set C, and open covering Vto, to E To, i.e., let us choose a S such that if t, t' E C,, e ( t , t l ) < 6, then there exists a to E To such that t,t' E K O . Let 0 < Ro 5 inf(Rt, : to E To}. Decreasing Ro and 6, if necessary, we may suppose that qL(776+ R o ) 5 E. Let t be an arbitrary element of C, and let t' be an element of C,-,(t,t,) for which ~ ( tt'), < 6. There exists a to E To for which t , t' E KO. Using the previous estimate and the fact that L and rj are integers, moreover the subadditivity of w, we obtain that for an arbitrary real number R for which 0 < R < Ro,

If e(t, t') 5 R, this estimate can be written in the form

where cl, c2, cg, and c4 do not depend on t, t', and R. Taking supremum first on the right-hand, and then on the left-hand side for all t E C, and t' E CE-,(t,t,) for which e(t, t') 5 r , we obtain the inequality

5 17. Local and global results on manifolds

165

< < < + <

whenever 0 r S R < Ro. If we choose an R which satisfies the condition c z R c3w(R) 112 - which can always be done by decreasing 6, if necessary, - we have

< <

whenever 0 r S. This proves that f is a locally Lipschitz function on C. For an arbitrary t E T , without loss of generality we may suppose that t is an interior point of C, because we may replace C with the union of C and a compact neighborhood o f t . Hence the proof is complete. We remark that this theorem, as Theorem 11.6, can be better used with some additional argument. (See the example after Theorem 11.6.) For equations having a more special form we can state more. 17.4. Theorem. Let T , Y, Z,, and Xi be manifolds (i = 1 , 2 , .. . , n ) . Let D be an open subset of T x Y and let Z be a Euclidean space. Consider the functions f : T + Z, gi : D + Xi, fi : Xi + Zi, hi : D x Zi -+ Z , (i = 1 , 2,... , n ) , and ho : D + Z. Let (to,yo) E D and suppose that

(1) for each ( t , y ) E D ,

(2) hi is continuously differentiable (0

< i 5 n);

(3) fi is continuous on Xi (i = 1,2, ... , n ) ; gi(to,y) is a submersion (4) gi is twice continuously differentiable and y at yo for 1 5 i < n. Then f is continuously differentiableon a neighborhood of to. Proof. The theorem can be reduced to Theorem 11.3 using local coordinates. If there are only three unknown functions and one of them is real-valued, then also we can say more: 17.5. Theorem. Let Z , T , Y , X I , Zo, and Z1 manifolds, Z1 with dimension 1. Let D be an open subset of T x Y . Consider the functions gl : D + X 1 , f : T + Z , f o : Y + Zo, f l : X I + Z1, a n d h : D x Zo x Z1 -+ Z. Suppose that to E T and the following conditions are satisfied:

166

Chapter V. Regularity theorems on manifolds

(1) for each ( t ,y) E D ,

(2) h is a locally Lipschitz function; (3) gl is twice continuously differentiable; (4) f o and f l are continuous;

(5) there exists a yo such that ( t o ,yo) E D and y t-+ ggl ( t o ,y) is a submersion at Yo. Then f is a locally Lipschitz function on a neighborhood o f t o . Proof. Using local coordinates we may reduce the statement t o Theorem 11.4. 17.6. Theorem: almost everywhere differentiable solutions are continuously differentiable. Let Z be a Banach manifold, Zi ( i = 1, . . . , n ) be separable Banach manifolds, Y , T , and Xi ( i = 1,2, . . . ,n ) manifolds. Let D be an open subset of T x Y . Consider the functions f : T + Z , f i : Xi -+ Z i , g i : D - + Xi ( i = 1,2, . . . , n ) , h : D x Z 1 x . - . x Z n -+ 2 . Suppose, that to E T and the following conditions are satisfied:

(1) for each ( t ,y) E D ,

dh dh (2) for any fixed y E Y the partial derivatives - and - ( i = 1,2,. . . ,n ) at azi are continuous in other variables; (3) fi is almost everywhere differentiable ( i = 1,2, . . . ,n ); (4) gi is continuously differentiableand there exists a yo, such that ( t o ,yo) E D and the mapping y t-+ gi(to,y) is a submersion at yo ( i = 1 , 2 , . . . ,n ) . Then f is continuously differentiable on a neighborhood of t o . Proof. Using local coordinates the statement may be reduced to Theorem 14.2.

