$H^1$ and BMO for certain locally doubling metric measure spaces of finite measure

Andrea Carbonaro¹ ; Giancarlo Mauceri¹ ; Stefano Meda²

¹ Dipartimento di Matematica Università di Genova via Dodecaneso 35, 16146 Genova, Italy
² Dipartimento di Matematica e Applicazioni Università di Milano-Bicocca via R. Cozzi 53 20125 Milano, Italy

Colloquium Mathematicum, Tome 118 (2010) no. 1, pp. 13-41

Résumé

In a previous paper the authors developed an $H^1\hbox{-BMO}$ theory for unbounded metric measure spaces $(M,\rho,\mu)$ 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 $(\mathbb R^d,\rho_\varphi, \mu_\varphi)$, where ${\rm d}\mu_\varphi = {\rm e}^{-\varphi}\,{\rm d} x$ and $\rho_\varphi$ is the Riemannian metric corresponding to the length element ${\rm d} s^2=(1+|\nabla\varphi|)^2 ({\rm d} x_1^2 +\cdots+{\rm d} x_d^2)$. This generalizes previous work of the last two authors for the Gauss space.

Détail

DOI : 10.4064/cm118-1-2

Keywords: previous paper authors developed hbox bmo theory unbounded metric measure spaces rho infinite measure locally doubling satisfy geometric properties called approximate midpoint property isoperimetric property paper develop similar theory spaces finite measure prove results infinite measure have their counterparts finite measure finally theory applies class unbounded complete riemannian manifolds finite measure class metric measure spaces form mathbb rho varphi varphi where varphi varphi rho varphi riemannian metric corresponding length element nabla varphi cdots generalizes previous work authors gauss space