Thesis Format
Monograph
Degree
Doctor of Philosophy
Program
Mathematics
Supervisor
Sinnamon, Gord
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.
Summary for Lay Audience
A fundamental feature of real numbers is that they form a total order, for any pair of distinct real numbers, one is bigger than the other. Consequentially, it is natural to define monotone functions as an assignation of numbers that preserve (or reverse) this order. In this thesis, we extend monotone functions to collections of elements that admit a notion of volume but do not have a predetermined order among the elements. Instead, we rely on a collection of subsets that take the role of the intervals {[0, x]}x>0 in the real line, called an ordered core. We use ordered cores to define monotone functions in this more abstract setting and extend some tools related to decreasing functions, previously only available on the real line, to this more abstract setting. We work with function spaces. For a fixed function, assign it a size by measuring its interaction with all the decreasing functions. The space we produce through this process is the down space. We describe them completely and study some of their properties. We focus on duality and interpolation. The dual space is a collection of functions over our original space that satisfy certain properties. In the case of finite dimensional spaces (collections of column vectors of n-entries), we may identify the dual space with the collection of row vectors of n-entries. For our Down spaces, we also give a concrete description of their duals. For interpolation, we consider a concrete pair of function spaces corresponding to the down spaces of L1 (L1 ↓) and L∞ (L∞ ↓). We consider intermediate collections of functions that can be written as the sum of a function that is not ’too wide’ and a function that is not ’too tall’. If we know the behavior of an operation on L1 and L∞, we also understand the operation on any intermediate collection. In this thesis, we give a complete characterization of the function spaces that are intermediate between L1↓ and L∞↓. We finish with an application of our theory of monotone functions in the study of Hardy inequalities.
Recommended Citation
Santacruz Hidalgo, Alejandro, "Monotone Functions On General Measure Spaces" (2024). Electronic Thesis and Dissertation Repository. 10297.
https://ir.lib.uwo.ca/etd/10297
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