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M. A. KRASNOSEI/SKrI AND YA. B. RUTIOKII vorond state University, Vorond'Institute of Oonstructional Engineering CONVEX FUNCTIONS AND ORLICZ SPACES Translaled from the first Russian edition By LEO F. B O RON Direclor, Universal Om'respondence School of llfathematics 1961 P. NOORDHOFF LTD. - GRONINGEN - THE NETHERLANDS DEDICATION This translation is dedicated to Prof. Dr. M. A. Krasnosel'skii and to Prof. Dr. Ya. B. Rutickii. L. F. B. Philadelphia 26, Pennsylvania, I96I. Copyright I96I by P. Noordhoff, Ltd. Groningen; J This book or parts thereof may not be reproduced in written permission of the pubLishers. Printed in The Netherlands. C O N T E N TS Page IV X XI DEDICATION . ANNOTATION FOREWORD . CHAPTER I Special Classes ot Convex Functions § . N-functions . . . . . . . . . . . . . . . Convex functions ( I ) . Integral representation of a convex function (3) . Definition of an N-function (6) . Properties of N-functions (7) . Second definition of an N-function (9). Compo­ sition of N-functions ( 0) . § 2. Complementary N-function . . . . . . . . . . 11 Definition ( I I ) . The Young inequality ( 2) . Examples ( 3) . Inequality for complementary functions ( 4) . § 3. The comparison of N-functions . . . . . . . . . 14 Definition ( 4) . Equivalent N-functions ( 5) . Principal part of an N-function ( 6) . An equivalence criterion ( 7) . The existence of various classes (20) . § 4. The L1 2-condition . . . . . . . . . . . . . . . . . 23 Definition (23) . Tests for the Lh-condition (24) . The Lh-condition for the complement to an N-function (25) . Examples (27) . § 5. The L1 '-condition . . . . . . . . . . . . . . . . . Definition (29). Sufficiency criteria for the satisfaction of the L1'-condition (3 ) . The LI'-condition for the complementary function (33) . Examples (33) . § 6. N-functions which increase more rapidly than power functions . . . . . . . . . . . . . . . . . . . . 29 35 The Lla-condition (35) . Approximations for the complementary function (36). The construction of N-functions which are equi­ valent to the complementary functions (37) . The composition of complementary functions (3 9) The Ll2-condition (40) . Properties of the complementary functions (44) . Test for the Ll2-condition for the complementary function (46) . Further discussion on the composition of N-functions (48) . . § 7. Concerning a class of N-functions . . . . . . . . . . Formulation of the problem (52) . The class 9R (52) . The class 91 (55) . Theorem on the complementary function (58) . 52 VIII CONTENTS CHAPTER II Orlicz SPaces Page § 8. Orlicz classes . . . 60 Definition (60) . The J ensen integral inequality (62) . The com­ parison of classes (63) . The structure of Orlicz classes (64) . § 9. The space LM . . . . . . . . . . . . . . . . . . 67 The Orlicz norm (67) . Completeness (70) . Norm of the charac­ teristic function (72) . HOlder's inequality (72) . The case of the Lh-condition (75) . Mean convergence (75) . The Luxemburg norm (78) . § 0. The space EM . . . . . . . . . . . . . . . . . . 80 § 1 . Compactness criteria . . . . . . . . . . . . . . . 94 § 2. Existence of a basis. . . . . . . . . . . . . . . . 101 Definition (80) . The separability of E M (8 ) . Disposition of the class L M with respect to the space E M (82) . Necessary conditions for separability of an Orlicz space (85) . On the definition of the norm (86) . The absolute continuity of the norm (87) . Calculation of the norm (88) . Another formula for the norm (9 ) . Vallee Poussin's theorem (94) . Steklov functions (95) . A. N. Kolmogorov's compactness criterion for the space E M (97) . A second criterion for compactness (98) . F. Riesz's criterion for compactness for the spaces E M (99) . Transition to the space of functions defined on a closed interval ( 1 ) . Haar functions ( 03) . Basis in E M (lOS). Further remarks on the conditions for separability ( 07) . § 3. Spaces determined by distinct N-functions . . . . . . 10 Comparison of spaces ( 1 0) . An inequality for norms ( 1 2) . Concerning a criterion for convergence in norm ( 1 4) . The product of functions in Orlicz spaces ( 1 7) . Su fficient con­ ditions ( 20) . § 4. Linear functionals. Linear functionals in L M ( 24) . General form of a linear function­ al on E M ( 28) . EN-weak convergence ( 30) . EN-weakly con­ tinuous linear functionals ( 33) . Norm of a functional and l!vll(N) ( 34) . 24 CHAPTER III Operators in Orlicz SPaces § 5. Conditions for the continuity of linear integral operators 37 Formulation of the problem ( 37) . General theorem ( 38) . Existence of the function !l>(u) ( 38) . Concerning a property of N-functions which satisfy the .d'-condition ( 40) . Sufficient conditions for continuity ( 45) . On splitting a continuous operator ( 46) . CONTENTS § 6. Conditions for the complete continuity of linear integral operators . . . . . . . . . . . . . . . . . . . . IX Page 49 The case of continuous kernels ( 49) . Fundamental theorem ( 50) . Complete continuity and EN-weak convergence ( 52) . Zaanen 's theorem ( 55) . Comparison of conditions ( 60) . On splitting a completely continuous operator ( 64) . On operators of potential type ( 65) . § 7. Simplest nonlinear operator . . . . . . . . . . . . 67 The CaratModory condition ( 67) . Domain of definition of the operator f ( 67) . Theorems on continuity ( 69) . Boundedness of the operator f ( 72) . General form of the operator f ( 74) . Sufficient conditions for the continuity and boundedness of the operator f ( 74) . The operator f and EN-weak convergence ( 75) . § 8. Differentiability. Gradient of the norm . . . . . . . . 76 Differentiable functionals ( 76) . Measurability of the function O(x) ( 77). Functional for the operator f ( 78) . The linear operator f ( 78) . The Frechet derivative ( 79) . Special condition for differentiability ( ) . Auxiliary lemma ( 86) . The Gateaux gradient ( 86) . Gradient of the Luxemburg norm ( 87) . Gradient of the Orlicz norm ( 89) . CHAPTER IV Nonlinear Integral Equations § 9. The P. S. Uryson operator . . . . . . . . . . . . . 