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Foundations of Symmetric Spaces of Measurable Functions: Lorentz, Marcinkiewicz and Orlicz Spaces 1st ed. 2016 [Kõva köide]

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  • Formaat: Hardback, 259 pages, kõrgus x laius: 235x155 mm, kaal: 639 g, 90 Illustrations, black and white; XVII, 259 p. 90 illus., 1 Hardback
  • Sari: Developments in Mathematics 45
  • Ilmumisaeg: 20-Dec-2016
  • Kirjastus: Springer International Publishing AG
  • ISBN-10: 3319427563
  • ISBN-13: 9783319427560
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Foundations of Symmetric Spaces of Measurable Functions: Lorentz, Marcinkiewicz and Orlicz Spaces 1st ed. 2016Suurem pilt
  • Formaat: Hardback, 259 pages, kõrgus x laius: 235x155 mm, kaal: 639 g, 90 Illustrations, black and white; XVII, 259 p. 90 illus., 1 Hardback
  • Sari: Developments in Mathematics 45
  • Ilmumisaeg: 20-Dec-2016
  • Kirjastus: Springer International Publishing AG
  • ISBN-10: 3319427563
  • ISBN-13: 9783319427560
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9783319427560.html
Märksõnad:

Key definitions and results in symmetric spaces, particularly Lp, Lorentz, Marcinkiewicz and Orlicz spaces are emphasized in this textbook. A comprehensive overview of the Lorentz, Marcinkiewicz and Orlicz spaces is presented based on concepts and results of symmetric spaces. Scientists and researchers will find the application of linear operators, ergodic theory, harmonic analysis and mathematical physics noteworthy and useful.

This book is intended for graduate students and researchers in mathematics and may be used as a general reference for the theory of functions, measure theory, and functional analysis. This self-contained text is presented in four parts totaling seventeen chapters to correspond with a one-semester lecture course. Each of the four parts begins with an overview and is subsequently divided into chapters, each of which concludes with exercises and notes. A chapter called “Complements” is included at the end of the text as supplementary material to assist students with independent work. 

Arvustused

This book represents a consistent reference on the basic theory of symmetric spaces of measurable functions also known as rearrangement invariant function spaces. The text contains many illustrations in order to clarify concepts and make the theory more comprehensive and understandable. The book is structured in such a way that it should correspond to a one-semester special course of lectures . (Santiago Boza, zbMATH 1361.42001, 2017)

