H 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, p. 543-582
Suppose that (M,ρ,μ) is a metric measure space, which possesses two “geometric” properties, called “isoperimetric” property and approximate midpoint property, and that the measure μ is locally doubling. The isoperimetric property implies that the volume of balls grows at least exponentially with the radius. Hence the measure μ is not globally doubling. In this paper we define an atomic Hardy space H 1 (μ), where atoms are supported only on “small balls”, and a corresponding space BMO(μ) of functions of “bounded mean oscillation”, where the control is only on the oscillation over small balls. We prove that BMO(μ) is the dual of H 1 (μ) and that an inequality of John–Nirenberg type on small balls holds for functions in BMO(μ). Furthermore, we show that the L p (μ) spaces are intermediate spaces between H 1 (μ) and BMO(μ), 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 (μ)-L 1 (μ) and L (μ)-BMO(μ) estimates for various interesting operators on Riemannian manifolds and symmetric spaces which are unbounded on L 1 (μ) and on L (μ).
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},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 8},
     number = {3},
     year = {2009},
     pages = {543-582},
     zbl = {1180.42008},
     mrnumber = {2581426},
     language = {en},
     url = {http://www.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://www.numdam.org/item/ASNSP_2009_5_8_3_543_0/

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