APPLICATIONS OF ORLICZ SPACES
M. M. Rao University of California, Riverside Riverside, California
Z. D. Ren Suzhou University Suzhou, People's Republic of China
u MARCEL
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PURE AND APPLIED MATHEMATICS A Program of Monographs, Textbooks, and Lecture Notes
EXECUTIVE EDITORS Earl J. Taft Rutgers University New Brunswick, New Jersey
Zuhair Nashed University of Delaware Newark, Delaware
EDITORIAL BOARD M. S. Baouendi University of California, San Diego Jane Cronin Rutgers University Jack K. Hale Georgia Institute of Technology
Anil Nerode Cornell University Donald Passman University of Wisconsin, Madison Fred S. Roberts Rutgers University
S. Kobayashi University of California, Berkeley
David L. Russell Virginia Polytechnic Institute and State University
Marvin Marcus University of California, Santa Barbara
Walter Schempp Universitdt Siegen
W. S. Massey Yale University
Mark Teply University of Wisconsin, Milwaukee
MONOGRAPHS AND TEXTBOOKS IN PURE AND APPLIED MATHEMATICS 1. K. Yano, Integral Formulas in Riemannian Geometry (1970) 2. S. Kobayashi, Hyperbolic Manifolds and Holomorpnic Mappings (1970) 3. V. S. Vladimirov, Equations of Mathematical Physics (A. Jeffrey, ed.; A. Littlewood, trans.) (1970) 4. B. N. Pshenichnyi, Necessary Conditions for an Extremum (L. Neustadt, translation ed.; K. Makowski, trans.) (1971) 5. L. Narici et a/., Functional Analysis and Valuation Theory (1971) 6. S. S. Passman, Infinite Group Rings (1971) 7. L Domhoff, Group Representation Theory. Part A: Ordinary Representation Theory. Part B: Modular Representation Theory (1971,1972) 8. W. Boothbyand G. L Weiss, eds., Symmetric Spaces (1972) 9. Y. Matsushima, Differentiate Manifolds (E. T. Kobayashi, trans.) (1972) 10. L E. Ward, Jr., Topology (1972) 11. A. Babakhanian, Cohomological Methods in Group Theory (1972) 12. R. Gilmer, Multiplicative Ideal Theory (1972) 13. J. Yeh, Stochastic Processes and the Wiener Integral (1973) 14. J. Bairos-Neto, Introduction to the Theory of Distributions (1973) 15. R. Larsen, Functional Analysis (1973) 16. K. Yano and S. Ishihara, Tangent and Cotangent Bundles (1973) 17. C. Procesi, Rings with Polynomial Identities (1973) 18. R. Hermann, Geometry, Physics, and Systems (1973) 19. N. R. Wallach, Harmonic Analysis on Homogeneous Spaces (1973) 20. J. Dieudonne, Introduction to the Theory of Formal Groups (1973) 21. /. Vaisman, Cohomology and Differential Forms (1973) 22. B.-Y. Chen, Geometry of Submanifolds (1973) 23. M. Marcus, Finite Dimensional Multilinear Algebra (in two parts) (1973,1975) 24. R. Larsen, Banach Algebras (1973) 25. R. O. Kujala and A. L Vitter, eds., Value Distribution Theory: Part A; Part B: Deficit and Bezout Estimates by Wilhelm Stoll (1973) 26. K. B. Stolarsky, Algebraic Numbers and Diophantine Approximation (1974) 27. A. R. Magid, The Separable Galois Theory of Commutative Rings (1974) 28. B. R. McDonald, Finite Rings with Identity (1974) 29. J. Satake, Linear Algebra (S. Koh et al., trans.) (1975) 30. J. S. Golan, Localization of Noncommutative Rings (1975) 31. G. Klambauer, Mathematical Analysis (1975) 32. M. K. Agoston, Algebraic Topology (1976) 33. K. R. Goodearl, Ring Theory (1976) 34. L. E. Mansfield, Linear Algebra with Geometric Applications (1976) 35. N. J. Pullman, Matrix Theory and Its Applications (1976) 36. B. R. McDonald, Geometric Algebra Over Local Rings (1976) 37. C. W. Groetsch, Generalized Inverses of Linear Operators (1977) 38. J. E. Kuczkowski and J. L Gersting, Abstract Algebra (1977) 39. C. O. Christenson and W. L Voxman, Aspects of Topology (1977) 40. M. Nagata, Field Theory (1977) 41. R. L. Long, Algebraic Number Theory (1977) 42. W. F. Pfeffer, Integrals and Measures (1977) 43. R. L Wheeden and A. Zygmund, Measure and Integral (1977) 44. J. H. Curtiss, Introduction to Functions of a Complex Variable (1978) 45. K. Hrbacek and T. Jech, Introduction to Set Theory (1978) 46. W. S. Massey, Homology and Cohomology Theory (1978) 47. M. Marcus, Introduction to Modern Algebra (1978) 48. E. C. Young, Vector and Tensor Analysis (1978) 49. S. B. Nadler, Jr., Hyperspaces of Sets (1978) 50. S. K. Segal, Topics in Group Kings (1978) 51. A. C. M. van Rooij, Non-Archimedean Functional Analysis (1978) 52. L. Corwin and R. Szczarba, Calculus in Vector Spaces (1979) 53. C. Sadosky, Interpolation of Operators and Singular Integrals (1979) 54. J. Cronin, Differential Equations (1980) 55. C. W. Groetsch, Elements of Applicable Functional Analysis (1980)
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D. L. Stand and M. L Stand, Real Analysis with Point-Set Topology (1987) T. C. Gard, Introduction to Stochastic Differential Equations (1988) S. S. Abhyankar, Enumerative Combinatorics of Young Tableaux (1988) H. Strade and R. Famsteiner, Modular Lie Algebras and Their Representations (1988) J. A. Huckaba, Commutative Rings with Zero Divisors (1988) W. D. Wallis, Combinatorial Designs (1988) W. IMps/aw, Topological Fields (1988) G. Karpilovsky, Field Theory (1988) S. Caenepeel and F. Van Oystaeyen, Brauer Groups and the Cohomology of Graded Rings (1989) W. Kozlowski, Modular Function Spaces (1988) E. Lowen-Colebunders, Function Classes of Cauchy Continuous Maps (1989) M. Pavel, Fundamentals of Pattern Recognition (1989) V. Lakshmikantham et a/., Stability Analysis of Nonlinear Systems (1989) R. Sivaramakrishnan, The Classical Theory of Arithmetic Functions (1989) N. A. Watson, Parabolic Equations on an Infinite Strip (1989) K. J. Hastings, Introduction to the Mathematics of Operations Research (1989) a Fine, Algebraic Theory of the Bianchi Groups (1989) D. N. Dikranjan et a/., Topological Groups (1989) J. C. Morgan II, Point Set Theory (1990) P. BilerandA. Witkowski, Problems in Mathematical Analysis (1990) H. J. Sussmann, Nonlinear Controllability and Optimal Control (1990) J.-P. Florens et a/., Elements of Bayesian Statistics (1990) N. Shell, Topological Fields and Near Valuations (1990) B. F. Doolin and C. F. Martin, Introduction to Differential Geometry for Engineers (1990) S. S. Holland, Jr., Applied Analysis by the Hilbert Space Method (1990) J. Okninski, Semigroup Algebras (1990) K. Zhu, Operator Theory in Function Spaces (1990) G. 8. Price, An Introduction to Multicomplex Spaces and Functions (1991) R. B. Darst, Introduction to Linear Programming (1991) P. L Sachdev, Nonlinear Ordinary Differential Equations and Their Applications (1991) T. Husain, Orthogonal Schauder Bases (1991) J. Foran, Fundamentals of Real Analysis (1991) W. C. Brown, Matrices and Vector Spaces (1991) M. M. Rao and Z. D. Ren, Theory of Oriicz Spaces (1991) J. S. Golan and T. Head, Modules and the Structures of Rings (1991) C. Small, Arithmetic of Finite Fields (1991) K. Yang, Complex Algebraic Geometry (1991) D. G. Hoffman et a/., Coding Theory (1991) M. O. Gonzalez, Classical Complex Analysis (1992) M. O. Gonzalez, Complex Analysis (1992) L W. Baggett, Functional