Rakotoson, Jean Michel Some properties of the relative rearrangement. (English) Zbl 0686.28003 J. Math. Anal. Appl. 135, No. 2, 488-500 (1988). Let \(\Omega \subset R^ N\) be a bounded Lebesgue measurable set of measure \(| \Omega |\). The decreasing rearrangement of a real measurable function u on \(\Omega\) is defined by \[ u_*(s)=\inf \{\theta \in R,| \{x,u(x)>\theta \}| \leq s\},\quad s\in {\bar \Omega}^*=[0,| \Omega |]. \] Given \(t\in R\) we put \(P(t)=\{x\in \Omega,\quad u(x)=t\}.\) If \(| P(t)| >0,\) then P(t) is called the flat region of value t of the function u. J. Mossino and R. Temam [Duke Math. J. 48, 475-495 (1981; Zbl 0476.35031)] introduced the notion of relative rearrangement: For \(v\in L^ p(\Omega),\quad 1\leq p\leq \infty,\) define a function w on \({\bar \Omega}{}^*\) by \[ w(s)=\int_{u>u_*(s)}v(x)dx+\int^{s-| \{u>u_*(s)\}|}_{0}(v|_{P(u_*(s))})_*(\sigma)d\sigma, \] where \(v|_ E\) is the restriction of v to the set E. The function \(v_{*u}=dw/ds\) satisfies \(\| v_{*u}\|_{L^ p(\Omega^*)}\leq \| v\|_{L^ p(\Omega)}\) and is called the relative rearrangement of v with respect to u. The author proves various properties of the relative rearrangement, among others the following generalization of the well-known Hardy-Littlewood inequality: If u is a real measurable function on \(\Omega\), \(v_ 1\in L^ p(\Omega),\quad v_ 2\in L^ q(\Omega^*),\quad 1/p+1/q=1,\) then \[ \int_{\Omega^*}v_{1*u}v_ 2 d\sigma \leq \int_{\Omega^*}v_{1*}v_{2*} d\sigma. \] Reviewer: J.Rákosník Cited in 1 ReviewCited in 13 Documents MSC: 28A20 Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence 26D15 Inequalities for sums, series and integrals 46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) Keywords:decreasing rearrangement; relative rearrangement; Hardy-Littlewood inequality Citations:Zbl 0476.35031 PDFBibTeX XMLCite \textit{J. M. Rakotoson}, J. Math. Anal. Appl. 135, No. 2, 488--500 (1988; Zbl 0686.28003) Full Text: DOI References: [1] Brocker, Th, Differential Germs anc Catastrophes, (London Math. Soc. Lecture Note Sér., Vol. 17 (1975), Cambridge Univ. Press: Cambridge Univ. Press New York/London) · Zbl 0302.58006 [2] Federer, H., Geometric Measure Theory (1969), Springer-Verlag: Springer-Verlag Berlin/Heidelberg/New York · Zbl 0176.00801 [3] Chiti, G., Rearrangements of functions and convergence in Orlicz spaces, Appl. Anal., 9 (1979) · Zbl 0424.46023 [4] Kranosel’skii, M. A., Topological Methods in the Theory of Nonlinear Integral Equations (1964), Pergamon: Pergamon Oxford/London/New York/Paris [5] Mossino, J., Inégalités Isopérimétriques et applications en physique, (Collection Travaux en cours (1984), Herman: Herman Paris) · Zbl 0537.35002 [6] J. Mossino and J. RakotosonAnn. Scuola Norm Sup. Pisa; J. Mossino and J. RakotosonAnn. Scuola Norm Sup. Pisa · Zbl 0652.35053 [7] Mossino, J.; Temam, R., Directional derivative of the increasing rearrangement mapping and application to a queer differential equation in plasma physics, Duke Math. J., 41, 475-495 (1981) · Zbl 0476.35031 [8] Rakotoson, J. M., Sur une synthèse des modèles locaux et non locaux en physique des plasmas, Thèse 3ème cycle (1984), Orsay [9] J. M. RakotosonAppl. Anal.; J. M. RakotosonAppl. Anal. · Zbl 0554.35068 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.