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Introductory Course in Lebesgue Spaces 1st ed. 2016 [Kõva köide]

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  • Formaat: Hardback, 461 pages, kõrgus x laius: 235x155 mm, kaal: 9198 g, 14 Illustrations, black and white; XII, 461 p. 14 illus., 1 Hardback
  • Sari: CMS Books in Mathematics
  • Ilmumisaeg: 06-Jul-2016
  • Kirjastus: Springer International Publishing AG
  • ISBN-10: 3319300326
  • ISBN-13: 9783319300320
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Introductory Course in Lebesgue Spaces 1st ed. 2016Suurem pilt
  • Formaat: Hardback, 461 pages, kõrgus x laius: 235x155 mm, kaal: 9198 g, 14 Illustrations, black and white; XII, 461 p. 14 illus., 1 Hardback
  • Sari: CMS Books in Mathematics
  • Ilmumisaeg: 06-Jul-2016
  • Kirjastus: Springer International Publishing AG
  • ISBN-10: 3319300326
  • ISBN-13: 9783319300320
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9783319300320.html
Märksõnad:
This book is devoted exclusively to Lebesgue spaces and their direct derived spaces. Unique in its sole dedication, this book explores Lebesgue spaces, distribution functions and nonincreasing rearrangement. Moreover, it also deals with weak, Lorentz and the more recent variable exponent and grand Lebesgue spaces with considerable detail to the proofs. The book also touches on basic harmonic analysis in the aforementioned spaces. An appendix is given at the end of the book giving it a self-contained character. This work is ideal for teachers, graduate students and researchers.

Convex Functions and Inequalities.- Lebesgue Sequence Spaces.- Lebesgue Spaces.- Distribution Function and Non-Increasing Rearrangement.- Weak Lebesgue Spaces.- Lorentz Spaces.- Non-Standard Lebesgue Spaces.- Interpolation of Operators.- Maximal Operator.- Integral Operators.- Convolution and Potentials.- Appendices.

Arvustused

The book under review is dedicated solely to Lebesgue spaces and their direct derivatives . The book may be recommended to graduate students and non-specialists in the area of function spaces. Each chapter is concluded with an extensive list of problems and short bibliographic notes. The book ends with a list of 86 references and with symbol and subject indices. (Alexei Yu. Karlovich, zbMATH 1352.46003, 2017)