Let Y , Z , T , Xi ( i = 17.7. Theorem: C P solutions are in C p + ' . 1 , 2 , . . . , n ) , and Zi (0 5 i 5 n ) be manifolds. Let D c T x Y be an open set. Consider the functions f : T -+ 2, gi : D + Xi ( i = 1,2,. . . , n ) , f i : X i : - + Z i ( i = 1 , 2,... , n ) , f o : Y - + Z o , h : D x Z o x Z 1 x ~ ~ ~ x Z n + Z . Suppose that p > 0, t o E T , and the following conditions hold:

§ 17. Local and global results on manifolds

167

(1) for each ( t ,y ) E D,

(2) the function h is p

+ 1 times continuously differentiable;

(3) fo is continuously differentiable and the functions f i , i = 1,2, . . . ,n are p times continuously differentiable;

+

(4) the functions gi are p 1 times continuously differentiable, and there exists a yo such that ( t o ,y o ) E D and y c, gi(to,y ) is a submersion at yo whenever i = 1,2,. . . ,n. Then f is p 1 times continuously differentiableon a neighborhood of t o .

+

Proof. Using local coordinates we may reduce this theorem to Theorem 15.2.

Chapter VI.

REGULARITY RESULTS WITH FEWER VARIABLES

As explained in the introduction (see 1.41), roughly speaking, all the results of 55 5-17 prove regularity of an r place function f that is the solution of a functional equation only if there are at least 2r variables in the functional equation. The last two examples of 1.21 show that the number of variables cannot be reduced without introducing some additional conditions. The first general regularity results which overcome this difficulty were given by ~ w i a t a k .She applied her distribution method to generalizations of the mean value equation

(hi E R,gi E Rr are fixed) with unknown function f and proved that continuous solutions are in CCO. First she investigated the generalized mean value equations

Chapter VI. Regularity results with fewer variables

170

(see [195], [196], [138], [197], [198]) and finally the equation (1) n

(see [199], [201]). Equation (1) is "almost linear", so, formally, it is much less general than equation (1) in problem 1.IS. However ~wiatak'stheorems can be applied even if the rank of

dgi is much less than the dimension of d~

-

the domain of the unknown function f. Roughly speaking, the method of ~ w i a t a kcan be applied even if f is an r place function and there are only 1 variables; this is the minimal number of variables if the equation is r not a functional equation in a "single variable". So the results of Swiatak suggest that the rank condition in the problem 1.18 is too strong, and the results concerning problem 1.18 can be extended for other cases, at least if we add some further conditions. In this chapter we discuss regularity results for functional equations with "few" variables, i. e. with r place unknown function but with less than 2r (but more than r) variables. First we briefly overview the main results of ~ w i a t a k in 5 18. Then new 'Lmeasurabilityimplies continuity", "Baire property implies continuity", and "continuity implies Cw" type results will be treated in 5s 19-21, respectively. Most of the new results can be applied to the nonlinear functional equation (1) of Problem 1.18.

+

Here we briefly overview ~wiatak'sdistribution method. Most of the material here is presented as in her paper [201], but her papers [196], [197], [198], [199], [200], and [202] and her joint paper [I381 with Lawruk were also used.

18.1. Parametric families of distributions. The essence of ~wiatak's method is to consider parametric families of distributions. Let T be an open subset of RS, and let us consider a Cw function h : T x IWr -+ R and a function g : T x Rr + IWr . If f E D'(Rr ), we consider the parametric family h(t, .) f 0 g(t, .) = h(t, .) (f o g(t, .)) , t E T of distributions and its differentiability with respect to t. To this end, let 0 < p j m and let us suppose

5 18. ~ w i a t a k ' smethod

171

that g and (t,s) e g;l(s) are in Cp(T x Rr ), for each t E T the function gt is a Cw diffeomorphism, and that for each multiindex a E N8, la1 5 p the function y e (drg)(t,y) is in Cw(Rr), where a? = dff1dy2- - - d:~, i. e. a partial differential operator acting only in the variable t. The next Lemma shows that calculating formally

we obtain a distribution, which in some sense acts as the derivative corresponding to of the family. To simplify our notation let us denote by d,ol (h(t, .) f o g(t, .)) the distribution obtained by calculating (1) formally. Of course, this expression depends an t. Let

a,ol(h(t,.) f o g(t, .)) 1 t=to denote the

distribution obtained by substituting t = to after the formal differentiation.

18.2. Lemma [Qwiatak]. Under the conditions of the above definition, for any test function cp E V ( F ) we have

whenever t E T .

