Duality and reflexivity in grand Lebesgue spaces

The grand $L^p$ space $L^{p)} (\Omega) (1 < p < +\infty)$ introduced by Iwaniec-Sbordone is defined as the \textit{Banach Function Space} of the measurable functions $f$ on $\Omega$ such that $$ \parallel f\parallel_{p)} = \sup_{0 < \varepsilon < p-1}\bigg(\varepsilon\frac{1}{\vert\Omega\vert}\int\limits_{\Omega}\vert f\vert^{p-\varepsilon}dx\bigg)^{1/(p-\varepsilon)} < +\infty.$$ We introduce the \textit{small} $L^{p'}$ \textit{space} denoted by $L^{p')}(\Omega)$ and we prove that the associate space of $L^{p)}(\Omega)$ is $L^{p)'}(\Omega)$. It turns out that $L^{p)'}(\Omega)$ is a \textit{Banach Function Space} whose norm satisfy the Fatou property, and that it is the dual of the closure of $L^\infty(\Omega)$ in $L^{p)}(\Omega)$ . Moreover, we give a characterization of $L^{p)}(\Omega)$ as dual space, and we prove that for any $1 < p < +\infty$ the spaces $L^{p)}(\Omega)$ and $L^{p)'}(\Omega)$ are not reflexive.