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 L p ( t 0 , t 1 ; X ) to the Banach space B M O ( t 0 , t 1 ; X ) ∩ K ( p − 1 ) / p ( t 0 , t 1 ; X ) . This improvement, refines a result by Hardy-Littlewood. 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 ≥ 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 ) .