 

Yongsheng Han,
Ji Li,
M. Cristina Pereyra, and
Lesley A. Ward
Atomic decomposition of product Hardy spaces via wavelet bases on spaces of homogeneous type
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print


Published: 
August 18, 2021. 
Keywords: 
Product Hardy spaces, spaces of homogeneous type, orthonormal wavelet basis,
test functions, distributions, Calderón reproducing formula. 
Subject: 
Primary 42B35; Secondary 43A85, 30L99, 42B30, 42C40. 


Abstract
We provide an atomic decomposition of the product Hardy spaces H^{p}(X) which were recently developed
by Han, Li, and Ward in the setting of product spaces of homogeneous type X = X_{1} x X_{2}. Here
each factor (X_{i},d_{i},μ_{i}), for i = 1, 2, is a space of homogeneous type in the sense of Coifman and Weiss. These Hardy spaces make use of the orthogonal wavelet bases of Auscher and Hytönen and their underlying reference dyadic grids. However, no additional assumptions on the quasimetric or on the doubling measure
for each factor space are made. To carry out this program, we introduce product (p,q)atoms on X and product atomic Hardy spaces H^{p,q}_{at}(X). As consequences of the atomic decomposition of H^{p}(X), we show that for all q > 1 the product atomic Hardy spaces coincide with the product Hardy spaces, and we show that the product Hardy spaces are independent of the particular choices of both the wavelet bases and the reference dyadic grids. Likewise, the product Carleson measure spaces CMO^{p}(X), the bounded mean oscillation space BMO(X), and the vanishing mean oscillation space VMO(X), as defined by Han, Li, and Ward, are also independent of the particular choices of both wavelets and reference dyadic grids.


Acknowledgements
The authors would like to thank the referees for careful reading and helpful comments, which made this paper more accurate. Ji Li and Lesley Ward were supported by the Australian Research Council Grant No. ARCDP160100153.


Author information
Yongsheng Han:
Department of Mathematics
Auburn University
Auburn, AL 368495310, USA
hanyong@auburn.edu
Ji Li:
Department of Mathematics and Statistics
Macquarie University
NSW 2019, Australia
ji.li@mq.edu.au
M. Cristina Pereyra:
Department of Mathematics and Statistics
University of New Mexico
Albuquerque, NM 87131, USA
crisp@math.unm.edu
Lesley A. Ward:
UniSA STEM
University of South Australia
Mawson Lakes SA 5095, Australia
lesley.ward@unisa.edu.au

