Monotone Functions On General Measure Spaces
Abstract
Given a measure space and a totally ordered collection of measurable sets, called an ordered core, the notion of a core decreasing function is introduced and used to generalize monotone functions to general measure spaces. The least core decreasing majorant construction, the level function construction, and the greatest core decreasing minorant, already known for functions on the real line, are extended to this general setting. A functional description of these constructions is provided and is shown to be closely related to the pre-order relation of functions induced by integrals over the ordered core.
For an ordered core, the down space construction of a Banach function space is defined as a variant of the Kothe dual restricted to core decreasing functions. Concrete descriptions ¨ of the duals of the down spaces are provided. The down spaces of L 1 and L ∞ are shown to form an exact Calderon couple with divisibility constant 1; a complete description of the exact ´ interpolation spaces for the couple is given in terms of level functions; and the down spaces of universally rearrangement invariant spaces (u.r.i.) are shown to be precisely those interpolation spaces that have the Fatou property. The dual couple is also an exact Calderon couple with ´ divisibility constant 1; a complete description of the exact interpolation spaces for the couple is given in terms of least core decreasing majorants; and the duals of down spaces of u.r.i. spaces are shown to be precisely those interpolation spaces that have the Fatou property.
Integral operators whose kernel operators satisfy a monotonicity condition on their level sets are shown to induce an ordered core. Certain weighted norm inequalities are shown to remain valid when the weights are replaced with core decreasing functions. Boundedness of an abstract formulation of Hardy operators between Lebesgue spaces over general measure spaces is studied and, when the domain is L 1, shown to be equivalent to the existence of a Hardy inequality on the half line with general Borel measures.