Abstract
In their 1977 Bulletin of AMS paper [CoWe77], Ronald Coifman and Guido Weiss managed to develop a theory of Hardy spaces on spaces of homogeneous type by taking the atomic characterization of H p(X) as a definition. This was the starting point in generalizing the theory of Hardy spaces in abstract settings. The main goal of this chapter is to explore the relationship between the Hardy spaces developed in this monograph and those in [CoWe77]. Understanding this connection is an important step towards unifying the theory of Hardy spaces.
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Notes
- 1.
The authors in [CoWe77] introduced the spaces \(H_{CW}^{p,q}(X,\rho,\mu )\) under the additional assumption that ρ is symmetric. This is an extraneous demand that we do not wish to make.
- 2.
Passing to ρ # was used in order to apply Theorem 1 in [MaSe79i] which only applies to symmetric quasi-distances.
- 3.
Ignoring momentarily whether this is well-defined.
- 4.
References
M. Bownik, Anisotropic hardy spaces and wavelets. Mem. Am. Math. Soc. 164(781), 122pp. (2003)
R.R. Coifman, G. Weiss, Extensions of Hardy spaces and their use in analysis. Bull. Am. Math. Soc. 83(4), 569–645 (1977)
C. Fefferman, Characterizations of bounded mean oscillation. Bull. Am. Math. Soc. 77, 587–588 (1971)
C. Fefferman, E. M. Stein, H p spaces of several variables. Acta Math. 129(3–4), 137–193 (1972)
G. Hu, D. Yang, Y. Zhou, Boundedness of singular integrals in Hardy spaces on spaces of homogeneous type. Taiwanese J. Math. 133(1), 91–135 (2009)
R.A. Macías, C. Segovia, Lipschitz functions on spaces of homogeneous type. Adv. Math. 33, 257–270 (1979)
R.A. Macías, C. Segovia, A Decomposition into atoms of distributions on spaces of homogeneous type. Adv. Math. 33(3), 271–309 (1979)
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Alvarado, R., Mitrea, M. (2015). Further Results. In: Hardy Spaces on Ahlfors-Regular Quasi Metric Spaces. Lecture Notes in Mathematics, vol 2142. Springer, Cham. https://doi.org/10.1007/978-3-319-18132-5_7
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DOI: https://doi.org/10.1007/978-3-319-18132-5_7
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