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.
A mathematician is a person who can find analogies between theorems; a better mathematician is one who can see analogies between proofs and the best mathematician can notice analogies between theories. Stefan Banach
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Castillo, R., Rafeiro, H. (2016). Distribution Function and Nonincreasing Rearrangement. In: An Introductory Course in Lebesgue Spaces. CMS Books in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-30034-4_4
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DOI: https://doi.org/10.1007/978-3-319-30034-4_4
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