Analysis (1992) M. Sniedovich, Dynamic Programming (1992) R. P. Agarwal, Difference Equations and Inequalities (1992) C. Brezinski, Biorthogonality and Its Applications to Numerical Analysis (1992) C. Swartz, An Introduction to Functional Analysis (1992) S. a Nadler, Jr., Continuum Theory (1992) M. A. AI-Gwaiz, Theory of Distributions (1992) E. Perry, Geometry: Axiomatic Developments with Problem Solving (1992) E. Castillo and M. R. Ruiz-Cobo, Functional Equations and Modelling in Science and Engineering (1992) A. J. Jerri, Integral and Discrete Transforms with Applications and Error Analysis (1992) A. Charlier et al., Tensors and the Clifford Algebra (1992) P. Bilerand T. Nadzieja, Problems and Examples in Differential Equations (1992) E. Hansen, Global Optimization Using Interval Analysis (1992) S. Guerre-Delabriere, Classical Sequences in Banach Spaces (1992) Y. C. Wong, Introductory Theory of Topological Vector Spaces (1992) S. H. Kulkami and B. V. Limaye, Real Function Algebras (1992) W. C. Brown, Matrices Over Commutative Rings (1993) J. Loustau and M. Dillon, Linear Geometry with Computer Graphics (1993) W. V. Petryshyn, Approximation-Solvability of Nonlinear Functional and Differential Equations (1993)
172. 173. 174. 175. 176. 177. 178. 179. 180. 181. 182. 183. 184. 185. 186. 187. 188. 189. 190. 191. 192. 193. 194. 195. 196. 197. 198. 199. 200. 201. 202. 203. 204. 205. 206. 207. 208. 209. 210. 211. 212. 213. 214. 215. 216. 217. 218. 219. 220. 221. 222. 223. 224. 225.
£ C. Young, Vector and Tensor Analysis: Second Edition (1993) T. A. Bick, Elementary Boundary Value Problems (1993) M. Pavel, Fundamentals of Pattern Recognition: Second Edition (1993) S. A. Albeverio et al., Noncommutative Distributions (1993) W. Fulks, Complex Variables (1993) M. M. Rao, Conditional Measures and Applications (1993) A. Janicki and A. Weron, Simulation and Chaotic Behavior of a-Stable Stochastic Processes(1994) P. Neittaanmaki and D. Tiba, Optimal Control of Nonlinear Parabolic Systems (1994) J. Cronin, Differential Equations: Introduction and Qualitative Theory, Second Edition (1994) S. Heikkila and V. Lakshmikantham, Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations (1994) X. Mao, Exponential Stability of Stochastic Differential Equations (1994) B. S. Thomson, Symmetric Properties of Real Functions (1994) J. E. Rubio, Optimization and Nonstandard Analysis (1994) J. L Bueso et al., Compatibility, Stability, and Sheaves (1995) A. N. Michel and K. Wang, Qualitative Theory of Dynamical Systems (1995) M. R. Darnel, Theory of Lattice-Ordered Groups (1995) Z. Naniewicz and P. D. Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities and Applications (1995) L. J. Corwin and R. H. Szczarba, Calculus in Vector Spaces: Second Edition (1995) L. H. Erbe et al., Oscillation Theory for Functional Differential Equations (1995) S. Agaian et al., Binary Polynomial Transforms and Nonlinear Digital Filters (1995) M. I. Gil', Norm Estimations for Operation-Valued Functions and Applications (1995) P. A. Grillet, Semigroups: An Introduction to the Structure Theory (1995) S. Kichenassamy, Nonlinear Wave Equations (1996) V. F. Krotov, Global Methods in Optimal Control Theory (1996) K. I. Beidaret al., Rings with Generalized Identities (1996) V. I. Amautov et al., Introduction to the Theory of Topological Rings and Modules (1996) G. Sierksma, Linear and Integer Programming (1996) R. Lasser, Introduction to Fourier Series (1996) V. Sima, Algorithms for Linear-Quadratic Optimization (1996) D. Redmond, Number Theory (1996) J. K. Beem et al., Global Lorentzian Geometry: Second Edition (1996) M. Fontana et al., Priifer Domains (1997) H. Tanabe, Functional Analytic Methods for Partial Differential Equations (1997) C. Q. Zhang, Integer Flows and Cycle Covers of Graphs (1997) E. Spiegel and C. J. O'Donnell, Incidence Algebras (1997) B. Jakubczyk and W. Respondek, Geometry of Feedback and Optimal Control (1998) T. W. Waynes et al., Fundamentals of Domination in Graphs (1998) T. W. Haynes et al., Domination in Graphs: Advanced Topics (1998) L. A. D'Alotto et al., A Unified Signal Algebra Approach to Two-Dimensional Parallel Digital Signal Processing (1998) F. Halter-Koch, Ideal Systems (1998) N. K. Govil et al., Approximation Theory (1998) R. Cross, Multivalued Linear Operators (1998) A. A. Martynyuk, Stability by Liapunov's Matrix Function Method with Applications (1998) A. FaviniandA. Yagi, Degenerate Differential Equations in Banach Spaces (1999) A. Illanes and S. Nadler, Jr., Hyperspaces: Fundamentals and Recent Advances (1999) G. Kato and D. Struppa, Fundamentals of Algebraic Microlocal Analysis (1999) G. X.-Z. Yuan, KKM Theory and Applications in Nonlinear Analysis (1999) D. Motreanu and N. H. Pavel, Tangency, Flow Invariance for Differential Equations, and Optimization Problems (1999) K. Hrbacek and T. Jech, Introduction to Set Theory, Third Edition (1999) G. E. Kolosov, Optimal Design of Control Systems (1999) N. L. Johnson, Subplane Covered Nets (2000) B. Fine and G. Rosenberger, Algebraic Generalizations of Discrete Groups (1999) M. Vath, Volterra and Integral Equations of Vector Functions (2000) S. S. Miller and P. T. Mocanu, Differential Subordinations (2000)
226. R. Li et a/., Generalized Difference Methods for Differential Equations: Numerical Analysis of Finite Volume Methods (2000) 227. H. LJ and F. Van Oystaeyen, A Primer of Algebraic Geometry (2000) 228. R. P. Agarwal, Difference Equations and Inequalities: Theory, Methods, and Applications, Second Edition (2000) 229. A. B. Kharazishvili, Strange Functions in Real Analysis (2000) 230. J. M. Appellet a/., Partial Integral Operators and Integra-Differential Equations (2000) 231. A. I. Prilepko et a/., Methods for Solving Inverse Problems in Mathematical Physics (2000) 232. F. Van Oystaeyen, Algebraic Geometry for Associative Algebras (2000) 233. D. L. Jagennan, Difference Equations with Applications to Queues (2000) 234. D. R. Hankerson et a/., Coding Theory and Cryptography: The Essentials, Second Edition, Revised and Expanded (2000) 235. S. DZsc&escu et a/., Hopf Algebras: An Introduction (2001) 236. R. Hagen et a/., C*-Algebras and Numerical Analysis (2001) 237. Y. Talpaert, Differential Geometry: With Applications to Mechanics and Physics (2001) 238. R. H. Villarreal, Monomial Algebras (2001) 239. A. N. Michel et a/., Qualitative Theory of Dynamical Systems: Second Edition (2001) 240. A. A. Samarskii, The Theory of Difference Schemes (2001) 241. J. Knopfmacher and W.-B. Zhang, Number Theory Arising from Finite Fields (2001) 242. S. Leader, The Kurzweil-Henstock Integral and Its Differentials (2001) 243. M. Biliotti et a/., Foundations of Translation Planes (2001) 244. A. N. Kochubei, Pseudo-Differential Equations and Stochastics over Non-Archimedean Fields (2001) 245. G. Sierksma, Linear and Integer Programming: Second Edition (2002) 246. A. A. Martynyuk, Qualitative Methods in Nonlinear Dynamics: Novel Approaches to Liapunov's Matrix Functions (2002) 247. B. G. Pachpatte, Inequalities for Finite Difference Equations (2002) 248. A. N. Michel and D. Liu, Qualitative Analysis and Synthesis of Recurrent Neural Networks (2002) 249. J. R. Weeks, The Shape of Space: Second Edition (2002) 250. M. M. Rao and Z. D. Ren, Applications of Oriicz Spaces (2002) Additional Volumes in Preparation
