# Distribution Function and Nonincreasing Rearrangement

• René Erlín Castillo
• Humberto Rafeiro
Chapter
Part of the CMS Books in Mathematics book series (CMSBM)

## Abstract

In this chapter we study the distribution function which is a tool that provides information about the size of a function but not about its pointwise behavior or locality; for example, a function f and its translation are the same in terms of their distributions. Based on the distribution function we study the nonincreasing rearrangement and establish its basic properties. We obtain sub-additive and sub-multiplicative type inequalities for the decreasing rearrangement. The maximal function associated with the decreasing rearrangement is introduced and some important relations are obtained, e.g., Hardy’s inequality. In the last section of this chapter we deal with the rearrangement of the Fourier transform.

## Keywords

Distribution Function Measure Space Maximal Function Lorentz Space Part Type
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© Springer International Publishing Switzerland 2016

## Authors and Affiliations

• René Erlín Castillo
• 1
• Humberto Rafeiro
• 2