On preduals and Köthe duals of some subspaces of Morrey spaces (Q1997236)

The \textit{Fofana space} \((L^{q}, l^{p})^{\alpha}(\mathbb{R}^{d})\) is introduced as \[ (L^{q}, l^{p})^{\alpha}(\mathbb{R}^{d})=\{f \in L_{\mathrm{loc}}^{q}(\mathbb{R}^{d}) : \|f\|_{q, p, \alpha}<\infty\} \] where \[ \|f\|_{q, p, \alpha}=\sup _{r>0} r^{d\left(\frac{1}{\alpha}-\frac{1}{q}\right)}{ }_{r}\| f \|_{q, p} \] with \[ {}_{r}\|f\|_{q, p}=r^{-\frac{d}{p}} \left\| \|f \chi_{J_{x}^{r}}\|_{q} \right\|_{p}=\left\{\begin{array}{ll} r^{-\frac{d}{p}}\left[\int_{\mathbb{R}^{d}}\left(\left\|f \chi_{J_{x}^{r}}\right\|_{q}\right)^{p} d x\right]^{\frac{1}{p}} & \text { if } p<\infty, \\ \operatorname{ess\, sup}_{x \in \mathbb{R}^{d}}\left\|f \chi_{J_{x}^{r}}\right\|_{q} & \text { if } p=\infty. \end{array}\right. \] Particular cases of Fofana spaces include the Lebesgue spaces, \((L^{q}, l^{\alpha})^{\alpha}(\mathbb{R}^{d})= L^{\alpha}(\mathbb{R}^{d})\), and the Morrey spaces \((L^{q}, l^{\infty})^{\alpha}(\mathbb{R}^{d})=\mathcal{M}_{q}^{\alpha}(\mathbb{R}^{d})\). It is proved that the space \(\mathcal{H}(q^{\prime}, p^{\prime}, \alpha^{\prime})(\mathbb{R}^{d})\) is a (ball) Banach function space which coincides with the Köthe dual space of \((L^{q}, l^{p})^{\alpha}(\mathbb{R}^{d})\) and represents the dual space of the closure \((L^{q}, l^{p})_{a}^{\alpha}(\mathbb{R}^{d})\) of \(L^{\alpha}(\mathbb{R}^{d})\) in \((L^{q}, l^{p})^{\alpha}(\mathbb{R}^{d}).\) As an application, existence and uniqueness of bounded linear extensions of some classical operators are obtained.