AN EXTENSION OF THE CARTWRIGHT–MCMULLEN THEOREM IN FRACTIONAL CALCULUS FOR THE STIELTJES CASE

In a 1978 article, Donald Cartwright and John McMullen characterized axiomatically the Riemann–Liouville fractional integral, thus answering affirmatively a conjecture stated by J. S. Lew in 1972. Essentially, they proved that the Riemann–Liouville fractional integral is the only continuous extension of the usual integral operator to positive real orders that satisfies the Index law, meaning that composition corresponds to addition of operator orders. We prove an analogous result for the uniqueness of the extension of the Stieltjes integral operator, in the case of a continuously differentiable integrator.