Book contents
- Frontmatter
- Contents
- Notation
- Foreword
- Preface
- Introduction
- 1 Rearrangements
- 2 Main Inequalities on Rn3
- 3 Dirichlet Integral Inequalities
- 4 Geometric Isoperimetric and Sharp Sobolev Inequalities
- 5 Isoperimetric Inequalities for Physical Quantities
- 6 Steiner Symmetrization
- 7 Symmetrization on Spheres, and Hyperbolic and Gauss Spaces
- 8 Convolution and Beyond
- 9 The ⋆-Function
- 10 Comparison Principles for Semilinear Poisson PDEs
- 11 The ⋆-Function in Complex Analysis
- References
- Index
2 - Main Inequalities on Rn3
Published online by Cambridge University Press: 22 February 2019
Book contents
- Frontmatter
- Contents
- Notation
- Foreword
- Preface
- Introduction
- 1 Rearrangements
- 2 Main Inequalities on Rn3
- 3 Dirichlet Integral Inequalities
- 4 Geometric Isoperimetric and Sharp Sobolev Inequalities
- 5 Isoperimetric Inequalities for Physical Quantities
- 6 Steiner Symmetrization
- 7 Symmetrization on Spheres, and Hyperbolic and Gauss Spaces
- 8 Convolution and Beyond
- 9 The ⋆-Function
- 10 Comparison Principles for Semilinear Poisson PDEs
- 11 The ⋆-Function in Complex Analysis
- References
- Index
Summary
Chapter2 covers the foundational inequalities for integrals of functions in Euclidean space. The two key results in this chapter are that symmetric decreasing rearrangement of a continuous function decreases the modulus of continuity, and that certain integral expressions increase when functions are replaced by their symmetric decreasing rearrangement. Other results include the Hardy-Littlewood inequalityand the contractivity of rearrangements in the L-infinity norm.
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- Symmetrization in Analysis , pp. 54 - 91Publisher: Cambridge University PressPrint publication year: 2019