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An inequality between $$u^*$$ and $$|\text{grad } u|^*$$. (English) Zbl 0783.26015
General inequalities 6, Proc. 6th Int. Conf., Oberwolfach/Ger. 1990, ISNM 103, 175-182 (1992).
[For the entire collection see Zbl 0746.00079.]
Suppose that $$u: \mathbb{R}^ n\to \mathbb{R}$$ is locally integrable and the distributional gradient of $$u$$ is integrable over $$\mathbb{R}^ n$$. Then $u^*(s)\leq n^{-1} \kappa^{-1/n}_ n \int^ \infty_ 0 |\text{grad }u|^* (t)(t+ s)^{-1+1/n} dt$ for every $$s\geq 0$$. Here $$u^*$$ and $$|\text{grad }u|^*$$ is the decreasing rearrangement of $$u$$ and $$|\text{grad }u|$$, respectively; $$\kappa_ n$$ stands for the volume of the unit $$n$$-dimensional ball. This is one of the main results of the paper. The author makes use of it to show that $$u$$ belongs to the Lorentz space $$L\Bigl({np\over n- p},q\Bigr)$$ provided that the gradient of $$u$$ belongs to the Lorentz space $$L(p,q)$$, $$1< p< n$$ and $$1\leq q\leq \infty$$.
Reviewer: B.Opic (Praha)

##### MSC:
 26D10 Inequalities involving derivatives and differential and integral operators 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems