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Characterizations of fractional operators via integral transforms

Daniel Cao Labora, Marc Jornet

Abstract

In 1972, J. S. Lew established a reasonable conjecture regarding an axiomatic characterization for the one-dimensional Riemann-Liouville integral. This conjecture was proved by Cartwright and McMullen in 1978. After that, little further work has been done on this topic, except some extensions for the Stieltjes case in one and several variables. In this paper, we prove the necessity of the axioms established in the conjecture of J. S. Lew using the Cauchy functional equation and Hamel bases. In addition, we give a proof for the characterization in several variables by employing Titchmarsh theorem, as a natural extension of the approach of Cartwright and McMullen. We also provide an alternative version and proof in one and several variables with Laplace transforms and the Cauchy functional equation, weakening parts of the continuity assumption. We show a similar result for the Riesz potential in terms of the Fourier transform. Finally, we illustrate how the theory can be used for characterization in the context of fractional calculus with respect to a non-smooth integrator, based on transmutation and measures.

Characterizations of fractional operators via integral transforms

Abstract

In 1972, J. S. Lew established a reasonable conjecture regarding an axiomatic characterization for the one-dimensional Riemann-Liouville integral. This conjecture was proved by Cartwright and McMullen in 1978. After that, little further work has been done on this topic, except some extensions for the Stieltjes case in one and several variables. In this paper, we prove the necessity of the axioms established in the conjecture of J. S. Lew using the Cauchy functional equation and Hamel bases. In addition, we give a proof for the characterization in several variables by employing Titchmarsh theorem, as a natural extension of the approach of Cartwright and McMullen. We also provide an alternative version and proof in one and several variables with Laplace transforms and the Cauchy functional equation, weakening parts of the continuity assumption. We show a similar result for the Riesz potential in terms of the Fourier transform. Finally, we illustrate how the theory can be used for characterization in the context of fractional calculus with respect to a non-smooth integrator, based on transmutation and measures.

Paper Structure

This paper contains 16 sections, 24 theorems, 12 equations.

Table of Contents

  1. Introduction
  2. Summary
  3. Some previous results
  4. Necessity of the axioms in the Cartwright-McMullen theorem
  5. The Cartwright-McMullen theorem in several variables
  6. Characterization of the Riemann-Liouville integral in dimension $1$
  7. The case of convolution operators
  8. The case of non-convolution operators
  9. Characterization of the Riemann-Liouville integral in dimension $n>1$
  10. The case of convolution operators
  11. The case of non-convolution operators
  12. Characterization of the Riesz potential in dimension $n\geq 1$
  13. Characterization of the Riemann-Liouville integral under transmutation
  14. Transmutation for integrals with a kernel using pushforward measures
  15. Application: Transmutation in fractional calculus using pushforward measures
  16. ...and 1 more sections

Key Result

Lemma 1.1

Let $(H, +)$ be a commutative group, and $(S, +)$ a subsemigroup of $(H, +)$. Then the group $G$ generated by $S$ is $G = S - S$. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (56)

  • Lemma 1.1: Theorem 4.5.1 in Kuczma
  • Lemma 1.2: Corollary 18.2.1 in Kuczma
  • Corollary 1.3
  • Lemma 1.4
  • proof
  • Theorem 2.1: Cartwright-McMullen, real version
  • Theorem 2.2: Cartwright-McMullen, complex version
  • Theorem 2.3
  • Theorem 2.4
  • proof
  • ...and 46 more