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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 $$[0,1]$$ $\| Mf\|_ X=\min\{\| f-h\|_ X: h\in X\cap{\mathcal N}\}.$

##### MSC:
 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