An extension of the Cartwright-McMullen theorem in fractional calculus for the smooth Stieltjes case

TL;DR

This paper extends the Cartwright-McMullen theorem to the Stieltjes integral, establishing a uniqueness result for its smooth case, analogous to the classical fractional integral characterization.

Contribution

It provides a novel axiomatic characterization of the Stieltjes integral operator in fractional calculus for smooth integrators, expanding the classical theorem.

Findings

Proves the uniqueness of the fractional Stieltjes integral extension for smooth integrators.

Establishes an axiomatic framework similar to the Cartwright-McMullen theorem for the Stieltjes case.

Generalizes the classical fractional integral characterization to a broader integral operator.

Abstract

In 1976, Donald Cartwright and John McMullen characterized axiomatically the Riemann-Liouvile fractional integral in a paper that was published in 1978. The motivation for their work was to answer affirmatively to a conjecture stated by J. S. Lew a few years before, in 1972. Essentially, their ``Cartwright-McMullen theorem in fractional calculus'' proved that the Riemann-Liouville fractional integral is the only continuous extension of the usual integral operator to positive real orders, in such a way that the Index Law holds. In this paper, we propose an analogous result for the uniqueness of the extension of the Stieltjes integral operator, in the case of a smooth integrator.