Multiplicative Inequalities of Carlson and Interpolation

Leo Larsson, Lech Maligranda, Josip Pecaric, Lars-Erik Persson9789812567086, 981-256-708-9

Collecting all the results on the particular types of inequalities, the coverage of this book is unique among textbooks in the literature. The book focuses on the historical development of the Carlson inequalities and their many generalizations and variations. As well as almost all known results concerning these inequalities and all known proof techniques, a number of open questions suitable for further research are considered. Two chapters are devoted to clarifying the close connection between interpolation theory and this type of inequality. Other applications are also included, in addition to a historical note on Fritz Carlson himself.

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Table of contents :
Contents……Page 12
Preface……Page 8
0. Introduction and Notation……Page 16
0.1.2 Constants……Page 18
0.1.3 Measure Spaces and Related Spaces……Page 19
0.1.5 Linear Mappings Between Normed Spaces……Page 20
0.1.6 Other……Page 21
1. Carlson’s Inequalities……Page 24
1.1 Carlson’s Proof……Page 25
1.2 Hardy’s Proofs……Page 29
1.4 Carlson’s Inequality for Finite Sums……Page 32
2.1 Gabriel……Page 36
2.2 Levin……Page 37
2.3 Caton……Page 39
2.5 Two Discrete Carlson By-products……Page 40
2.6 Landau and Levin-Steckin……Page 41
2.7 Some Extensions of the Landau and Levin-Steckin Inequalities……Page 43
2.7.1 The Case p = 1……Page 44
2.7.2 General p……Page 45
2.8 Proofs……Page 46
2.9 Levin-Godunova……Page 51
2.10 More About Finite Sums……Page 56
3. The Continuous Case……Page 62
3.1 Beurling……Page 70
3.2 Kjellberg……Page 72
3.3 Bellman……Page 77
3.4 Sz. Nagy……Page 80
3.5 Klefsjo……Page 82
3.6 Hu……Page 83
3.7 Yang-Fang……Page 84
3.8 A Continuous Landau Type Inequality……Page 85
3.9 Integrals on Bounded Intervals……Page 87
4. Levin’s Theorem……Page 92
5.1 Some Preliminaries……Page 100
5.2 A Sharp Inequality for Cones in Rn……Page 104
5.3.1 Kjellberg Revisited……Page 110
5.3.2 Andrianov……Page 111
5.3.3 Pigolkin……Page 113
5.3.4 Bertolo-Fernandez……Page 114
5.3.5 Barza et al……Page 115
5.3.6 Kamaly……Page 116
5.4 Some Further Generalizations……Page 117
5.4.1 A Multi-dimensional Extension of Theorem 3.6……Page 118
5.4.2 An Extension of Theorem 5.8……Page 122
6.1 The Basic Case……Page 126
6.2 The Product Measure Case – Two Factors……Page 135
6.3 The General Product Measure Case……Page 142
7.1 Interpolation of Normed Spaces……Page 144
7.2 The Real Interpolation Method……Page 145
7.2.2 The J-method……Page 146
7.2.4 The Classes CJ and CK……Page 147
7.2.5 Reiteration……Page 148
7.3 Embeddings of Real Interpolation Spaces……Page 149
8.1 Introduction……Page 154
8.2 Carlson Type Inequalities as Sharpenings of Jensen’s Inequality……Page 157
8.3 The Peetre Interpolation Method and Interpolation of Orlicz Spaces……Page 162
8.4 A Carlson Type Inequality with Blocks……Page 165
8.5 The Calderon-Lozanovskii Construction on Banach Lattices……Page 173
9.1 A Generalization of Redheffer……Page 184
9.2 Sobolev Type Embeddings……Page 186
9.3 A Local Hausdorff-Young Inequality……Page 187
9.4 Optimal Sampling……Page 188
9.5 More on Interpolation the Peetre Parameter Theorem……Page 189
9.6 Carlson Type Inequalities with Several Factors……Page 192
9.7 Reverse Carlson Type Inequalities……Page 193
9.8.1 Other Function Spaces……Page 195
9.9 Necessity in the Case of a General Measure……Page 196
Appendix A A Historical Note on Fritz David Carlson (1888-1952)……Page 198
Appendix B A Translation of the Original Article by Carlson from French to English……Page 202
Bibliography……Page 208
Index……Page 214