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Some extremal problems related to majorization. (English) Zbl 0736.46008
Topics in matrix and operator theory, Proc. Workshop, Rotterdam/Neth. 1989, Oper. Theory, Adv. Appl. 50, 83-92 (1991).
[For the entire collection see Zbl 0722.00022.]
Some notations: \(f^*\) denotes the non-increasing rearrangement of a Lebesgue measurable function \(f\). The set of all functions of the form \(f^*\) constitutes the cone denoted by \(\mathcal M\). For \(f,g\in L^ 1(0,1)\) the notation \(f\prec g\) means that \(\int^ 1_ 0 f^*(t) dt=\int^ 1_ 0 g^*(t) dt\), for all \(x\in [0,1]\). The cone \(\mathcal N\) is defined as the class of all \(h\in L^ 1(0,1)\) such that \(h\prec 0\). For a function \(f\in L^ 1(0,1)\) the least concave majorant of the function \(F(x)=\int^ x_ 0 f(t) dt\) is denoted by \(\hat{F}(x)\). The derivative of \(\hat F\) is denoted by \(Mf\). Finally all rearrangement invariant spaces (RIS) in this work are assumed to have Fatou property.
A typical result of the present paper is the following
Theorem 1. Let \(f\in L^ 1(0,1)\). Then \(f=Mf+(f-Mf)\) with \(Mf\in {\mathcal M}\), \(f-Mf\in {\mathcal N}\) and \(Mf\prec {f-h}\) for each \(h\in {\mathcal N}\). Therefore in any RIS \((X,\|\cdot \|_ X)\) on \([0,1]\) \[ \| Mf\|_ X=\min\{\| f-h\|_ X: h\in X\cap{\mathcal N}\}. \]

46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
26D15 Inequalities for sums, series and integrals