Suppose that is a metric measure space, which possesses two “geometric” properties, called “isoperimetric” property and approximate midpoint property, and that the measure is locally doubling. The isoperimetric property implies that the volume of balls grows at least exponentially with the radius. Hence the measure is not globally doubling. In this paper we define an atomic Hardy space , where atoms are supported only on “small balls”, and a corresponding space of functions of “bounded mean oscillation”, where the control is only on the oscillation over small balls. We prove that is the dual of and that an inequality of John–Nirenberg type on small balls holds for functions in . Furthermore, we show that the spaces are intermediate spaces between and , and we develop a theory of singular integral operators acting on function spaces on . Finally, we show that our theory is strong enough to give - and - estimates for various interesting operators on Riemannian manifolds and symmetric spaces which are unbounded on and on .
@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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