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

In this manuscript, we examine the continuity properties of the Riemann-Liouville fractional integral for order $\alpha =1\mathrm{/}p$, where $p>1$, mapping from $L^p(t_0,t_1;X)$ to the Banach space $BMO(t_0,t_1;X)\cap K_{(p-1)\mathrm{/}p}(t_0,t_1;X)$. This improvement, in some sense, refines a result by Hardy-Littlewood ([12]). To achieve this, we study properties between spaces $BMO(t_0,t_1;X)$ and $K_{(p-1)\mathrm{/}p}(t_0,t_1;X)$. Additionally, we obtained the boundedness of the fractional integral of order $\alpha \ge 1$ from $L^1(t_0,t_1;X)$ into the Riemann-Liouville fractional Sobolev space $W_{RL}^{s,p}(t_0,t_1;X)$.