Multiplicative Inequalities of Carlson Type and Interpolation

The following sections are included:

• Preface

• Contents

The following sections are included:

• Notational Conventions
  • Indices and Exponents
  • Constants
  • Measure Spaces and Related Spaces
    • Lebesgue Spaces
    • Weighted Lebesgue Spaces
  • Interpolation Spaces
  • Linear Mappings Between Normed Spaces
  • Other

The following sections are included:

• Carlson's Proof

• Hardy's Proofs

• An Alternate Proof

• Carlson's Inequality for Finite Sums

The following sections are included:

• Gabriel

• Levin

• Caton

• Bellman

• Two Discrete Carlson By-products

• Landau and Levin–Stečkin

• Some Extensions of the Landau and Levin–Stečkin Inequalities
  • The Case p = 1
  • General p

• Proofs

• Levin–Godunova

• More About Finite Sums

The following sections are included:

• Beurling

• Kjellberg

• Bellman

• Sz. Nagy

• Klefsjö

• Hu

• Yang–Fang

• A Continuous Landau Type Inequality

• Integrals on Bounded Intervals

In this chapter, we will present and discuss a remarkable result of V. I. Levin [56]. In 1948, he found the following generalization of (3.1), which is, in a way, the most general inequality of its type one can hope for (see Remark 4.5 below)…

The following sections are included:

• Some Preliminaries

• A Sharp Inequality for Cones in ℝⁿ

• Some Variations on the Multi-dimensional Theme
  • Kjellberg Revisited
  • Andrianov
  • Pigolkin
  • Bertolo—Fernandez
  • Barza et al
  • Kamaly

• Some Further Generalizations
  • A Multi-dimensional Extension of Theorem 3.6
  • An Extension of Theorem 5.8

The following sections are included:

• The Basic Case

• The Product Measure Case – Two Factors

• The General Product Measure Case

The following sections are included:

• Interpolation of Normed Spaces

• The Real Interpolation Method
  • The K-method
  • The J-method
  • The Equivalence Theorem
  • The Classes and
  • Reiteration
  • Interpolation of Weighted Lebesgue Spaces

• Embeddings of Real Interpolation Spaces

The following sections are included:

• Introduction

• Carlson Type Inequalities as Sharpenings of Jensen's Inequality

• The Peetre Interpolation Method and Interpolation of Orlicz Spaces

• A Carlson Type Inequality with Blocks

• The Calderón–Lozanovskiĭ Construction on Banach Lattices

The following sections are included:

• A Generalization of Redheffer

• Sobolev Type Embeddings

• A Local Hausdorff–Young Inequality

• Optimal Sampling

• More on Interpolation, the Peetre Parameter Theorem

• Carlson Type Inequalities with Several Factors

• Reverse Carlson Type Inequalities

• Some Further Possibilities
  • Other Function Spaces
  • Matrix Weights

• Necessity in the Case of a General Measure

The following sections are included:

• Appendices
  • A Historical Note on Fritz David Carlson (1888–1952)
  • A Translation of the Original Article by Carlson from French to English

• Bibliography

• Index