# H¹ and BMO for certain locally doubling metric measure spaces of finite measure

Andrea Carbonaro; Giancarlo Mauceri; Stefano Meda

Colloquium Mathematicae (2010)

- Volume: 118, Issue: 1, page 13-41
- ISSN: 0010-1354

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topAndrea Carbonaro, Giancarlo Mauceri, and Stefano Meda. "H¹ and BMO for certain locally doubling metric measure spaces of finite measure." Colloquium Mathematicae 118.1 (2010): 13-41. <http://eudml.org/doc/284037>.

@article{AndreaCarbonaro2010,

abstract = {In a previous paper the authors developed an H¹-BMO theory for unbounded metric measure spaces (M,ρ,μ) of infinite measure that are locally doubling and satisfy two geometric properties, called “approximate midpoint” property and “isoperimetric” property. In this paper we develop a similar theory for spaces of finite measure. We prove that all the results that hold in the infinite measure case have their counterparts in the finite measure case. Finally, we show that the theory applies to a class of unbounded, complete Riemannian manifolds of finite measure and to a class of metric measure spaces of the form $(ℝ^\{d\},ρ_\{φ\}, μ_\{φ\})$, where $dμ_\{φ\} = e^\{-φ\} dx$ and $ρ_\{φ\}$ is the Riemannian metric corresponding to the length element $ds² = (1+|∇φ|)² (dx₁² + ⋯ + dx²_\{d\})$. This generalizes previous work of the last two authors for the Gauss space.},

author = {Andrea Carbonaro, Giancarlo Mauceri, Stefano Meda},

journal = {Colloquium Mathematicae},

keywords = {atomic Hardy space; BMO; singular integrals; Riemannian manifolds Riesz transform},

language = {eng},

number = {1},

pages = {13-41},

title = {H¹ and BMO for certain locally doubling metric measure spaces of finite measure},

url = {http://eudml.org/doc/284037},

volume = {118},

year = {2010},

}

TY - JOUR

AU - Andrea Carbonaro

AU - Giancarlo Mauceri

AU - Stefano Meda

TI - H¹ and BMO for certain locally doubling metric measure spaces of finite measure

JO - Colloquium Mathematicae

PY - 2010

VL - 118

IS - 1

SP - 13

EP - 41

AB - In a previous paper the authors developed an H¹-BMO theory for unbounded metric measure spaces (M,ρ,μ) of infinite measure that are locally doubling and satisfy two geometric properties, called “approximate midpoint” property and “isoperimetric” property. In this paper we develop a similar theory for spaces of finite measure. We prove that all the results that hold in the infinite measure case have their counterparts in the finite measure case. Finally, we show that the theory applies to a class of unbounded, complete Riemannian manifolds of finite measure and to a class of metric measure spaces of the form $(ℝ^{d},ρ_{φ}, μ_{φ})$, where $dμ_{φ} = e^{-φ} dx$ and $ρ_{φ}$ is the Riemannian metric corresponding to the length element $ds² = (1+|∇φ|)² (dx₁² + ⋯ + dx²_{d})$. This generalizes previous work of the last two authors for the Gauss space.

LA - eng

KW - atomic Hardy space; BMO; singular integrals; Riemannian manifolds Riesz transform

UR - http://eudml.org/doc/284037

ER -

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