On separating the submajorization order into majorization and pointwise inequality

Abstract

As is known, the Hardy–Littlewood–Polya submajorization preorder among integrable real-valued functions separates into the concatenation of pointwise inequality and majorization, in this order, i.e., if \\( x\\prec \\prec y\\), then there is a z with \\(x\\le z\\prec y\\). Submajorization also separates, in the other order, into majorization and inequality, i.e., if \\(x\\prec \\prec y\\), then there is a w with \\(x\\prec w\\le y\\) and, as is shown here, such a w can be chosen to be nonnegative if both x and y are. It is also shown that the former separation result (existence of z) can be deduced from the latter one (existence of w) by using a doubly stochastic operator on the Banach space \\(L^{\\varrho }\\left( T\\right) \\), where T is a finite measure space and \\(\\varrho \\in \\left[ 1,+\\infty \\right] \\). The results are applied to a \\(\\prec \\prec \\)-isotone real-valued function C on the nonnegative cone \\( L_{+}^{\\varrho }\\left( T\\right) \\) and to its positive-part extension to all of \\(L^{\\varrho }\\left( T\\right) \\), defined by \\(C^{\\dagger }\\left( y\\right) =C\\left( y^{+}\\right) \\), whose economic interpretation, when \\( C\\left( y\\right) \\) is the joint cost of producing quantities \\(\\left( y\\left( t\\right) \\right) _{t\\in T}\\) of a spectrum of commodities, is that of adding free disposal to the technology.