Czechoslovak Mathematical Journal, Vol. 67, No. 3, pp. 715-732, 2017


Interpolation and duality of generalized grand Morrey spaces on quasi-metric measure spaces

Yi Liu, Wen Yuan

Received February 21, 2016.  First published July 13, 2017.

Abstract:  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.
Keywords:  grand Lebesgue space; grand Morrey space; Gagliardo-Peetre method; quasi-metric measure space; Calderón product; predual space; $\pm$ interpolation method
Classification MSC:  46B70, 46B10
DOI:  10.21136/CMJ.2017.0081-16


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Affiliations:   Yi Liu, Wen Yuan (corresponding author), School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, No. 19, Xinjiekouwai Street, Beijing 100875, Haidian, People's Republic of China, e-mail: liuyimath1991@163.com, wenyuan@bnu.edu.cn

 
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