Abstract: Monotone functions over the real numbers are very well-behaved compared to general measurable functions. Consequently, a wide variety of techniques and applications are in place for working with them. In this talk, we explore the notion of an ordered core, which allows us to define core decreasing functions and generalize monotone functions to general measure spaces without reliance on a strict ordering among elements.

We will begin by introducing a definition of monotone functions compatible with the ordered core. This allows us to extend the down space construction, a variant of the Köthe dual restricted to positive decreasing functions, to all measure spaces. We will look at their associate spaces and their relationship with a suitable version of the least decreasing majorant construction in this more general setting. We will discuss the interpolation structure of these spaces and find strong similarities to the real line case; the down spaces corresponding to L1 and L∞ form an exact Calderón couple and as a consequence, we can describe all their exact interpolation spaces in terms of the K−functional.

We illustrate the versatility and adaptability of this generalized perspective on decreasing functions by proving a new characterization for the boundedness of an abstract Hardy operator between L^1 to L^q with general measures.