The Riemann-Liouville fractional integral in Bochner-Lebesgue spaces I

TL;DR

This paper investigates the properties of the Riemann-Liouville fractional integral as a linear operator on Bochner-Lebesgue spaces, including conditions for compactness, semigroup behavior, and bounds on its norm.

Contribution

It provides necessary and sufficient conditions for compactness, shows it forms a $C_0$-semigroup but not a uniformly continuous one, and establishes bounds on its norm.

Findings

Characterizes compactness conditions for the fractional integral

Demonstrates the fractional integral forms a $C_0$-semigroup

Establishes bounds for the operator's norm

Abstract

In this paper we study the Riemann-Liouville fractional integral of order $\alpha>0$ as a linear operator from $L^p(I,X)$ into itself, when $1\leq p\leq \infty$, $I=[t_0,t_1]$ (or $I=[t_0,\infty)$) and $X$ is a Banach space. In particular, when $I=[t_0,t_1]$, we obtain necessary and sufficient conditions to ensure its compactness. We also prove that Riemann-Liouville fractional integral defines a $C_0-$semigroup but does not defines a uniformly continuous semigroup. We close this study by presenting lower and higher bounds to the norm of this operator.