Speaker: Marc Jornet, Escuela Superior de Ingenierıa y Tecnologıa, Universidad Internacional de La Rioja, Logrono, Spain

Time : 15:00 - 16:00 CEST (Rome/Paris)

Hosted at: SISSA, International School of Advanced Studies, Trieste, Italy

Zoom : A Zoom link will appear here, an hour before the talk

Organizers : Pavan Pranjivan Mehta* (pavan.mehta@sissa.it) and Arran Fernandez** (arran.fernandez@emu.edu.tr)

* SISSA, International School of Advanced Studies, Italy

** Eastern Mediterranean University, Northern Cyprus

Keywords: fractional integral operator; characterization; Cartwright-McMullen theorem

Abstract: The Riemann-Liouville integral is a very important operator in mathematical analysis and, in particular, in fractional calculus. I review the first proof on an axiomatic characterization for the one-dimensional Riemann-Liouville integral, given by Cartwright and McMullen in 1978, based on factorization and Titchmarsh convolution theorem. Then I study an extension by Cao Labora (2025) for the Stieltjes case in one variable, which uses transmutation. I also present my paper from 2025, which generalizes Cao Labora's contribution to several variables and weighted fractional calculus with respect to a function. Then I focus on my recent paper with Cao Labora (2026), where we develop several aspects of the characterization of the Riemann-Liouville integral: (i) the necessity of the axioms using the Cauchy functional equation and Hamel bases; (ii) a proof for the characterization in several variables by employing Titchmarsh theorem as a natural extension of the approach of Cartwright and McMullen; (iii) an alternative version and proof in one and several variables with Laplace transforms and the Cauchy functional equation, weakening parts of the continuity assumption; (iv) a characterization in the context of fractional calculus with respect to a non-smooth integrator, based on transmutation and pushforward measures; and (v) the adaptation to other integral operators, such as the Riesz potential, in terms of the Fourier transform.

Biography: Marc Jornet Sanz has graduated in Mathematics from the University of Valencia and holds a PhD in applied mathematics from the Technical University of Valencia, Spain, since 2020. Currently, he works as a professor at the International University of La Rioja, Spain. His research is focused on random, stochastic, and fractional differential equations and their applications in modeling (epidemiology, population growth, physics, etc.). You can check his publications at https://orcid.org/0000-0003-0748-3730.

Bibliography

[1] D.I. Cartwright and J.R. McMullen. “A note on the fractional calculus”. In: Proc. Edinburgh Math. Soc. 21.1 (1978), pp. 79–80.
[2] D. Cao Labora. “An extension of the Cartwright-McMullen theorem in fractional calculus for the Stieltjes case”. In: J. Integral Equations Appl. 37.1 (2025), pp. 1–6. Fractional Calculus Seminar Series 2025
[3] M. Jornet. “An axiomatic characterization for the multidimensional weighted Riemann-Liouville integral with respect to a function”. In: J. Integral Equations Appl. 37.2 (2025), pp. 149–161.
[4] D. Cao Labora and M. Jornet. “Characterizations of fractional operators via integral transforms”. In: arXiv:2604.00714 (2026).