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Keywords:
grand Lebesgue space; grand Morrey space; Gagliardo-Peetre method; quasi-metric measure space; Calderón product; predual space; $\pm $ interpolation method
grand Lebesgue space; grand Morrey space; Gagliardo-Peetre method; quasi-metric measure space; Calderón product; predual space; $\pm $ interpolation method
Summary:
Let $\theta \in (0,1)$, $\lambda \in [0,1)$ and $p,p_0,p_1\in (1,\infty ]$ be such that ${(1-\theta )}/{p_{0}}+{\theta }/{p_{1}}={1}/{p}$, and let $\varphi , \varphi _0, \varphi _1 $ be some admissible functions such that $\varphi , \varphi _0^{{p}/{p_0}}$ and $\varphi _1^{{p}/{p_1}}$ are equivalent. We first prove that, via the $\pm $ interpolation method, the interpolation $\langle L^{p_0),\lambda }_{\varphi _0}(\mathcal {X}), L^{p_1),\lambda }_{\varphi _1}(\mathcal {X}), \theta \rangle $ of two generalized grand Morrey spaces on a quasi-metric measure space $\mathcal {X}$ is the generalized grand Morrey space $L^{p),\lambda }_{\varphi }(\mathcal {X})$. Then, by using block functions, we also find a predual space of the generalized grand Morrey space. These results are new even for generalized grand Lebesgue spaces.
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