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 $$\alpha \in (0,1+1/p)$$.

MSC:

 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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References:

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