To the memories of Professors W. Orlicz and M. A. Krasnoselskii for originating and deepening the subject on which this book and our earlier one are based
Preface The following account consists of some important applications in which the theory of Orlicz spaces plays a crucial role. A general account of Orlicz spaces was given in considerable detail in our companion volume in 1991. The applications discussed here often specialize and sharpen the general theory as well as show a need for further new investigations. These include specific results with the underlying measure spaces restricted to purely discrete or purely diffuse cases and also concentrating on subclasses of TV-functions from general Young functions, to study some finer aspects of the problems. This motivated us to consider various properties of Orlicz sequence and function spaces useful for current and future applications. In fact, the material covers a very wide spectrum including geometric, Fourier analytic, stochastic, PDE, composition, and Frechet metric function space results among others. A brief description of the contents of various chapters gives a better idea of our treatment, planned and executed. More detailed summations appear at the beginnings of the chapters. First we recall some background material from the earier volume in Chapter I with several additions, in a form for immediate use. These include quantitative indices of N-functions, properties of general Young functions as well as certain interpolation inequalities. Then the next four chapters present aspects of quantitative geometry, specialized from the general Banach space theory, for Orlicz sequence and function spaces the latter signifying that the underlying measure is diffuse or nonatomic. These include nonsquare constants in the sense of James, von Neumann-Jordan constants in Chapter II, weakly convergent sequence and normal structure coefficients in Chapter III, Jung constants in Chapter IV as well as Kottman constants and packing problems in Chapter V all of which prove very convenient technical results for many applications here and elsewhere. Thus they have some independent interest. The coefficients and these constants constitute geometric invariants which find an immediate use of the quantitative indices of TV-functions and the interpolation results included in Chapter I. The following four chapters, almost independent of each other, deal with substantial accounts of applications to different parts of analysis where in each case Orlicz spaces play a significant role, demanding much additional
vi
Preface
work. Thus Chapter VI is devoted to Fourier analysis including conjugate functions with Ryan's extension of the classical M. Riesz theorem, a Hausdorff-Young theorem on compact and locally compact abelian groups as well as some results on generalized (or nonlocally convex) Orlicz spaces. Chapter VII treats prediction analysis including work on vector valued functions and nonreflexive Orlicz spaces as well as a discussion of conditions for linearity of predictors. Next Chapter VIII, which is one of the longest chapters of the book, deals with an analysis of large deviations of probability theory and exponential (vector) Orlicz spaces. The analysis here illustrates the methods by showing how the basic concepts are closely related to a class of Fenchel-Orlicz spaces, thereby enlarging the scope of the subject. Also (vector and operator) martingale convergence and regularity properties of stochastic functions are considered. In Chapter IX, similar ideas play a key role in applications of this work to partial differential equations by studying Orlicz-Sobolev spaces for removable singularities, extensions of (strong and weak) Hardy-Littlewood type inequalities and continuity properties of composition and Nemitsky operators on these spaces. We conclude the work with Chapter X, containing supplements to the previous chapters. These include a section on Beurling-Orlicz classes of convolution algebras, geometric invariants with F. Riesz angles, embedding theorems for Orlicz sequence spaces as well as sharp norm differentiability properties of Orlicz spaces when the measures are purely discrete or diffuse. We end the book with a detailed analysis of nonlocally convex Orlicz spaces, complementing the last third of Chapter VI, when the measures are restricted as above, since it is possible to give a more refined analysis of the subjects considered. There is much new material in these chapters, and the details appear here for the first time. We have given credit to authors in a bibliographical notes section for each chapter, as far as possible. Many of the applications can be studied independently. But we observe that they are all based on the fundamental and common properties of Orlicz spaces included here and in our companion volume. We have repeated a few concepts at different places, instead of referring the reader to sources where they first appeared, for better comprehension. The work presented has been in preparation for nearly four years. During this period, we have tried to unify, refine and generalize the available original contributions. We learned a great deal in this investigation and tried to show
Preface
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the fascination of these applications. We also attempted to exemplify the saying that the life of a theory is derived from real applications. We shall be happy if our readers share this feeling with us. We never met Professor W. Orlicz (1903-1990). One of us had a brief correspondence with him in earlier times. We sadly learned of his passing when our companion volume was in press. The very interesting life history and mathematical work of Professor Orlicz have been provided to us by Dr. Lech Maligranda, one of his former doctoral students and an able associate, along with his own important research and a book (1989). Although for space reasons we could not include much of this material, we are very grateful to him for this information and detail. Then again Professor M. A. Krasnoselskii (1920-1997) passed away [we learned this from the notices of the journal Russian Math. Surveys, 53:1 (1998), 195-198] while this book was over half completed. He had been provided with a copy of our first volume soon after it appeared, and one of us met him later at a conference in 1996, but could not communicate well because of language difficulties. However, it is clear that we and much of the world mathematical