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

TL;DR

This paper investigates the continuity of the Riemann-Liouville fractional integral in specific function spaces, refining previous results and establishing boundedness properties for fractional integrals of order greater than or equal to one.

Contribution

It refines existing results on the continuity of fractional integrals in Bochner-Lebesgue spaces and introduces boundedness results into fractional Sobolev spaces.

Findings

Continuity of Riemann-Liouville fractional integral for order 1/p in certain spaces.

Refinement of Hardy-Littlewood result on fractional integrals.

Boundedness of fractional integral of order ≥ 1 into fractional Sobolev spaces.

Abstract

In this manuscript, we examine the continuity properties of the Riemann-Liouville fractional integral for order $\alpha = 1/p$, where $p > 1$, mapping from $L^p(t_0, t_1; X)$ to the Banach space $BMO(t_0, t_1; X)\cap K_{(p-1)/p}(t_0, t_1; X)$. 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 $K_{(p-1)/p}(t_0, t_1; X)$. Additionally, we obtained the boundedness of the fractional integral of order $\alpha \geq 1$ from $L^1(t_0, t_1; X)$ into the Riemann-Liouville fractional Sobolev space $W^{s,p}_{RL}(t_0, t_1; X)$.