Rearrangements of functions in Besov spaces (Q2762260)

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 NEWLINE\[NEWLINEZ_{\theta,p}(0,l)= (L_\infty(0,l), BV(0,l))_{\theta,p},NEWLINE\]NEWLINE the real interpolation space between \(L_\infty(0,l)\) and \( BV(0,l)\). The Banach space \(V_p(0,l)\) is defined by NEWLINE\[NEWLINEV_p(0,l)=\{ u\in W_p^1(0,l): u'\in Z_{1/p,p}(0,l)\}NEWLINE\]NEWLINE and endowed with the norm NEWLINE\[NEWLINE\|u\|_{V_p(0,l)}=\|u\|_{ W_p^1(0,l)}+\|u'\|_{ Z_{1/p,p}(0,l)}.NEWLINE\]NEWLINE Then NEWLINE\[NEWLINEB_{p,q}^{\alpha}(0,l)=(L_p(0,l), V_p(0,l))_{\alpha p/(p+1),q}NEWLINE\]NEWLINE 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 NEWLINE\[NEWLINE\||u|\|_{B_{p,q}^{\alpha}(0,l)}= \|u\|_{(L_p(0,l), V_p(0,l))_{\alpha p/(p+1),q}}.NEWLINE\]NEWLINE The main result in the paper is the following:NEWLINENEWLINENEWLINETheorem 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 NEWLINE\[NEWLINE \||u^*|\|_{B^\alpha_{p,q}(0,l)} \leq \||u|\|_{B^\alpha_{p,q}(0,l)},NEWLINE\]NEWLINE where \(u^*\) is the decreasing rerrangement of \(u\).NEWLINENEWLINENEWLINEAs 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 \(\alpha \in (0,1+1/p)\).