Hardy space theory on spaces of homogeneous type via orthonormal wavelet bases

By Yongsheng Han, Ji Li, Lesley A. Ward. Applied and Computational Harmonic Analysis, 2015

52 citations recorded by the source corpus.

In this paper, using the remarkable orthonormal wavelet basis constructed recently by Auscher and Hyt\"onen, we establish the theory of product Hardy spaces on spaces ${\widetilde X} = X_1\times X_2\times\cdot \cdot\cdot\times X_n$, where each factor $X_i$ is a space of homogeneous type in the sense of Coifman and Weiss. The main tool we develop is the Littlewood--Paley theory on $\widetilde X$, which in turn is a consequence of a corresponding theory on each factor space. We define the square function for this theory in terms of the wavelet coefficients. The Hardy space theory developed in this paper includes product~$H^p$, the dual $\cmo^p$ of $H^p$ with the special case $\bmo = \cmo^1$, and the predual $\vmo$ of $H^1$. We also use the wavelet expansion to establish the Calder\'on--Zygmund decomposition for product $H^p$, and deduce an interpolation theorem. We make no additional assumptions on the quasi-metric or the doubling measure for each factor space, and thus we extend to the full generality of product spaces of homogeneous type the aspects of both one-parameter and multiparameter theory involving the Littlewood--Paley theory and function spaces. Moreover, our methods would be expected to be a powerful tool for developing wavelet analysis on spaces of homogeneous type.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f3aa9f320c16e95fe970013635e927bfc8ccff13

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In Applied Mathematics

1. Wen Yuan
2. Dachun Yang
3. Xianjie Yan
4. Ziyi He
5. Jun Liu
6. Yoshihiro Sawano
7. Ciqiang Zhuo
8. Yangyang Zhang

In Mathematical Physics

1. Wen Yuan
2. Dachun Yang
3. Xianjie Yan
4. Ziyi He
5. Yoshihiro Sawano
6. Ciqiang Zhuo
7. Jun Liu
8. Yangyang Zhang

Citations in the corpus

• Hardy spaces associated with ball quasi‐Banach function spaces on spaces of homogeneous type: Characterizations of maximal functions, decompositions, and dual spaces (2021, background): “We refer the reader to [8, 9, 12, 22, 28, 31, 34, 45, 49, 72] for more studies on the real-variable theory of function spaces over and to [18, 46, 51, 73] for their applications in the boundedness of operators; particularly, we refer the reader to [6, 7, 10] for some recent progress on the real-var”