community have gained considerably by the fundamental research of both these stalwarts. As an expression of our deep appreciation we dedicate this volume to the memories of both these fine mathematicians. The preparation of the monograph has taken considerable time and effort. The typing was aided by Jan Carter in the early Chapters. It was then continued with the able assistance of Ren's doctoral students in the Department of Mathematics, Suzhou University, in the People's Republic of China. They are Jincai Wang, Tao Zhang and Yaqiang Yan whose help is very much appreciated. Ren's visits to UCR have been enthusiastically supported by his wife Qingzeng Qiu. Also in 1999-2001, Ren was supported financially by Suzhou University ("211" program) and Jiangsu Province (Natural Science Fund), enabling him to spend the year at UCR. Some problems with AMS-LaTeX in obtaining the final version were resolved with some help by our colleague Y. Kakihara and earlier by the UCR systems group member L. Timmermans. We shall be gratified if this volume helps further the analysis of the important subjects covered here. M. M. Rao Z. D. Ren
Contents Preface
v
Chapter I. Introduction and Background Material
1
1.1 TV-functions and quantitative indices 1.2 Some basic results on L$-spaces 1.3 Notes on Young functions and general measures 1.4 Interpolation results on Orlicz spaces Bibliographical notes
1 13 28 35 47
Chapter II. Nonsquare and von Neumann-Jordan Constants 2.1 Introduction 2.2 Lower bounds for J(L^) and cNJ(L*} 2.3 Upper bounds for J(L^) and J(L*) 2.4 Sharp bounds for CNJ(L^^} and c NJ(L?S} Bibliographical notes Chapter III. Normal Structure and WCS Coefficients 3.1 Introduction 3.2 WCS and iV-coefficients of Orlicz function spaces 3.3 WCS coefficients of Orlicz sequence spaces 3.4 More on Orlicz sequence spaces Bibliographical notes Chapter IV. Jung Constants of Orlicz Spaces 4.1 Introduction to Jung constants 4.2 Lower bounds of JC(L^) and JC(L*) 4.3 Bounds for JC(L^) and JC(L*°) 4.4 Bounds for JC(l^) and JC(l*)
IX
49 49 53 59 66 72 73 73 78 90 107 116 117 117 124 135 138
x
Contents Bibliographical notes
145
Chapter V. Packing in Orlicz Spaces 5.1 Preliminaries 5.2 Packing in Orlicz sequence spaces 5.3 Packing in Orlicz function spaces 5.4 Packing in reflexive Orlicz spaces Bibliographical notes
147 147 150 167 175 183
Chapter VI. Fourier Analysis in Orlicz Spaces
184
6.1 6.2 6.3 6.4 6.5
Preliminaries on Fourier series 184 Conjugate functions and Orlicz spaces 190 Conjugate series and convergence in subsets of Orlicz spaces . . . . 199 A Hausdorff-Young theorem for Orlicz spaces 209 Fourier analysis on generalized Orlicz spaces 227 Bibliographical notes 234
Chapter VII. Applications to Prediction Analysis 7.1 7.2 7.3 7.4
236
Best predictions with convex loss When is a prediction operator linear ? Nonlinear prediction for nonreflexive spaces Some extensions to vector valued functions Bibliographical notes
236 244 249 260 271
Chapter VIII. Applications to Stochastic Analysis
273
8.1 8.2 8.3 8.4
Large deviations and Young functions Infinite dimensional extension and vector Orlicz spaces Regularity of stochastic functions and Orlicz spaces Martingales and Orlicz spaces Bibliographical notes
273 293 303 313 325
Contents
xi
Chapter IX. Nonlinear PDEs and Orlicz Spaces 9.1 9.2 9.3 9.4
Orlicz-Sobolev spaces for PDEs Removable singularities, PDE and Orlicz spaces Strong and weak type inequalities in Orlicz spaces Composition and Nemitsky operators in Orlicz spaces Bibliographical notes
Chapter X. Miscellaneous Applications 10.1 10.2 10.3 10.4 10.5
Beurling-Orlicz algebras Riesz angles of Orlicz spaces Embedding theorems for sequence spaces Differentiability properties of Orlicz spaces L* -spaces and applications Bibliographical notes
327 327 345 356 368 378 380 380 384 392 402 411 436
References
437
Notation
457
Index
461
Chapter I Introduction and Background Material
This chapter contains an account of the background material on Orlicz spaces to be used in the rest of the book. Since for applications some additional and specialized results other than those presented in our earlier volume are needed, such material is included here with details. In particular, a new quantitative index and inequalities based on certain interpolation theorems for a class of Orlicz spaces (the L*sspaces) are established. A characterization of the Lebesgue Lp spaces, from Z/*, is also given. These results are utilized in a study of particular geometrical aspects of Orlicz spaces, especially in the next four chapters. They are useful in several applications. Since the primary goal here is to apply the material to various interesting problems, we concentrate on ("smooth" and) nice Young functions, or the ./V-mnctions, for most of the following analysis. Extensions of these results for general Young functions and arbitrary measure spaces need further work, and we shall often omit it, referring the interested reader to our earlier volume where such a treatment was considered.
1.1 JV-functions and quantitative indices A mapping • R+ is termed an N-function (nice Young function) if (i) $ is even and convex, (ii) $(x) = 0 iff x = 0, and
/. Introduction and Background Material (iii) lim v
' x _> 0
x
= 0,
lim
x-»oo
= +00.
x
The left (right) derivative 0}, y € R.
(3)
A classification of TV-functions based on their growth rates is facilitated by the following: Definition 1. An TV-function $ is said to obey the A2-condition for large x (for small £, or for all #), written often as $ € A2(oo)($ e A2(0), or $ € A2), if there exist constants XQ > Q,K > 2 such that 3>(2x) < K3>(x) for x > XQ (for 0 < x < XQ, or for all x > 0); and it obeys the V2-condition for large x (for small x, or for all x), denoted symbolically as $ € V2(oo) ($ e V2(0), or $ G V2) if there are constants XQ > 0 and c > 1 such that $(rc) < -^ x0 (for 0 < a; < XQ, or for all re > 0). Quantitative indices of an TV-function <E> are provided by the following six constants.
= lim inf t->oo
$(£)
,
5* = lim supF *
^oo
$(i)
,
(4) V } (5)
1.1 N -functions and quantitative indices Clearly 1 < A$> < mm(A^,A^) and max(jB$,Sj) < £$ < oo. Analogously we define Ay, By, Ay, By, Ay and jB# for the complementary TV-function ^ of to, i.e., B$> < oo (B$ < oo, -6$ < oo); (Hi) there exist constants a > 1 and SQ > 0 such that 8^y/ > a, /or s > s0, i.e., Ay > 1(A^ > 1, Ay > 1); ^ * € V 2 (oo) (* e V 2 (0), ^ € V 2 ). Proof, (i) =4> (ii). $ G A2(oo) =» (by Definition 1) the existence of K > 2, t0 > 0 such that $(2i) < K$(t),t > t 0 - Let ^0 > K. Then for it /
f (p(u)du < I (p(u}du ./o
so that B$> < fi < oo. (ii) => (iii). Let a = -^y for (3 > 1 of (ii). Choose SQ > 0 such that V>( s o) > *o- Since s, ^y 4, as s t and ^j 4, as x t, taking t = ip(s) we get
. = a > 1,
so that Ay > a > 1. (iii) => (iv). For o; and SQ of (iii), let c > 1 be chosen to satisfy ca-i > 2. Then , In
This implies ^(w) < ^^(cu),v > s0 so that ^ e V 2 (oo).
