Book contents
- Frontmatter
- Contents
- Introduction
- 1 Measure and integral
- 2 The Cauchy–Schwarz inequality
- 3 The AM–GM inequality
- 4 Convexity, and Jensen's inequality
- 5 The Lp spaces
- 6 Banach function spaces
- 7 Rearrangements
- 8 Maximal inequalities
- 9 Complex interpolation
- 10 Real interpolation
- 11 The Hilbert transform, and Hilbert's inequalities
- 12 Khintchine's inequality
- 13 Hypercontractive and logarithmic Sobolev inequalities
- 14 Hadamard's inequality
- 15 Hilbert space operator inequalities
- 16 Summing operators
- 17 Approximation numbers and eigenvalues
- 18 Grothendieck's inequality, type and cotype
- References
- Index of inequalities
- Index
7 - Rearrangements
Published online by Cambridge University Press: 06 July 2010
- Frontmatter
- Contents
- Introduction
- 1 Measure and integral
- 2 The Cauchy–Schwarz inequality
- 3 The AM–GM inequality
- 4 Convexity, and Jensen's inequality
- 5 The Lp spaces
- 6 Banach function spaces
- 7 Rearrangements
- 8 Maximal inequalities
- 9 Complex interpolation
- 10 Real interpolation
- 11 The Hilbert transform, and Hilbert's inequalities
- 12 Khintchine's inequality
- 13 Hypercontractive and logarithmic Sobolev inequalities
- 14 Hadamard's inequality
- 15 Hilbert space operator inequalities
- 16 Summing operators
- 17 Approximation numbers and eigenvalues
- 18 Grothendieck's inequality, type and cotype
- References
- Index of inequalities
- Index
Summary
Decreasing rearrangements
Suppose that (E, ∥.∥E ) is a Banach function space and that f ∈ E. Then ∥f∥E = ∥|f|∥E , so that the norm of f depends only on the absolute values of f. For many important function spaces we can say more. Suppose for example that f ∈ Lp , where 1 < p < ∞. By Proposition 1.3.4, ∥f∥p = (p ∫ t p−1µ(|f| > t)dt)1/p , and so ∥f∥p depends only on the distribution of |f|. The same is true for functions in Orlicz spaces. In this chapter, we shall consider properties of functions and spaces of functions with this property.
In order to avoid some technical difficulties which have little real interest, we shall restrict our attention to two cases:
(i) (Ω, Σ, µ) is an atom-free measure space;
(ii) Ω = N or {1, …, n}, with counting measure.
In the second case, we are concerned with sequences, and the arguments are usually, but not always, easier. We shall begin by considering case (i) in detail, and shall then describe what happens in case (ii), giving details only when different arguments are needed.
Suppose that we are in the first case, so that (Ω, Σ, µ) is atom-free. We shall then make use of various properties of the measure space, which follow from the fact that if A ∈ Σ and 0 < t < µ(A) then there exists a subset B of A with µ(B) = t (Exercise 7.1).
- Type
- Chapter
- Information
- Inequalities: A Journey into Linear Analysis , pp. 78 - 102Publisher: Cambridge University PressPrint publication year: 2007