Contents -- Preface -- 0. Introduction and Notation -- 0.1 Notational Conventions -- 0.1.1 Indices and Exponents -- 0.1.2 Constants -- 0.1.3 Measure Spaces and Related Spaces -- 0.1.4 Interpolation Spaces -- 0.1.5 Linear Mappings Between Normed Spaces -- 0.1.6 Other -- 1. Carlson's Inequalities -- 1.1 Carlson's Proof -- 1.2 Hardy's Proofs -- 1.3 An Alternate Proof -- 1.4 Carlson's Inequality for Finite Sums -- 2. Some Extensions and Complements of Carlson's Inequalities -- 2.1 Gabriel -- 2.2 Levin -- 2.3 Caton -- 2.4 Bellman -- 2.5 Two Discrete Carlson By-products -- 2.6 Landau and Levin-Steckin -- 2.7 Some Extensions of the Landau and Levin-Steckin Inequalities -- 2.7.1 The Case p = 1 -- 2.7.2 General p -- 2.8 Proofs -- 2.9 Levin-Godunova -- 2.10 More About Finite Sums -- 3. The Continuous Case -- 3.1 Beurling -- 3.2 Kjellberg -- 3.3 Bellman -- 3.4 Sz. Nagy -- 3.5 Klefsjo -- 3.6 Hu -- 3.7 Yang-Fang -- 3.8 A Continuous Landau Type Inequality -- 3.9 Integrals on Bounded Intervals -- 4. Levin's Theorem -- 5. Some Multi-dimensional Generalizations and Variations -- 5.1 Some Preliminaries -- 5.2 A Sharp Inequality for Cones in Rn -- 5.3 Some Variations on the Multi-dimensional Theme -- 5.3.1 Kjellberg Revisited -- 5.3.2 Andrianov -- 5.3.3 Pigolkin -- 5.3.4 Bertolo-Fernandez -- 5.3.5 Barza et al -- 5.3.6 Kamaly -- 5.4 Some Further Generalizations -- 5.4.1 A Multi-dimensional Extension of Theorem 3.6 -- 5.4.2 An Extension of Theorem 5.8 -- 6. Some Carlson Type Inequalities for Weighted Lebesgue Spaces with General Measures -- 6.1 The Basic Case -- 6.2 The Product Measure Case - Two Factors -- 6.3 The General Product Measure Case -- 7. Carlson Type Inequalities and Real Interpolation Theory -- 7.1 Interpolation of Normed Spaces -- 7.2 The Real Interpolation Method -- 7.2.1 The K-method -- 7.2.2 The J-method.

7.2.3 The Equivalence Theorem -- 7.2.4 The Classes CJ and CK -- 7.2.5 Reiteration -- 7.2.6 Interpolation of Weighted Lebesgue Spaces -- 7.3 Embeddings of Real Interpolation Spaces -- 8. Further Connection to Interpolation Theory the Peetre s Method -- 8.1 Introduction -- 8.2 Carlson Type Inequalities as Sharpenings of Jensen's Inequality -- 8.3 The Peetre Interpolation Method and Interpolation of Orlicz Spaces -- 8.4 A Carlson Type Inequality with Blocks -- 8.5 The Calderon-Lozanovskii Construction on Banach Lattices -- 9. Related Results and Applications -- 9.1 A Generalization of Redheffer -- 9.2 Sobolev Type Embeddings -- 9.3 A Local Hausdorff-Young Inequality -- 9.4 Optimal Sampling -- 9.5 More on Interpolation the Peetre Parameter Theorem -- 9.6 Carlson Type Inequalities with Several Factors -- 9.7 Reverse Carlson Type Inequalities -- 9.8 Some Further Possibilities -- 9.8.1 Other Function Spaces -- 9.8.2 Matrix Weights -- 9.9 Necessity in the Case of a General Measure -- Appendix A A Historical Note on Fritz David Carlson (1888-1952) -- Appendix B A Translation of the Original Article by Carlson from French to English -- Bibliography -- Index.