${H}^{\mathbf{1}}$ and $BMO$ for certain locally doubling metric measure spaces
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 8 (2009) no. 3, pp. 543-582.

Suppose that $\left(M,\phantom{\rule{-0.166667em}{0ex}}\rho ,\phantom{\rule{-0.166667em}{0ex}}\mu \right)$ is a metric measure space, which possesses two “geometric” properties, called “isoperimetric” property and approximate midpoint property, and that the measure $\mu$ is locally doubling. The isoperimetric property implies that the volume of balls grows at least exponentially with the radius. Hence the measure $\mu$ is not globally doubling. In this paper we define an atomic Hardy space ${H}^{1}\left(\mu \right)$, where atoms are supported only on “small balls”, and a corresponding space $BMO\left(\mu \right)$ of functions of “bounded mean oscillation”, where the control is only on the oscillation over small balls. We prove that $BMO\left(\mu \right)$ is the dual of ${H}^{1}\left(\mu \right)$ and that an inequality of John–Nirenberg type on small balls holds for functions in $BMO\left(\mu \right)$. Furthermore, we show that the ${L}^{p}\left(\mu \right)$ spaces are intermediate spaces between ${H}^{1}\left(\mu \right)$ and $BMO\left(\mu \right)$, and we develop a theory of singular integral operators acting on function spaces on $M$. Finally, we show that our theory is strong enough to give ${H}^{1}\left(\mu \right)$-${L}^{1}\left(\mu \right)$ and ${L}^{\infty }\left(\mu \right)$-$BMO\left(\mu \right)$ estimates for various interesting operators on Riemannian manifolds and symmetric spaces which are unbounded on ${L}^{1}\left(\mu \right)$ and on ${L}^{\infty }\left(\mu \right)$.

Classification : 42B20,  42B30,  46B70,  58C99
@article{ASNSP_2009_5_8_3_543_0,
author = {Carbonaro, Andrea and Mauceri, Giancarlo and Meda, Stefano},
title = {$H^{\bf 1}$ and $BMO$ for certain locally doubling metric measure spaces},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
pages = {543--582},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 8},
number = {3},
year = {2009},
zbl = {1180.42008},
mrnumber = {2581426},
language = {en},
url = {archive.numdam.org/item/ASNSP_2009_5_8_3_543_0/}
}
Carbonaro, Andrea; Mauceri, Giancarlo; Meda, Stefano. $H^{\bf 1}$ and $BMO$ for certain locally doubling metric measure spaces. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 8 (2009) no. 3, pp. 543-582. http://archive.numdam.org/item/ASNSP_2009_5_8_3_543_0/

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