/. Introduction and Background Material (iv) =» (i). Let §(v) - £*M. Then ^(*) - ^(cs) and the complementary TV-function $ of ^ is given by: U
SQ- It is well-known that (cf. Rao and Ren [1], p. 15) then $(w) > $(u),it > UQ for some u0 > 0, whence $(2u) < K$(u), U>UQ with # = 2c, i.e., $ € A 2 (oo). D Remark. Theorem 2 was essentially given in Krasnoselskii and Rutickii ([1], Thms. 4.1 and 4.3), see also Rao and Ren ([1], p. 26). The case for A2(0) and V2(0) was noted by Lindberg [1]. The following consequences are of some interest. Corollary 3. For a pair of complementary N -functions ($, ^) one has:
Proof. The first two equations were proved in Rao and Ren ([1], p. 27) and in Linderstrauss and Tzafriri ([1], p. 148). We now verify the last: (A$)~l + (By)~l = 1. If By < oo, then for any e > 0, and any s > 0, < By + e. By the proof that (ii) =» (iii), we have for t > 0 -
t 0 is arbitrary, (8) implies A<s> + By.
(9)
\ I / G A 2 < ^ £ e V 2 and hence A$> > 1 by the theorem and ^4$ < ^iU < oo for any N- function. Hence given 0 < 6 < A® — 1, and t > 0, we have -^W > A$— 5, by definition of A$. Letting s = y?(i), this gives stjj(s)
A$ — 6
*(s) < A* -6-1' Since 6 > 0 is arbitrary, this implies
A*BV 0 such that $(2u) > (2 + <J)$(u), t* > «o,
(11)
(for 0 < u < UQ, or for all u > 0). This was proved in Rao and Ren ([1], p. 23) for $ £ V2 and the argument is the same for the other cases. Remark. In 1946, Lozinski [2] noted that $ e V2(oo) iff liminf ^ffi^ > 2. This is equivalent to (11). Similarly $ e V 2 (0) iff liminf W£ > 2, v u u—*Q
\ )
and $ € V2 iff inf{^^ : 0 < u < 00} > 2. Also by Definition 1, we note that 2 n $(v n ) > 2$(v n ),
/
n > 1.
Setting un = $(vn) and using this expression, we get 1 \ L ^
^
- .
n
'.
$-l(2un]
\ ^
1+ n
.
-n "> 1 I i ^ i.
~
so that (3$ = 1. Regarding the second part, if $ e V 2 (oo), then its complementary function ^ 6 A 2 (oo) by Theorem 2, and by (15), there exist 0 < 6 < 1 and v0 > 0 such that ty((l + 6)v) < 2^f(v},v > VQ. But it is then known that (cf. Rao and Ren [1], p. 16) 2$ (^) < $ ( l + j ) ,w >WQ for some WQ > 0. Taking x = y, and a = ^^, one has 2 un =
VJn
>0
Letting x — $ -1 (u) here, one has $~ 1 (2w) < (2 —a)$~ 1 (w), u > $(WQ)It follows that 77^ < 77^< a*. ^ — Q,
LI N-functions and quantitative indices
7
The other result if $ ^ V2(oo) is similarly obtained with Corollary 4, and (ii), and (iii) can likewise be established. The details are omitted (or see our book [1], pp. 25-26). D Remark. Using the indices a$,/3$, Ren ([1], Sec. 7) gives a short proof of Goposkin's Theorem ([1], [2]) with a theorem of Semenov [1], which will be given in Sec. VI.4. It was reported in Chen ([1], Thm. 1.98). See e.g., Olevskii ([1], p. 73) and Banach ([1], p. 165). Chen ([1], pp. 48-50) also includes another form of Theorem 7 (i) above. The relations between (4) and (12), (5) and (13), and (6) and (14) can be given as follows: Theorem 8. Let $ be an N-function. Then we have 2
A
* < a$ < fe < 2
B
(16)
*,
and
Proof. We present the argument for just one of them, the others being analogous. Thus we establish /3$ < 2-B*" in (16). Since /3$ < 1 always, we may assume that B& < oo. Then given e > 0, by (4), there exists a to > 0 such that ^^f < B& + e for all t > to. Consequently, for trt < ti < t-z we have
/*«2
. /D , \ / < (B$ + £)
Jtl
Jj.
/ -t-
\ '
"£ i I t2\ — i = log I — I
\*1/
Let ti = $ -1 (u),^2 — $-1(2u) and UQ = UQ.
Thus the desired inequality obtains because e > 0 is arbitrary. Since | < a$ < /5$ < 1 holds always we may assume 1 < A$ < oo, and show that
8
/. Introduction and Background Material
to establish (16) completely. Thus given 0 < 6 < A$ — 1, one can find a t0 > 0 such that § > A$> - 5 for t > to. Hence for t t\> to,
Letting ti = $~ 1 (u),t 2 = $ -1 (2u) and UQ = $(io)> we deduce that __ L_ 3>-l(u] 2 ^*-* < ^ 1,/ \ , $~ (2u)'
w > uu0 . ~
The desired inequality follows from this since 6 > 0 is arbitrary. As noted before, the proof of the other inequalities is similar, (cf. also Rao and Ren [2]). D Corollary 9. Let $, ^ be complementary N -functions. If Cy<J>
==
lim
_ , $(*)
exists, then the following quantities exist and the relations hold as indicated: (i) 7$ = lim^oo ^-"iff) = 2~^; tV'f*) o -f^,
-1
* (u) r>— c 7=^~ 7^ = Tlimu-^oo ^-i (y} = 2 * ;
— 1. Similarly, if C$ = lim^o -§T^\ exists, then the corresponding limits below exist and equalities hold as indicated:
Proof, (i) If C$ is defined, then from (4) we find that A$ = B$> = C7$. __ i_ Hence a$ = /3$ = 2 c* holds and 7$ exits, 7$ = a$ = /3$. (ii) Since yl$ = B$ = (7$, by Corollary 3, we get A# = By so that C# exists. Using the preceding theorem for \I/ we find
and hence ay = 3 =
__ _ = 2 c* .
1.1 N -functions and quantitative indices (iii) By (i) and (ii) above, we have C$l 4- C^1 — 1 so that
as asserted.
D
We now include several examples to explain the utility of the preceding indices. Example 10. Consider the (Lebesgue) complementary TV-functions: P
p
< oo; Q
Then ^^ = p, 0 < t < oo, and from Corollary 9 we deduce that
C* = Cg = p = A* = B$;^ = 7g = a$ = fa = 2 ~ p ; C* = Cg = »• l
where F(u) — [log(l+ii)][log(l+2u)] , it follows from a differentiation of F relative to u > 0, obtaining dF
=
(1 + 2u) log(l + 2u) - 2(1 + u} log(l + «)
> Q
(19)
that
a$r = /3$r — 7*.
