Rearrangements of functions in Besov spaces. (English) Zbl 1022.46021

Let \(BV(0,l)\) be the space of functions of bounded variation in \((0,l)\). For \(\theta \in(0,1)\) and \(p\in[1,\infty]\) define \(Z_{\theta,p}(0,l)\) as \[ Z_{\theta,p}(0,l)= (L_\infty(0,l), BV(0,l))_{\theta,p}, \] the real interpolation space between \(L_\infty(0,l)\) and \( BV(0,l)\). The Banach space \(V_p(0,l)\) is defined by \[ V_p(0,l)=\{ u\in W_p^1(0,l): u'\in Z_{1/p,p}(0,l)\} \] and endowed with the norm \[ \|u\|_{V_p(0,l)}=\|u\|_{ W_p^1(0,l)}+\|u'\|_{ Z_{1/p,p}(0,l)}. \] Then \[ B_{p,q}^{\alpha}(0,l)=(L_p(0,l), V_p(0,l))_{\alpha p/(p+1),q} \] for \(p\in [1,\infty)\), \(q\in [1,\infty)\) and \(\alpha \in (0,1+1/p)\). Moreover, the norm on \(B_{p,q}^{\alpha}(0,l)\) is given by \[ \||u|\|_{B_{p,q}^{\alpha}(0,l)}= \|u\|_{(L_p(0,l), V_p(0,l))_{\alpha p/(p+1),q}}. \] The main result in the paper is the following:
Theorem 1.1. Let \(p\in [1,\infty)\), \(q\in [1,\infty)\) and \(\alpha \in (0,1+1/p)\). Let \(u\) be a nonnegative function from \(B^\alpha_{p,q}(0,l)\). Then \(u^*\in B^\alpha_{p,q}(0,l)\) and \[ \||u^*|\|_{B^\alpha_{p,q}(0,l)} \leq \||u|\|_{B^\alpha_{p,q}(0,l)}, \] where \(u^*\) is the decreasing rerrangement of \(u\).
As a consequence of this theorem, it is obtained that the decreasing rearrangement operator \(*\) is bounded on \(B_{p,q}^{\alpha}(0,l)\) for every \(p\in [1,\infty)\), \(q\in [1,\infty)\) and \(\alpha \in (0,1+1/p)\).


46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
26D10 Inequalities involving derivatives and differential and integral operators
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