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

TL;DR

This paper investigates the properties of the Riemann-Liouville fractional integral operator acting between Bochner-Lebesgue spaces, providing necessary and sufficient conditions for its compactness in specific function space settings.

Contribution

It establishes a comprehensive characterization of the compactness of the Riemann-Liouville fractional integral operator between $L^p$ and $L^q$ spaces in Banach space-valued functions.

Findings

Characterization of the fractional integral operator's boundedness.

Necessary and sufficient conditions for compactness.

Applicability to different interval types.

Abstract

In this work we study the Riemann-Liouville fractional integral of order $\alpha\in(0,1/p)$ as an operator from $L^p(I;X)$ into $L^{q}(I;X)$, with $1\leq q\leq p/(1-p\alpha)$, whether $I=[t_0,t_1]$ or $I=[t_0,\infty)$ and $X$ is a Banach space. Our main result give necessary and sufficient conditions to ensure the compactness of the Riemann-Liouville fractional integral from $L^p(t_0,t_1;X)$ into $L^{q}(t_0,t_1;X)$, when $1\leq q< p/(1-p\alpha)$.