[Paper Review] Some Properties of Prabhakar-type Operators

Federico Polito, Živorad Tomovski|arXiv (Cornell University)|Aug 13, 2015

Fractional Differential Equations Solutions1 references3 citations

TL;DR

This paper investigates Prabhakar-type integral and derivative operators, including their regularized and Hilfer extensions, deriving new Opial- and Hardy-type inequalities. It establishes analytical properties and connects them to probability theory, contributing to fractional calculus with applications in mathematical physics and stochastic processes.

ABSTRACT

In the paper we study some properties of Prabhakar integrals and derivatives and of some of their extensions such as the regularized Prabhakar derivative or the Hilfer--Prabhakar derivative. Some Opial- and Hardy-type inequalities are derived. In the last section we point out relationships with probability theory.

Motivation & Objective

• To analyze the analytical properties of Prabhakar integrals and derivatives, extending their theoretical foundation.
• To investigate the regularized Prabhakar derivative and the Hilfer–Prabhakar derivative as generalizations of fractional calculus operators.
• To derive new inequalities of Opial- and Hardy-type for these operators, enhancing their analytical utility.
• To establish connections between Prabhakar-type operators and probability theory, particularly in the context of stochastic processes.
• To provide a comprehensive framework for applications in mathematical physics and applied analysis.

Proposed method

• Employing the generalized Mittag-Leffler function as the kernel in integral and differential operators to define Prabhakar-type operators.
• Applying functional analysis techniques to study boundedness and mapping properties of the operators in suitable function spaces.
• Deriving Opial-type inequalities through integral estimates involving the generalized Mittag-Leffler function.
• Establishing Hardy-type inequalities using weighted norm inequalities and properties of the Prabhakar kernel.
• Utilizing the Laplace transform and operational calculus to analyze the regularized and Hilfer–Prabhakar derivatives.
• Linking the operators to probability distributions via the distributional properties of the Mittag-Leffler function.

Experimental results

Research questions

• RQ1 What are the fundamental analytical properties of Prabhakar integrals and derivatives in function spaces?
• RQ2 How do the regularized Prabhakar and Hilfer–Prabhakar derivatives extend classical fractional calculus?
• RQ3 What new Opial- and Hardy-type inequalities can be derived for Prabhakar-type operators?
• RQ4 In what ways do Prabhakar operators relate to probability distributions and stochastic processes?
• RQ5 How do these operators preserve or generalize known inequalities in fractional calculus?

Key findings

• New Opial-type inequalities are established for Prabhakar-type operators, extending classical results to a broader class of fractional integrals.
• Hardy-type inequalities are derived for the Prabhakar integral operator, providing bounds in weighted Lebesgue spaces.
• The regularized Prabhakar derivative is shown to preserve key properties of fractional derivatives while improving regularity in solution spaces.
• The Hilfer–Prabhakar derivative is characterized as a generalization that unifies Riemann–Liouville and Caputo-type derivatives under the Prabhakar kernel.
• A connection is demonstrated between the generalized Mittag-Leffler function and probability distributions, particularly in the context of continuous-time random walks.
• The paper provides a theoretical framework linking fractional operators with stochastic processes through the distributional behavior of the Prabhakar kernel.