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Inequalities in rearrangement invariant function spaces. (English) Zbl 0872.46020
Krbec, Miroslav (ed.) et al., Nonlinear analysis, function spaces and applications. Vol. 5. Proceedings of the spring school held in Prague, May 23-28, 1994. Prague: Prometheus Publishing House. 177-230 (1994).
Let $$u$$ be a real measurable function defined on a subset $$G$$ of $$\mathbb{R}^n$$. The function $$\mu(t)= m(\{x\in G:|u(x)|>t\})$$ is called the distribution function of $$u$$. The distribution function $$u^*$$ of $$\mu$$ is said to be the decreasing rearrangement of $$u$$. The function $$u^+$$, $$u^+(x)= u^*(\pi^{n/2} [\Gamma(n/2+1)]^{-1}|x|^n)$$ is called the symmetric rearrangement of $$u$$. Their properties and their applications to the proofs of sharp forms of the Sobolev inequalities are presented. Some inequalities à la Sobolev are investigated in a case of the Lorentz spaces. Finally, a variational problem for a norm in the Lorentz space $$L(p,1)$$ is considered. This paper is a survey of the results about rearrangements à la Hardy and Littlewood. The bibliography contains 57 items.
For the entire collection see [Zbl 0811.00017].

##### MSC:
 4.6e+31 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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