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

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 $1\le q\le p/(1-p\alpha )$, 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 $1\le q<p/(1-p\alpha )$.