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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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[1] Almgren, J. Amer. Math. Soc. 2 pp 683– (1989)
[2] : A Unified Approach to Symmetrization. In: Partial Diff. Eq. of Elliptic Type, A. Alvino, E. Fabes and G. Talenti eds., Symposia Math. 35, Cambridge Univ. Press, 1994
[3] and : Interpolation of Operators, Academic Press, Boston, 1988
[4] Bourdaud, J. Funct. Anal. 91 pp 351– (1991) · Zbl 0737.46011
[5] Chiti, Appl. Anal. 9 pp 23– (1979) · Zbl 0424.46023
[6] Cianchi, Duke Math. J. 105 pp 355– (2000) · Zbl 1017.46023
[7] Garsia, Ann. Inst. Fourier 24 pp 67– (1974) · Zbl 0274.26006
[8] : Rearrangements and Convexity of Level Sets in PDE, Lecture Notes in Math. 1150, Springer-Verlag, Berlin-New York, 1985 · Zbl 0593.35002
[9] Oswald, Comment. Math. Univ. Carolinae 33 pp 57– (1992)
[10] Savaré, Nonlinear Anal. 27 pp 1055– (1996) · Zbl 0871.47040
[11] Tartar, J. Funct. Anal. 9 pp 468– (1972) · Zbl 0241.46035
[12] : Theory of Function Spaces II, Birkhäuser, Basel, 1992
[13] Triebel, Forum Math. 12 pp 731– (2000) · Zbl 0957.46029
[14] : Weakly Differentiable Functions, Springer-Verlag, New York, 1989
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.