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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)

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