Part I Symmetric Spaces. The Spaces Lp, L1 ∩ L∞, L1 + L∞
1 Definition of Symmetric Spaces
5(12)
1.1 Distribution Functions, Equimeasurable Functions
5(4)
1.2 Generalized Inverse Functions
9(2)
1.3 Decreasing Rearrangements
11(1)
1.4 Integrals of Equimeasurable Functions
12(1)
1.5 Definition of Symmetric Spaces
13(1)
1.6 Example. Lp, 1 ≤ p ≤ ∞
14(3)
2 Spaces Lp, 1 ≤ p ≤ ∞
17(12)
2.1 Holder's and Minkowski's Inequalities
17(4)
2.2 Completeness of Lp
21(2)
2.3 Separability of Lp, 1 ≤ p < ∞
23(1)
2.4 Duality
24(5)
3 The Space L1 ∩ L∞
29(12)
3.1 The Intersection of the Spaces L1 and L∞
29(1)
3.2 The Space L0∞
30(3)
3.3 Approximation by Step Functions
33(2)
3.4 Measure-Preserving Transformations
35(3)
3.5 Approximation by Simple Integrable Functions
38(3)
4 The Space L1 + L∞
41(18)
4.1 The Maximal Property of Decreasing Rearrangements
41(4)
4.2 The Sum of L1 and L∞
45(4)
4.3 Embeddings L1 ⊂ L1 + L∞ and L∞ ⊂ L1 + L∞. The Space R0
49(10)
Exercises
51(4)
Notes
55(4)
Part II Symmetric Spaces. The Embedding Theorem. Properties (A), (B), (C)
5 Embeddings L1 ∩ L∞ ⊂ X ⊂ L1 + L∞ ⊂ L0
59(12)
5.1 Fundamental Functions
59(2)
5.2 The Embedding Theorem L1 ∩ L∞ ⊂ X ⊂ L1 + L∞
61(5)
5.3 The Space L0 and the Embedding L1 + L∞ ⊂ L0
66(5)
6 Embeddings. Minimality and Separability. Property (A)
71(12)
6.1 Embedded Symmetric Spaces
71(2)
6.2 The Intersection and the Sum of Two Symmetric Spaces
73(2)
6.3 Minimal Symmetric Spaces
75(1)
6.4 Minimality and Separability
76(3)
6.5 Separability and Property (A)
79(4)
7 Associate Spaces
83(12)
7.1 Dual and Associate Spaces
83(2)
7.2 The Maximal Property of Products f*g*
85(5)
7.3 Examples of Associate Spaces
90(2)
7.4 Comparison of X1 and X*
92(3)
8 Maximality. Properties (B) and (C)
95(20)
8.1 The Second Associate Space
95(2)
8.2 Maximality and Property (B)
97(1)
8.3 Embedding X ⊂ X11 and Property (C)
98(5)
8.4 Property (AB). Reflexivity
103(12)
Exercises
106(3)
Notes
109(6)
Part III Lorentz and Marcinkiewicz Spaces
9 Lorentz Spaces
115(12)
9.1 Definition of Lorentz Spaces
115(4)
9.2 Maximality. Fundamental Functions of Lorentz Spaces
119(1)
9.3 Minimal and Separable Lorentz Spaces
120(4)
9.4 Four Types of Lorentz Spaces
124(3)
10 Quasiconcave Functions
127(12)
10.1 Fundamental Functions and Quasiconcave Functions
127(1)
10.2 Examples of Quasiconcave Functions
128(2)
10.3 The Least Concave Majorant
130(5)
10.4 Quasiconcavity of Fundamental Functions
135(1)
10.5 Quasiconvex Functions
136(3)
11 Marcinkiewicz Spaces
139(12)
11.1 The Maximal Function f**
139(4)
11.2 Definition of Marcinkiewicz Spaces
143(1)
11.3 Duality of Lorentz and Marcinkiewicz Spaces
144(3)
11.4 Examples of Marcinkiewicz Spaces
147(4)
12 Embedding Λ0~V ⊂ X ⊂ Mv*
151(20)
12.1 The Embedding Theorem
151(5)
12.2 The Renorming Theorem
156(1)
12.3 Examples of Lorentz and Marcinkiewicz Spaces
157(5)
12.4 Comparison of Lorentz and Marcinkiewicz Spaces
162(9)
Exercises
163(3)
Notes
166(5)
Part IV Orlicz Spaces
13 Definition and Examples of Orlicz Spaces
171(12)
13.1 Orlicz Functions
171(2)
13.2 Orlicz Spaces
173(4)
13.3 Fundamental Functions of Orlicz Spaces
177(1)
13.4 Examples of Orlicz Spaces
178(5)
14 Separable Orlicz Spaces
183(12)
14.1 Young Classes YΦ and Subspaces HΦ
183(2)
14.2 Separability Conditions for Orlicz Spaces
185(5)
14.3 The (Δ2) Condition
190(2)
14.4 Examples of Orlicz Spaces with and Without the (Δ2) Condition
192(3)
15 Duality for Orlicz Spaces
195(12)
15.1 The Legendre Transform
195(2)
15.2 The Geometric Interpretation
197(3)
15.3 Duality for Orlicz Spaces
200(5)
15.4 Duality and the (Δ2) Condition. Reflexivity
205(2)
16 Comparison of Orlicz Spaces
207(10)
16.1 Comparison of Orlicz Spaces
207(2)
16.2 The Embedding Theorem for Orlicz Spaces
209(3)
16.3 The Coincidence Theorem for Orlicz Spaces
212(1)
16.4 Zygmund Classes
213(4)
17 Intersections and Sums of Orlicz Spaces
217(20)
17.1 The Intersection and the Sum of Orlicz Spaces
217(3)
17.2 The Spaces LΦ + L∞ and LΨ ∩ L1
220(3)
17.3 The Spaces LΦ + L1 and LΨ ∩ L∞
223(1)
17.4 The Spaces Lp ∩ Lq and Lp + Lq, 1 ≤ p ≤ q ≤ ∞
224(13)
Exercises
229(5)
Notes
234(3)
Complements
237(14)
1 Symmetric Spaces on General Measure Spaces
237(2)
2 Symmetric Spaces on [ 0, 1]
239(2)
3 Symmetric Sequence Spaces
241(2)
4 The Spaces Lp, 0 < p < 1
243(2)
5 Weak Sequential Completeness. Property (AB)
245(1)
6 The Least Concave Majorant
246(1)
7 The Minimal Part M0v of the Marcinkiewicz Space Mv
247(1)
8 Lorentz Spaces Lp,q and Orlicz--Lorentz Spaces
248(3)
References 251(4)
Index 255
Ben-Zion A. Rubshtein is a Professor in the Department of Mathematics at Ben Gurion University. His research interests are ergodic theory and probability.





Genady Ya. Grabarnik is an Associate Professor in the Mathematics and Computer Science Department at St. John's University.  His research interest is in functional analysis.





Mustafa A. Muratov is a Professor in the Mathematics and Computer Science Department at Taurida National University. His research interest is in functional analysis.





Yulia S. Pashkova is a Professor in the Mathematics and Computer Science Department at Taurida National University. Her research interest is in dynamical systems and ergodic theory.