# Tempered and Hadamard-type fractional calculus with respect to functions

## TL;DR

This paper unifies various fractional calculus operators, including Hadamard-type and tempered, through a general framework based on fractional calculus with respect to functions, and explores their fundamental properties and applications.

## Contribution

It introduces a unifying generalization of multiple fractional operators using fractional calculus with respect to functions, connecting and extending existing definitions.

## Key findings

- Established a connection between Hadamard-type and tempered fractional calculus.
- Developed a general framework unifying Riemann--Liouville, Caputo, Hadamard, and tempered operators.
- Derived properties, function spaces, and integration by parts formulae for the new operators.

## Abstract

Many different types of fractional calculus have been defined, which may be categorised into broad classes according to their properties and behaviours. Two types that have been much studied in the literature are the Hadamard-type fractional calculus and tempered fractional calculus. This paper establishes a connection between these two definitions, writing one in terms of the other by making use of the theory of fractional calculus with respect to functions. By extending this connection in a natural way, a generalisation is developed which unifies several existing fractional operators: Riemann--Liouville, Caputo, classical Hadamard, Hadamard-type, tempered, and all of these taken with respect to functions. The fundamental calculus of these generalised operators is established, including semigroup and reciprocal properties as well as application to some example functions. Function spaces are constructed in which the new operators are defined and bounded. Finally, some formulae are derived for fractional integration by parts with these operators.