实直线上分数积分算子的唯一公理化
Unique axiomatisation of fractional integral operators on the real line
AI总结：
该研究改进了Cartwright–McMullen定理，移除了正性公理，并借助Fréchet空间理论和新的平移交换公理，将实直线上分数积分算子的唯一公理化结果推广到左有界支撑的函数与分布空间。
AI中文摘要：
1978年的Cartwright–McMullen定理确立了紧区间$[0,1]$上Riemann–Liouville型分数积分算子族在一组自然公理下的唯一性：包含经典积分、半群性质、正性与连续性。通过拆解该定理的证明步骤并明确其结构，我们对原结果进行了改进与推广。首先，利用拓扑群的部分性质，我们证明可移除正性公理且几乎不影响结果；其次，针对1978年论文中提及但未证明的、积分常数为$-\boldsymbol{\times}$而非$0$的实直线$\boldsymbol{\times}$上的定理版本，将其推广到该场景并非易事，需用到Fréchet空间理论，且我们引入了新的平移交换公理，以获得推广证明所需的密度结果，从而可将定理扩展到具有左有界支撑的$\boldsymbol{\times}$上函数与分布空间。
英文摘要：
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 Fréchet 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.
