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

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