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arXiv · 2108.02291

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

Paulo Mendes Carvalho-Neto·Renato Fehlberg Júnior
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Abstract

In this paper we study the Riemann-Liouville fractional integral of order $\alpha>0$ as a linear operator from $L^p(I,X)$ into itself, when $1\leq p\leq \infty$, $I=[t_0,t_1]$ (or $I=[t_0,\infty)$) and $X$ is a Banach space. In particular, when $I=[t_0,t_1]$, we obtain necessary and sufficient conditions to ensure its compactness. We also prove that Riemann-Liouville fractional integral defines a $C_0-$semigroup but does not defines a uniformly continuous semigroup. We close this study by presenting lower and higher bounds to the norm of this operator.

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Paulo Mendes Carvalho-Neto, Renato Fehlberg Júnior. 2021-08-04. The Riemann-Liouville fractional integral in Bochner-Lebesgue spaces I. https://arxiv.org/abs/2108.02291

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