$H^{\bf 1}$ and $BMO$ for certain locally doubling metric measure spaces

Carbonaro, Andrea; Mauceri, Giancarlo; Meda, Stefano

H^{1}

and

B M O

for certain locally doubling metric measure spaces

Carbonaro, Andrea; Mauceri, Giancarlo; Meda, Stefano

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 8 (2009) no. 3, p. 543-582

Abstract

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 $B M O (μ)$ of functions of “bounded mean oscillation”, where the control is only on the oscillation over small balls. We prove that $B M O (μ)$ is the dual of $H^{1} (μ)$ and that an inequality of John–Nirenberg type on small balls holds for functions in $B M O (μ)$ . Furthermore, we show that the $L^{p} (μ)$ spaces are intermediate spaces between $H^{1} (μ)$ and $B M O (μ)$ , 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 $L^{\infty} (μ)$ - $B M O (μ)$ estimates for various interesting operators on Riemannian manifolds and symmetric spaces which are unbounded on $L^{1} (μ)$ and on $L^{\infty} (μ)$ .

MR 2581426 | Zbl 1180.42008

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, Serie 5, Volume 8 (2009) no. 3, pp. 543-582. http://www.numdam.org/item/ASNSP_2009_5_8_3_543_0/

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