Multiplicative Inequalities of Carlson Type and Interpolation Book

Preface vii

0 Introduction and Notation 1

0.1 Notational Conventions 3

0.1.1 Indices and Exponents 3

0.1.2 Constants 3

0.1.3 Measure Spaces and Related Spaces 4

0.1.3.1 Lebesgue Spaces 4

0.1.3.2 Weighted Lebesgue Spaces 5

0.1.4 Interpolation Spaces 5

0.1.5 Linear Mappings Between Normed Spaces 5

0.1.6 Other 6

1 Carlson's Inequalities 9

1.1 Carlson's Proof 10

1.2 Hardy's Proofs 14

1.3 An Alternate Proof 17

1.4 Carlson's Inequality for Finite Sums 17

2 Some Extensions and Complements of Carlson's Inequalities 21

2.1 Gabriel 21

2.2 Levin 22

2.3 Caton 24

2.4 Bellman 25

2.5 Two Discrete Carlson By-products 25

2.6 Landau and Levin-Ste&ccheck;kin 26

2.7 Some Extensions of the Landau and Levin-Ste&ccheck;kin Inequalities 28

2.7.1 The Case p = 1 29

2.7.2 General p 30

2.8 Proofs 31

2.9 Levin-Godunova 36

2.10 More About Finite Sums 41

3 The Continous Case 47

3.1 Beurling 55

3.2 Kjellberg 57

3.3 Bellman 62

3.4 Sz. Nagy 65

3.5 Klefsj&oddot; 67

3.6 Hu 68

3.7 Yang-Fang 69

3.8 A Continuous Landau Type Inequality 70

3.9 Integrals on Bounded Intervals 72

4 Levin's Theorem 77

5 Some Multi-dimensional Generalizations and Variations 85

5.1 Some Preliminaries 85

5.2 A Sharp Inequality for Cones in Rn 89

5.3 Some Variations on the Multi-dimensional Theme 95

5.3.1 Kjellberg Revisted 95

5.3.2 Andrianov 96

5.3.3 Pigolkin 98

5.3.4 Bertolo-Fernandez 99

5.3.5 Barza et al 100

5.3.6 Kamaly 101

5.4 Some Further Generalizations 102

5.4.1 A Multi-dimensional Extension of Theorem 3.6 103

5.4.2 An Extension of Theorem 5.8 107

6 Some Carlson Type Inequalities for Weighted Lebesgue Spaces with General Measures 111

6.1The Basic Case 111

6.2 The Product Measure Case - Two Factors 120

6.3 The General Product Measure Case 127

7 Carlson Type Inequalities and Real Interpolation Theory 129

7.1 Interpolation of Normed Spaces 129

7.2 The Real Interpolation Method 130

7.2.1 The K-method 131

7.2.2 The J-method 131

7.2.3 The Equivalence Theorem 132

7.2.4 The Classes mathcalCJ and mathcalCK 132

7.2.5 Reiteration 133

7.2.6 Interpolation of Weighted Lebesgue Spaces 134

7.3 Embeddings of Real Interpolation Spaces 134

8 Further Connection to Interpolation Theory, the Peetre <.>ϕ Method 139

8.1 Introduction 139

8.2 Carlson Type Inequalities as Sharpenings of Jensen's Inequality 142

8.3 The Peetre Interpolation Method and Interpolation of Orlicz Spaces 147

8.4 A Carlson Type Inequality with Blocks 150

8.5 The Calderón-Lozanovski&icheck; Construction on Banach Lattices 158

9 Related Results and Applications 169

9.1 A Generalization of Redheffer 169

9.2 Sobolev Type Embeddings 171

9.3 A Local Hausdorff-Young Inequality 172

9.4 Optimal Sampling 173

9.5 More on Interpolation, the Peetre Parameter Theorem 174

9.6 Carlson Type Inequalities with Several Factors 177

9.7 Reverse Carlson Type Inequalities 178

9.8 Some Further Possibilities 180

9.8.1 Other Function Spaces 180

9.8.2 Matrix Weights 181

9.9 Necessity in the Case of a General Measure 181

Appendix A A Historical Note on Fritz David Carlson (1888-1952) 183

Appendix B A Translation of the Original Article by Carlson from French to English 187

Bibliography 193

Index 199