Unique axiomatisation of fractional integral operators on the real line

Abstract

The Cartwright--McMullen theorem (1978) establishes the uniqueness of the Riemann--Liouville family of fractional integral operators on a compact interval $[0,1]$ under a natural set of axioms: inclusion of the classical integral, a semigroup property, positivity, and continuity. By unpacking the proof into its constituent steps and understanding its structure clearly, we provide improvements and extensions of the original result. Firstly, by using some properties of topological groups, we demonstrate that the positivity axiom can be removed with almost no effect on the result. Secondly, we consider a version of the theorem on the whole real line $\mathbb{R}$, with constant of integration $-\infty$ instead of $0$, which was mentioned without proof in the 1978 paper. The theorem can be extended to this setting, but it is not trivial to do so: the theory of Fr\'echet spaces must be used, and we introduced a new shift-commutativity axiom to get the density result that we need to extend the proof to spaces of functions and distributions on $\mathbb{R}$ with left-bounded support.