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

Yi Liu; Wen Yuan

Czechoslovak Mathematical Journal (2017)

  • Volume: 67, Issue: 3, page 715-732
  • ISSN: 0011-4642

Abstract

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Let θ ∈ ( 0 , 1 ) , λ ∈ [ 0 , 1 ) and p , p 0 , p 1 ∈ ( 1 , ∞ ] be such that ( 1 - θ ) / p 0 + θ / p 1 = 1 / p , and let ϕ , ϕ 0 , ϕ 1 be some admissible functions such that ϕ , ϕ 0 p / p 0 and ϕ 1 p / p 1 are equivalent. We first prove that, via the ± interpolation method, the interpolation 〈 L ϕ 0 p 0 ) , λ ( 𝒳 ) , L ϕ 1 p 1 ) , λ ( 𝒳 ) , θ 〉 of two generalized grand Morrey spaces on a quasi-metric measure space 𝒳 is the generalized grand Morrey space L ϕ p ) , λ ( 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.

How to cite

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Liu, Yi, and Yuan, Wen. "Interpolation and duality of generalized grand Morrey spaces on quasi-metric measure spaces." Czechoslovak Mathematical Journal 67.3 (2017): 715-732. <http://eudml.org/doc/294082>.

@article{Liu2017,
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.},
author = {Liu, Yi, Yuan, Wen},
journal = {Czechoslovak Mathematical Journal},
keywords = {grand Lebesgue space; grand Morrey space; Gagliardo-Peetre method; quasi-metric measure space; Calderón product; predual space; $\pm $ interpolation method},
language = {eng},
number = {3},
pages = {715-732},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Interpolation and duality of generalized grand Morrey spaces on quasi-metric measure spaces},
url = {http://eudml.org/doc/294082},
volume = {67},
year = {2017},
}

TY - JOUR
AU - Liu, Yi
AU - Yuan, Wen
TI - Interpolation and duality of generalized grand Morrey spaces on quasi-metric measure spaces
JO - Czechoslovak Mathematical Journal
PY - 2017
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 67
IS - 3
SP - 715
EP - 732
AB - 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.
LA - eng
KW - grand Lebesgue space; grand Morrey space; Gagliardo-Peetre method; quasi-metric measure space; Calderón product; predual space; $\pm $ interpolation method
UR - http://eudml.org/doc/294082
ER -

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