•u—>-oo
Then by Theorem 7, $r € A 2 (0) n V 2 (0), and &L £ V 2 (oo) but it is not in A2(oo). From (19) we find that o;$r — 7$r, ^<j.r = 7$r. It will be seen in Chapter 5 that the "packing constants" of the Orlicz sequence space for $r and the Lebesgue sequence space lr are the same! Example 13. Consider the function M p ( - ) , p > 2, defined by u IP Mp(u) = log(e+ \u\)' This was shown to be an TV-function of class A2(oo), in Krasnoselskii and Rutickii ([1], p. 30), for p = 2. The same assertion is true for p > 2 as well. In fact, since Mp(0) = 0, and for u > 0 dMp _ pup~1(e + u) log(e + u) - up > ~d^(U} ~ (e + n)[log(e + u)] 2 ' 2 2 2 d Mp = puP- {[(e + u} log(e + u) - u} + g ( u ) } 2 (U) du (e + u) 2 [log(e + n)]3 where 2 1 g(u) = (p ~ 2}(e + w) 2 [log(e + n)] 2 - w2 + -u2 + -u2 log(e + u)
> (p~ 2)u2 - u2 + —u2 + -n2 P P u2 P -- + - > 0. y
1.1 N-functions and quantitative indices
11
Thus Mp(-) is convex increasing, and is an AT-function for p > 2 also. Since clearly limK_).00 M fu) ~ ^P' ^ 1S m ^2(00) as well. We shall see in Chapter 3 that the Orlicz sequence space ^(MP) and the Lebesgue sequence space lp have the same corresponding weakly convergent sequence coefficients. Example 14. Consider the function Mp defined by (1 < p < oo)
This is an TV-function for c > (2p — l}/p(p — 1), (cf., Gribanov [1], p. 346). However, for c = 1 and p > 1, Mp is the principal part of an TV-function for \u\ > 1, for instance,
u\p, Mp(u),
\u\ < 1, \u\ > 1,
defines 3>p as an TV-function. This distinction was not detailed in our book ([1], p. 26) or in Krasnoselskii and Rutickii ([1], pp. 15, 27, 33). [Recall that a function M(-) is called the principal part of an TVfunction TV(-), if it (is convex and) coincides with TV(-) for large values of the argument. This Mp above is a principal part of 0.] The following result will often be used in Chapters 2-5, and it is recently verified by Yan [1]. Theorem 15. Let 0 such that $-1(w) > (a$ — £-)$~ 1 (2u) for u > KQ, or equivalently u > $[(a$ - £:)$~ 1 (2u)],u > u0. Let t = $~ 1 (2u),i 0 = $~ 1 (2u 0 ), then - e)t],
t > t0
(24)
/. Introduction and Background Material
12
From some elementary results on ./V-functions (cf. e.g., Krasnoselskii and Rutickii [1], p. 12 and p. 14, Theorem 2.1), (2) implies that there exists s0 > 0 such that S > SQ
or equivalently,
,-1 2 J1
O/^ 2(a$ — ,-N £)
,
S ^ SQ.
Letting v — |^(s), we get for v > VQ = |\I
Thus 2(a$ — e}8jf < 1 since /8&* = limsup . T ,_i/v\, / ' * • ^ W l A V J x
so
(25) On the other hand, for any given e > 0 there is a VQ — VQ(E) > 0 such that . < fa +e for v > VQ. Let s — ^~ 1 (2v), then ^(s) < ifs > SQ = ty~l(2vo). By the same results on TV-functions, there is a to > 0 such that
> 2$
, or
t L 2
for £ > £Q- Letting u = ^$(£) we have for u > UQ — |^(*0;
Hence 2a$ (/3* + e) > 1, and since e is arbitrary one has
(26) Finally, (23) follows from (25) and (26). Similarly, we can verify (21) and (22). D
1.2 Some basic results on L* -spaces
13
1.2 Some basic results on L*-spaces We set down the basic concepts and some results of Orlicz spaces for convenience and constant use below. Hereafter $ usually denotes an JV-function and (fi, S, /z) a measure space to be specialized from time to time. Definition 1. The set of functions L* is denned on (£2, S, //) by: L* = {/ : £} ->• R, measuarable,
p$(/) = / 3>(f)dfj, Jsi
< 00} .
If 0 < n(A) < oo, A £ X)> then we have the Jensen inequality:
In case // is a discrete measure on A, then one has the corresponding inequality of (1), replacing the integral by sums with appropriate weights. If n is discrete on Q itself, where fi is also countable, and (j, is a counting measure (take fi = {1,2,...} for definiteness, and so p>({i}) = 1) the space L* becomes the Orlicz sequence class, denoted ^*. These classes are not necessarily linear. We have: Proposition 2. ft) //$ e A2(oo) [$ e A 2 ], i/ien L* is a vector space when /Li(fJ) < oo[/Lt(fi) = oo]. The converse holds if fj, is diffuse on a set of positive measure. (ii) ^* is a vector space i f f $ £ A2(0). Proof. Consider the case that /^(fi) < oo, and $ e A2(oo) so that there are K > 2, UQ > 0, such that $(2u) < K$(u], for all u > UQ. For linearity let / 6 L® and c e R. Then we can find an integer no > 1 such that |c| < 2n° and so $(cu) < $(2n°u) < K n °$(u),u > u0. Hence
r
cfrl it 0, U —>• OO.
But we also have ||/n||* =UnH(Gn)V-
U
Hence (iii) fails. A similar argument proves that (iv) => (i) . D We now introduce the gauge norm, with Theorem 6 (iii), using the classical Minkowski gauge and then present its equivalence with the Orlicz norm given by (2). The gauge norm || • ||($) is defined by:
/GL*.
(8)
1.2 Some basic results on L* -spaces
17
One can always normalize the Young pair ($, \I>) so that $(1) + ^(1) = 1, and in this case the corresponding functional (denoted AT$(-)) is given by >0 : p* < $ ( ! ) , / € L*, (9) and (8) and (9) are clearly equivalent. This was used systematically in our companion volume ([1], p. 56) after its introduction. For applications the unnormalized form (8) is convenient, and we use it unless otherwise explicitly stated, with the simpler notation of (8). Some basic properties, often used in the following, are given by: Theorem 8. Let $, \f be a complementary pair of N -functions and L*,L* be the corresponding Orlicz spaces on (Q, £,//). Then Q ^ f E L*, p*(/) < 1 iff H/llw < 1 so that
so that these are equivalent (norm) functional; (Hi) (Improved Holder's inequality) f E L*, g e L* =>• fg 6 Ll and
We note that (i) is an immediate consequence of the convexity of $ and (ii) and (iii) are proved in (Rao and Ren [1], pp. 61-62) and will not be repeated here. We add a computation for comparison. Example 9. If E e S, 0 < n(E) < oo, then
This follows from the fact that in (8) if there is a k0 > 0 such that P& (*M = 1, then ||/||(<j>) = ko. Thus for XE we ^n^ A;0 > 0 such that P& (if) — 1 where k0 = \3>~l f ^ y )
, whence (12) obtains.
Example 10. If $p(w) = \u\p, 1 < p < oo, then the complementary JV-mnction ^ is seen to be with q = -^-,
18
/. Introduction and Background Material
Then &0 = ||/||P < oo, is the Lebesgue Lp-norm, and
The infimum is attained for k = ko so that ||/||($ ) = ||/||P- However, the same is not true for || • ||(* ), but we have for g € L*9, \\9\\*q = *up
\fg\d» : \ \ f \ \ ( * p ) < H|A.