Preface vii
1 Convex Functions and Inequalities
1(22)
1.1 Convex Functions
1(6)
1.2 Young Inequality
7(7)
1.3 Problems
14(5)
1.4 Notes and Bibliographic References
19(4)
Part I Function Spaces
2 Lebesgue Sequence Spaces
23(20)
2.1 Holder and Minkowski Inequalities
23(3)
2.2 Lebesgue Sequence Spaces
26(5)
2.3 Space of Bounded Sequences
31(4)
2.4 Hardy and Hilbert Inequalities
35(5)
2.5 Problems
40(2)
2.6 Notes and Bibliographic References
42(1)
3 Lebesgue Spaces
43(96)
3.1 Essentially Bounded Functions
43(2)
3.2 Lebesgue Spaces with p ≥ 1
45(18)
3.3 Approximations
63(7)
3.4 Duality
70(18)
3.5 Reflexivity
88(2)
3.6 Weak Convergence
90(11)
3.7 Continuity of the Translation
101(2)
3.8 Weighted Lebesgue Spaces
103(5)
3.9 Uniform Convexity
108(6)
3.10 Isometries
114(6)
3.11 Lebesgue Spaces with 0 < p < 1
120(9)
3.12 Problems
129(8)
3.13 Notes and Bibliographic References
137(2)
4 Distribution Function and Nonincreasing Rearrangement
139(44)
4.1 Distribution Function
139(3)
4.2 Decreasing Rearrangement
142(26)
4.3 Rearrangement of the Fourier Transform
168(12)
4.4 Problems
180(2)
4.5 Notes and Bibliographic References
182(1)
5 Weak Lebesgue Spaces
183(32)
5.1 Weak Lebesgue Spaces
183(4)
5.2 Convergence in Measure
187(4)
5.3 Interpolation
191(11)
5.4 Normability
202(11)
5.5 Problems
213(1)
5.6 Notes and Bibliographic References
214(1)
6 Lorentz Spaces
215(54)
6.1 Lorentz Spaces
215(11)
6.2 Normability
226(8)
6.3 Completeness
234(2)
6.4 Separability
236(3)
6.5 Duality
239(11)
6.6 L1 + L∞ Space
250(5)
6.7 L exp and L log L Spaces
255(5)
6.8 Lorentz Sequence Spaces
260(7)
6.9 Problems
267(1)
6.10 Notes and Bibliographic References
268(1)
7 Nonstandard Lebesgue Spaces
269(44)
7.1 Variable Exponent Lebesgue Spaces
269(28)
7.1.1 Luxemburg-Nakano Type Norm
272(5)
7.1.2 Another Version of the Luxemburg-Nakano Norm
277(1)
7.1.3 Holder Inequality
278(2)
7.1.4 Convergence and Completeness
280(1)
7.1.5 Embeddings and Dense Sets
281(3)
7.1.6 Duality
284(1)
7.1.7 Associate Norm
285(6)
7.1.8 More on the Space Lp(·)(Ω) in the Case p + = ∞
291(1)
7.1.9 Minkowski Integral Inequality
292(1)
7.1.10 Some Differences Between Spaces with Variable Exponent and Constant Exponent
293(4)
7.2 Grand Lebesgue Spaces
297(11)
7.2.1 Banach Function Spaces
297(3)
7.2.2 Grand Lebesgue Spaces
300(6)
7.2.3 Hardy's Inequality
306(2)
7.3 Problems
308(2)
7.4 Notes and Bibliographic References
310(3)
Part II A Concise Excursion into Harmonic Analysis
8 Interpolation of Operators
313(18)
8.1 Complex Method
316(3)
8.2 Real Method
319(10)
8.3 Problems
329(1)
8.4 Notes and Bibliographic References
330(1)
9 Maximal Operator
331(28)
9.1 Locally Integrable Functions
331(1)
9.2 Vitali Covering Lemmas
332(2)
9.3 Hardy-Littlewood Maximal Operator
334(10)
9.4 Maximal Operator in Nonstandard Lebesgue Spaces
344(6)
9.5 Muckenhoupt Weights
350(5)
9.6 Problems
355(3)
9.7 Notes and Bibliographic References
358(1)
10 Integral Operators
359(24)
10.1 Some Inequalities
359(9)
10.2 The Space L2
368(5)
10.2.1 Radon-Nikodym Theorem
371(2)
10.3 Problems
373(9)
10.4 Notes and Bibliographic References
382(1)
11 Convolution and Potentials
383(36)
11.1 Convolution
383(10)
11.2 Support of a Convolution
393(2)
11.3 Convolution with Smooth Functions
395(7)
11.3.1 Approximate Identity Operators
397(5)
11.4 Riesz Potentials
402(4)
11.5 Potentials in Lebesgue Spaces
406(8)
11.6 Problems
414(3)
11.7 Notes and Bibliographic References
417(2)
A Measure and Integration Theory Toolbox
419(8)
A.1 Measure Spaces
419(1)
A.2 Measurable Functions
420(1)
A.3 Integration and Convergence Theorems
421(1)
A.4 Absolutely Continuous Norms
422(1)
A.5 Product Spaces
422(1)
A.6 Atoms
423(1)
A.7 Convergence in Measure
423(1)
A.8 CT-Homomorphism
424(1)
A.9 References
425(2)
B A Glimpse on Functional Analysis
427(4)
C Eulerian Integrals
431(12)
C.1 Beta Function
431(5)
C.2 Gamma Function
436(3)
C.3 Some Applications
439(4)
D Fourier Transform
443(6)
E Greek Alphabet
449(2)
References 451(6)
Symbol Index 457(2)
Subject Index 459
René Erlín Castillo obtained his PhD from Ohio University. He has been professor in Universidad de Oriente, Venezuela, Visiting Professor in Ohio University and nowadays he is a professor in the Universidad Nacional de Colombia. His research, spread across  around 50 papers, is done mainly in functional analysis, real analysis, complex analysis, harmonic analysis, operator theory, potential theory and partial Differential Equations. He has authored a book and a textbook both in Spanish.

 





Humberto Rafeiro received his PhD from Algarve University, Portugal and did postdoctoral work in the research group Centro de Análise Funcional e Aplicações in the Instituto Superior Técnico, Universidade de Lisboa, Portugal. Nowadays he is a professor in the Pontifical Universidad Javeriana. His research interests range from harmonic analysis, function spaces, operator theory in non-standard function spaces, potential type operators, hypersingular integrals and fractional calculus, having published near 40 papers on these subjects. He has co-authored a textbook in Spanish.