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

Paulo Mendes de Carvalho Neto; Renato Fehlberg Júnior

paper

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

arXiv:2408.10180

Abstract

In this manuscript, we examine the continuity properties of the Riemann-Liouville fractional integral for order $α = 1/ p$ , where $p > 1$ , mapping from $L^{p} (t_{0}, t_{1}; X)$ to the Banach space $B M O (t_{0}, t_{1}; X) \cap K_{(p - 1) / p} (t_{0}, t_{1}; X)$ . This improvement, in some sense, refines a result by Hardy-Littlewood ([12]). To achieve this, we study properties between spaces $B M O (t_{0}, t_{1}; X)$ and $K_{(p - 1) / p} (t_{0}, t_{1}; X)$ . Additionally, we obtained the boundedness of the fractional integral of order $α \geq 1$ from $L^{1} (t_{0}, t_{1}; X)$ into the Riemann-Liouville fractional Sobolev space $W_{R L}^{s, p} (t_{0}, t_{1}; X)$ .