Abstract
To shed some light on the John—Nirenberg space, the authors of this article introduce the John—Nirenberg-Q space via congruent cubes, \(JNQ_{p,q}^\alpha \left( {{\mathbb{R}^n}} \right)\), which, when p = ∞ and q = 2, coincides with the space Qα (ℝn) introduced by Essén, Janson, Peng and Xiao in [Indiana Univ Math J, 2000, 49(2): 575-615]. Moreover, the authors show that, for some particular indices, \(JNQ_{p,q}^\alpha \left( {{\mathbb{R}^n}} \right)\) coincides with the congruent John—Nirenberg space, or that the (fractional) Sobolev space is continuously embedded into \(JNQ_{p,q}^\alpha \left( {{\mathbb{R}^n}} \right)\). Furthermore, the authors characterize \(JNQ_{p,q}^\alpha \left( {{\mathbb{R}^n}} \right)\) via mean oscillations, and then use this characterization to study the dyadic counterparts. Also, the authors obtain some properties of composition operators on such spaces. The main novelties of this article are twofold: establishing a general equivalence principle for a kind of ‘almost increasing’ set function that is here introduced, and using the fine geometrical properties of dyadic cubes to properly classify any collection of cubes with pairwise disjoint interiors and equal edge length.
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Acknowledgements
The authors would like to thank Professor Feng DAI for some useful discussions regarding Proposition 2.13, and also Professor Yuan ZHOU for some useful discussions on Proposition 4.3, as well as the topic of quasiconformal mappings.
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This project was partially supported by the National Natural Science Foundation of China (12122102 and 11871100) and the National Key Research and Development Program of China (2020YFA0712900).
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Tao, J., Yang, Z. & Yuan, W. John—Nirenberg-Q Spaces via Congruent Cubes. Acta Math Sci 43, 686–718 (2023). https://doi.org/10.1007/s10473-023-0214-4
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DOI: https://doi.org/10.1007/s10473-023-0214-4
Key words
- John—Nirenberg space
- congruent cube
- Q space
- (fractional) Sobolev space
- mean oscillation
- dyadic cube
- composition operator