(13)
The reason for this difference is that ($p, \£g) is not a normalized pair. Thus alternating the gauge and Orlicz norms we get a classical form of some results such as the Holder inequality. We next consider the M* space of Definition 3 and describe its adjoint or dual, to be used in applications. Recall that the set M* is given as: M* = {/ : p*(a/) < oo for all a > 0}. (14) If 0. Since $ is an AT-function, it follows from standard Real Analysis results that / vanishes off a 0, we have (||/||($) = 1) P*((l + e)/) > 11(1 + £)/!!(*) = (1 + e), if we define the truncated functions fmn as fmn =
JXAmr\[f oo and by (18), there exists an no such that for n, m > HQ one has mn
£• g
_
Recalling that (f> is the left derivative of $, if we set ^(C 1 + g)/mn)
-
9mn — -
/
//-
\ /.
, on x
v^W
\\ j
then ^mn is bounded and has support in Am so that gmn e M® for each m,n. Moreover, by (2) and the Young inequality, ||pmn||# < 1. However, one has : g G M*, ||^||^ < 1
> sup
: gmn G M*, as in (20)
m,n>n 0 3—— SUp / (1 + e)/ -1- + £ m,n>n 0 7fi 0
; SmC6 0 < fmn < /,
> j - : , by (19).
Since e > 0 is arbitrary, we get the desired inequality, and hence (16). The proof of (17) is similar. In fact it is also given in Krasnoselskii and Rutickii ([1], p. 86) for the case that [J>(Q) < oo, and the argument extends easily for the general case as in this work. D The above result will be used in obtaining the adjoint space (M*)* of M* on a cr-finite space, where (M*)* is the set of continuous linear functionals on M$. The general L* space is more complicated. Thus the following result is a specialized version of (Rao and Ren [l], p. 125) which suffices for the applications here. Theorem 13. Let $, ^ be a pair of complementary N -functions and (Q, S, /Lt) a a-finite space on which M* and L® are defined. Then (i) (M*)* = L^\ (isometric isomorphism), this means, for each I € (M $ )*, there is a unique g G L* such that
t
Jn.
f € M*,
(21)
1.2 Some basic results on L* -spaces
21
and \\l\\ = sup{|^(/)| : ||/||$ < 1} = ||^||(*), t «->• g is a one-to-one and onto correspondence: (ii) (M^*))* = L*. The correspondence being given by (21), and \\l\\ = \\g\\y, the Orlicz norm. The proof even of this result is fairly long and we refer the reader to the above book and we shall not repeat the specialization here. Recall that a Banach space X is reflexive if X** can be identified with X. Using this concept, and taking ($7, E) as (R+,/3) or ([0,1], B] and n as the Lebesgue or counting measure, we have the following consequence: (L* is for L*(R + ),L*([0, 1]), or t® in these cases.) Corollary 14. (i) The space L*([0, 1])(L(*)([0, 1])) is reflexive iff® e (A 2 nV 2 )(oo), (ii) L*(R+)(L(*>(R+)) is reflexive iff® G A 2 n V 2 , (Hi) £*(#*)) is reflexive iff $ £ (A2 n V 2 )(0). The proof follows immediately from the preceding theorem and the classifications of TV-functions given in the preceding section. We remark that if $ is not A2-regular, then (L*)* D L^ properly, and the corresponding characterization of (L*)* extending Theorem 13 is more difficult. That result and relevant references are given in Rao and Ren ([1], Chapter 4), and will not be discussed here. The following result is used in Chapter 3. Theorem 15. Let (Q, E, /it) be a measure space with fj, as a counting measure, and l®(£^) be the corresponding Orlicz sequence space with $ as an N -function, and m*(m^) the Morse-Transue subspace of it. Then a bounded sequence { f n ^ n > 1} C ra*(ra(*)) converges weakly to Oiff (i) fn(t) —> 0, os n —» oo, for each t G Q, and (ii) lim ^ p ^ ( k f n ) = 0, uniformly in n. Proof. Since p, is a counting measure, fn(t) -> 0 for each t with 0 < jj>({t}) < oo. Hence,
{t}
But /„(£) -> f ( i ) = 0 for each t, 0 < //({*}) < °°) an^ then fn -* 0 on each set of finite ^-measure. Now the characteristic functions of such sets form a norm determining subspace of m^(m*). Thus fn -> 0 weakly. By a well-known result (cf. Rao and Ren [1], p. 150, Prop.
22
/. Introduction and Background Material
7), the set {/n, n > 1} is relatively weakly sequentially compact iff (ii) holds. The converse is obtained by the converse part of the result just referred to. D To have a better idea of the Orlicz and gauge norms, we include a few sharper properties here. Thus if 0 such that (kf usually depends on $ as well)
1,
(22)
where \I/ is complementary to $, then by the equality conditions in Young's inequality and (22), one has ll/ll* = / \f\V>(kf\f\)dn
= -j-(l + P*(fc//)).
(23)
It is also possible to present the Orlicz norm || • ||$ in terms of <E> alone: ||* = inf
fc>0 K
(l + p*(*/)),
(24)
(cf. Krasnoselskii and Rutickii [1], Thm. 10.5; also Hudzik and Maligranda [1]). The following result, established by Wu, Zhao and Chen [1], presents conditions for the existence of a single kf to satisfy (24). Note that both the complementary AT- functions $, ^ come into play as in (23). Theorem 16. For each 0 ^ f G L*; there exists a fc0(= ^o(/> $) > 0) such that (24) becomes
o
(25)
< k$ < kz, where k\ = infjA; > 0 : pv((k\f\)) > 1} fc2 - supjfc > 0 : p*(yj(fe|/|)) < 1}. For a proof of this result we refer the reader to (Orlicz [2], pp. 156158), Chen ([1], Theorem 1.30) or Rao and Ren ([1], pp. 69-72). With this, equality in Holder's inequality for Orlicz spaces can be given:
1.2 Some basic results on L* -spaces
23
Theorem 17. For a pair of complementary N-functions $, ^ and 0 7^ / € L(*),0 / g G L* equality in Holder's inequality holds, i.e.,
L M i/iere exists a 0 < k* < oo such that — oo.
The proof of the parenthetical statement is similar.
D
Lemma 22. Let f e L*, ||/||((») = UQ. If ») = 1 and |/| > $~! (26) holds, and a(f) — p*(y(|/|)), */ien (sgn(x) = Q,x = 0; — A, a;
o;
26
/. Introduction and Background Material
Proof. By lemmas 20 (iv) and 21, it follows that M* = L* and p$(f) = 1, since ||/||($) = 1. Also 0 < a(f) < oo. Define
(32) l ' By choice |/| > UQ and so with (30), we get
P*(kgi) = / * f ./ft
V
so that ||fc / 92/d» - 1—i— r / \f\v(\f\)dn Jo. +«(/)7o
and
Thus there is equality in the above inequalities implying gtdf* =
l
J- "T
tt^/J
{ |/|^(i/|)^ = 1 - ||/||(*).
