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 α = 1/p, where $p > 1$, mapping from Lᵖ(t₀, t₁; X) to the Banach space BMO(t₀, t₁; X)∩ K(p-1)/p(t₀, t₁; X). This improvement, in some sense, refines a result by Hardy-Littlewood ([12]). To achieve this, we study properties between spaces BMO(t₀, t₁; X) and K(p-1)/p(t₀, t₁; X). Additionally, we obtained the boundedness of the fractional integral of order α ≥ 1 from L¹(t₀, t₁; X) into the Riemann-Liouville fractional Sobolev space Ws,pRL(t₀, t₁; X).