Proof. This is Lemma 4.1 in [201]. Although it is very important, we do not repeat here the proof, which is of technical nature. 18.3. Theorem [Qwiatak]. We use the notation of the previous definition. Let T be an open subset of PSS, let to E T , and let 0 < p 5 ca. Let f : IlBT + R, gi : T x Rr -+ Rr (i = 1 , 2, . . . , n ) , gi : Rr + Rr (i = n + l , . . . , m ) , hi : T xPST + R (i = 0,1, . . . , n ) , h : I F xRm-n be functions, and suppose that the following conditions are satisfied: (1) whenever t E T and y E Rr we have

(2) h is continuous and hi is in Cw whenever 0 5 i 5 n; (3) if n < i 5 m then the function gi is continuous; (4) if1 5 i 5 n then for each t E T themappinggi,t i s a C w diffeomorphism of F onto itself, the mappings gi and (t, s) ri g,? (s) are in CP ( T x IWT ) , and afgi,t is in Cw(PST) whenever /3 E N8, 1/31 5 1);

Chapter VI. Regularity results with fewer variables

172

(5) gi(to,y)

-- y whenever I I i I n;

(6) there exists an ct. E Ns for which 0 < jal 5 p such that each distribution solution f of the partial differential equation

is represented by a function in C". Then every continuous solution f of (1) is in C". Note that there are well-known conditions for partial differential equations which imply that distribution solutions are in Cm. For example, for any Cm function on the right hand side in (6) if the differential operator on then the left hand side is of constant strength, hypoelliptic at an yo E F, any distribution solution of (6) is in Cm; see Hormander [73], p. 176.

Proof. Let f be a continuous function. For any fixed t E T both sides of (1) can be identified with a distribution. From (1) we obtain that for any test function cp E D ( R T )we have

Using the previous Lemma we obtain that f as a distribution satisfies

whenever t E T. Substituting t = t o and using (5) we obtain that f as a distribution satisfies the partial differential equation in (6). Hence by (6) we obtain that f is almost everywhere equal to a CW function. Since f is continuous, it is equal to this Cm function. The following corollary (Theorem 6.4 in ~ w i a t a k[201]) is less general but easy to apply.

5 18. Swiatak's

method

173

18.4. Corollary [~wiatak]. Suppose that the conditions of the previous theorem are satisfied with s = 1 and p = 2, but instead of (6) we have a (to,y) span the space PST and hi (to,y) > (6') for every y E Rr the vectors (-gi) at 0 for 15 i 5 n. Then every continuous solution f is in C" .

Proof. We have to prove that (6) of the previous theorem is satisfied. The principal part of the equation

can be written as

where gi = (gi,. . . , gr ). The corresponding quadratic form for

< = (J1, . . . , <,)

To prove Q(E) > 0 for ( # 0 notice that Q ( < )= 0 if and agi y), i = 1 , . . . , n . But this is impossible only if is orthogonal to all -(to, at because these vectors span Rr. Therefore Q([) > 0 for [ # 0, i. e. (7) is an elliptic differential equation. Using regularity theorems of the theory of partial differential equations (see Hormander 1731, pp. 176-177) we obtain from the previous theorem that f E Cm. because hi(to,y)

<

> 0.

18.5. Remarks. (1) From the previous general theorem, which is sometimes difficult to apply, further much more special, but easier to apply results can be deduced. Applications of the previous general theorem to generalized mean value type equations are Theorem 6.2 and Theorem 6.3 in ~ w i a t a k [201]. In these cases differentiation twice and four times with respect to the (scalar) variable t is used, respectively. It is remarkable, that in the second case the proof that the resulting linear partial differential equation with constant coefficients is hypoelliptic is far from being trivial (10 pages).

174

Chapter VI. Regularity results with fewer variables

(2) The method of Swiatak is capable of treating locally integrable solutions, too, but in this case we only obtain that each solution is almost everywhere equal to a Cm function. Adding not very restrictive assumptions one can prove that all the locally integrable solutions are equal almost everywhere to another solution which is in Cm (see Swiatak [197], [199], [202]). The fact that every locally integrable solution of 18.3.(1) is a Cm solution was proved by ~ w i a t a kunder a rather strong assumption implying s 2 r, i. e. that there are at least 2r variables, if the unknown function f is an r place function (see [197], [199], [202]). (3) With the method of ~ w i a t a kwe may deduce that continuous solutions are analytic (see Lawruk and ~ w i a t a k[138]). For example, if the linear partial differential operator on the left hand side of 18.3.(6) is elliptic with analytic coefficients and the right hand side is analytic, too, then Theorem 7.5.1 in Hormander [73], p. 178 implies, that all continuous solutions f are analytic. (4) The method can be easily extended to treat functional-differential equations; see ~ w i a t a k[200]. (5) Among applications we compare ~wiatak'smethod with ours on some examples, see 22.16, 22.17.

19. BETWEEN MEASURABILITY AND

CONTINUITY What seems to be most important for equations with "few" variables is to prove "measurability implies continuity" type results, because by the method of Swiatak we may only start with continuous solutions - a consequence of the distribution method. To the best knowledge of the author such "measurability implies continuity" type results without the strong rank condition in 1.18.(3) or some abstract version of it were known