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Second-order derivatives and rearrangements. (English) Zbl 1017.46023

Monotonic and symmetric rearrangements (here, as in most of the paper under review, only the former will be addressed) are important in analysis. (For instance, the Lebesgue integral of a function is the Riemann integral of its monotonic rearrangement.) The decreasing rearrangement \(u^\ast\) of a nonnegative function \(u\) from a Sobolev space \(W^{1,p}(0,m)\) is also in \(W^{1,p}(0,m)\) and \(\|(u^*)'\|_{L^p(0,m)}\leq\|u'\|_{L^p(0,m)}.\) Things are not so simple with rearrangements of functionals containing second derivatives. The author offers, among others, the following result: Let \(u\) be a nonnegative function from BV\(^2(0,m)\). Then so is \(u^*\) and \(\|D^2 (u^*)\|_{L^p(0,m)}\leq \|D^2 u\|_{L^p(0,m)}\).

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
26D10 Inequalities involving derivatives and differential and integral operators
46A45 Sequence spaces (including Köthe sequence spaces)
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