94 The P. S. Uryson operator ( 94) . Boundedness of the Uryson operator ( 96) . Transition to a simpler operator ( 97) . A second transition to a simpler operator ( 98) . A third transition to a simpler operator (200) . Fundamental theorem on the complete continuity of Uryson's operator (20 ) . The case of weak non­ linearities (203) . Hammerstein operators (207) . § 20. Some existence theorems . . . . . . . . . . . . . . 208 Problems under consideration (208) . The existence of solutions (209) . Positive characteristic functions (2 3) . Characteristic functions of potential operators (2 4) . Theorem on branch points (2 6) . SUMMARY OF FUNDAMENTAL RESULTS. BIBLIOGRAPHICAL NOTES LITERATURE. INDEX . . . . . . . . 217 230 239 248 ANN OTATION In the present book are discussed the theory of extensive classes of convex functions which play an important role in many branches of mathematics. The theory of Orlicz spaces (i. e. normed spaces of which the Lp spaces are a special case ) is developed in detail and applications are pointed out. The book is intended for mathematicians ( students in upper level courses, aspirants for the doctoral degree and scientific workers ) , who deal with functional analysis and its applications and also with various problems in the theory of functions. FO R EW O R D The present monograph consists of four chapters. In the first of these chapters we study various classes of convex functions. The fundamental content of this chapter has been published, up to this time, only in various j ournal articles. It appears to the authors that the material in the first chapter is of interest independently of the remainder of the book inasmuch as convex functions are applied extensively in the most diversified branches of mathematics. In the second chapter, we discuss the general theory of Orlicz spaces-spaces which are the direct extensive generalization of Lp spaces . Here, we consider the usual problems of functional analysis with regard to applications to Orlicz spaces : completeness, separability conditions, the existence of a basis, equivalent normizations, com­ pactness conditions, properties of linear functions, and so on. It is made clear that Orlicz spaces are, in many cases, similar to Lp spaces. In the third chapter, we study operators and functionals defined on Orlicz spaces. The authors found it necessary to apply Orlicz spaces in the consideration of nonlinear integral equations of the form Atp(X) J k(x, y)1 [tp(y)]dy, = o where I(u) is a function which increases more rapidly than an arbitrary power function. The operator defined by the right member of this integral e­ quation does not operate in any of the Lp spaces. Therefore, the investigation of the indicated integral equations by the methods of nonlinear functional analysis turns out to be difficult . The results of the second and third chapters enable us to investigate wide classes of nonlinear equations . The fourth, and last, chapter of the book is devoted to the in­ vestigation of some nonlinear problems. The authors take i his opportunity to express their gratitude to G. E. � ilov whose many constructive criticisms were used in pre­ paring the present book. BIBLIOGRAPHICAL NOTES 234 interval in such a way that under this mapping the measure of every subset remains invariant see ROHLIN [ ] . § 13. Some of the results of this section were discussed earlier in KRASNOSEL'SKI! and RUTICKI! [ , 4, 6J . § 14. The theorem on the general form of a linear functional on the space in the case when the N-function M(u) satisfies the L1 -condition was proved in ORLlCZ [ ] . Theorem 4. is proved in ORLlCZ [2] . For the case when the condition is not satisfied, the theorem on the general form of a linear functional on was proved in KRASNOSEL'SKII and RUTICKII [5J (also see LUXEMBURG [ I J ) . The problem of the general form of a linear functional on for the case when M(u) does not satisfy the L1 -condition remains open. The theorem on the connection between the norm of a linear functional and the Luxemburg norm, generated by the Luxemburg function, is proved in LUXEMBURG [ I J . The function k(v) is studied in SALEHOV [ , 2] . In the paper by AMEMIYA [ ] , in connection with the theory of modulared spaces, the relation between the Luxemburg norm and the norm defined by formula ( 0. 1 ) is studied. It is assumed that these norms differ by a constant factor and it is proved that in this case the spaces considered are Lp. In virtue of Theorem 0.5, formula ( 0. 1 ) defines the ordinary Orlicz norm, and the Luxem­ burg norm coincides with the norm of the linear functional generated by it (see subsection 5) . Therefore, D. V. Salehov's theorem, introduced on p. 26, also follows from Amemiya's results. The consideration of ordinary weak convergence is inconvenient since the general form of a linear functional on is unknown. In this connection, it turned out to be convenient to consider that weak convergence which arises if is assumed to be the space of linear functionals on LM EM LM LM EN. LM §§ 15, 16. We shall point out some of the numerous papers in which linear integral operators are studied. The most detailed analysis of linear integral operators for the case of space of continuous functions was carried out by RADON [ ] . The simplest theorems in the case of the spaces Lp are given in BANACH [ I J and RIEsz-Sz . -NAGY [ I J . Linear integral operators of a special form (the so-called potential- C 235 BIBLIOGRAPHICAL NOTES type operators) are studied in the works by S. V . SOBOLEV and V . I . KONDRASOV (see S. SOBOLEV [ ] ) . Strong results in the study of integral operators, acting in LP, were obtained by KANTOROVI