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arXiv · 2408.10180

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

Paulo Mendes de Carvalho Neto·Renato Fehlberg Júnior
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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 $W^{s,p}_{RL}(t_0, t_1; X)$.

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Paulo Mendes de Carvalho Neto, Renato Fehlberg Júnior. 2024-08-19. The Riemann-Liouville fractional integral in Bochner-Lebesgue spaces III. https://arxiv.org/abs/2408.10180

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