^Q
(34)
Consequently there is equality in Holder's inequality [(33) and (34)], and since (p is continuous, M* = L*, it follows that pi = g% a.e. which is the same as (31). D Proof of Theorem 19. We now can quickly complete the proof. Let MI, u-2 > uQ = 6, i.e., ^ ^ /, a contradiction since we assumed ^ G / n //. Next we show $ G IV. If not, then
0. therefore, ^(^) = 0 for 0 < f < a and $(v) = Jjj1'1 ^(s)ds = 0 if 0 < |v < a, i.e., ^ ^ II. A contradiction. By Proposition 3 (iii), $ is an AT-function. Finally, \I> is also an JV-function, since the complementary function of an JV-function is an AT-function as pointed out in Section 1.1. D Now we give some examples, which are Young functions but not AT-functions. Example 5. Let $i(u) = jj e*dt = e\u\ — 1, then its complementary Young function is given by
— v
v v\ > I.
By Definition 2, $1 G / n II n /// but $1 ^ IV since 2, *!(2t;) - 0 - KVi(v) if 0 < v < J, and lim ^ - 1, i.e., $1 ^ V2(0). A basic property for AT-complementary pair ($, \l/) is that $ G V 2 (0)($ G V 2 (oo), or $ G V 2 ) iff * G A 2 (0)(# G A2(oo), or \I> G A 2 ). This example shows that the Young pair does not satisfy the above assertions. We shall give a classification of these growth conditions for Young functions (see Definition 8 below). Example 6. Let $2(«) = ^ if w < 1; = \u - \ if \u\ > 1. Then its complementary Young function \l>2 is given by ^ r 2 (v) = -^j- if |v| < 1; = +00 if \v > 1. By Definition 2, $2 G / n // n IV but $2 g ///,
30
/. Introduction and Background Material
and #2 e II n III n IV but *2 i I- Note that for all K > 0, # 2 (2v) = oo = KV2(v) if |v| > 1, and Jim - 1, i.e., $2 £ V2(oo). t—yoo
Example 7. Let $3(n) = \u , then * 3 (v) = 0 if v\ < 1;= +00 if v\ > 1, which follows from (1) and (2). We see that $3 G I n II but $2 £ /// u IV, and W 3 ^ I U II but ^3 G III n IV. Note that ^^ = 2 for all u > 0, i.e., $3 ^ V 2 (0) U V 2 (oo) but $3 G A 2 . Let us introduce the following: Definition 8. Let $ be a Young function. (i) $ G A 2 (oo) means $ G I and limsup AJy < oo, u—-too
(ii) Q
(iii) $ G A 2 means $ G I n II and sup ^vv < oo, •u>0
^ '
(iv) $ G V 2 (oo) means $ G I and lim inf ^Ly v > 2, (v) $ G V 2 (0) means $ G II and lim inf J, v > 2, and finally u->0
*W
(vi) $ G V 2 means $ G I Pi II and inf v? v wv > 2. w>0
v ;
Prom this definition, we find that G A 2 iff $ G A 2 (oo)nA 2 (0), and $ G V 2 iff $ G V 2 (oo) n V 2 (0). Let ($;, ^i), i = 1, 2,3 be given as in Examples 5-7. Then $! G A 2 (0)\V 2 (0), $1 G V 2 (oo)\A 2 (oo), *i G A 2 (oo)\V 2 (oo), and tyl £ A 2 (0) U V 2 (0). For i = 2,3,$i G A 2 but ^i ^ V 2 . Thus, the basic property for JV-functions, recalled in Example 5, need not be true for the Young pair. Remark 9. For a Young function $, Maligranda [2, p. 98] defined the A 2 condition of $ somewhat differently. However, our Definition 8 (i)-(iii) is seen to be the same as his definition. By Proposition 4 and Definition 8 we have the following: Corollary 10. Let ($, ^) be a Young pair. If $ G A 2 and $ G A 2; then $ and \£ are N-functions. If (ft, S, fj,) is an abstract measure space and $ is a Young function, we let L® and I/*) denote, as before, for measurable functions:
{/ (. Jn
= < f : ft -> R, |/||$ = sup { \ fgdfj, : py(g} < 1 >
0 : / $ (() dp < 1\ < 00} . I I 7n \ f c / J J It is verified that || • ||$ and || • ||($) are norms if a.e. equal functions are identified and then L® = L^ as sets, and these are Banach spaces. We again call them Orlicz spaces. Same proofs hold, as may be seen from our earlier volume [1]. We recall the result of Wu et al. [1], stating that for an TV-function • the existence of a k* > 0 such that 11/11* =
[l + P*(**/)].
(3)
If $ = $3 ($3(14) = |u|), which is a Young (not an N-) function, then (3) cannot hold unless k* = oo and -&rp$>3(k*f) is taken p®3(f). As another example, let $ be a continuous Young function that vanishes only at the origin and suppose that the complementary Young function ^ of $ has the same properties (by Proposition 4, $ is an TV-function), then it is shown that ||/||$ = ||/||($) iff / = 0 (cf. Rao [5], Lemma 1). Since for $3 of the above Young function, which is not an TV-function, this result is not applicable (it cannot hold since, in fact, the equality is valid for L 1 ). One of the important features of Orlicz spaces is that the two norms || • ||$ and || • ||($) are equivalent and rarely equal, in contrast to the Lebesgue spaces. The spaces L® and L^ are isomorphic. In some ways this is a weaker property, but it is compensated by the existence of a rich collection of isomorphically equivalent Orlicz spaces. This is exemplified by the following result of considerable importance in applications. Theorem 11. Let $ be a continuous Young function such that $(u) = 0 iffu — 0, i.e., e I nil. Then there exists a strictly convex continuous Young function $1 such that L*(JJ,) = L*1^), on any measure space (fi, £, /Lt), and their norms are equivalent. Thus the spaces are isomorphic (but not isometric). Moreover, z/i(t) = 0 and set /• / Joo
l) < 2||/||w. Thus, the spaces are isomorphic. The A2-regularity is immediate from construction. D In the above reference an uncountable collection of distinct Young functions generating equivalent norms is given. Note in particular, if $(w) = w|, then $1 constructed above is strictly convex and A2regular, and hence even though L^(IJL) = Ll(p,} is not strictly convex, its isomorphically equivalent (collection) L^l^(n) is strictly convex (cf. our [1], p. 272). On the other hand, 4>(u) = \u\ and its (Young) complementary function ^ are not ./NT-functions, but are necessary to consider these "boundary" functions if the Orlicz space theory is to generalize the Lebesgue theory. Moreover, L1 and L°° are some of the most important concrete Banach spaces which are also algebras. These spaces cannot be included in Orlicz spaces based on iV-functions alone. Let us start with the following: Definition 12. A vector space X is called a commutative Banach algebra if (X, \\ • ||) is a Banach space and for each or, y in X a product xy is defined and the latter satisfies xy = yx as well as the inequality \\xy\\ ^ 11^ II I \y I 5 so that multiplication is a continuous operation in the norm topology of X. As examples, (i) X = Co(R), the space of continuous functions vanishing at ±00, with uniform norm; (ii) L°°(R), under pointwise multiplication; (iii) X = L 1 (M), the Lebesgue space of integrable functions on (R, S, //) with (j, as Lebesgue measure and multiplication as convolution, i.e., for /, g E -^1(R), the convolution / * g is given by (/ * g)(x) = f f(x- y)g(y)dn(y),
x e R.
(6)
JR
These are commutative, and if X — B(y) ,the space of all bounded linear operators on a Banach space 3^ with operator norm, then it is a noncommutative Banach algebra. A change of variables, and translation invariance of ^ shows that f *g = g* f m (6), and ||/*
