Annales Academiæ Scientiarum Fennicæ

Mathematica

Volumen 32, 2007, 13-26

*BMO* ON SPACES OF HOMOGENEOUS TYPE:
A DENSITY RESULT ON *C-C* SPACES

## A.O. Caruso and M.S. Fanciullo

Dipartimento di Matematica e Informatica, Università di Catania

Viale A. Doria 6 - I, 95125, Catania, Italy; aocaruso 'at' dmi.unict.it

Dipartimento di Matematica e Informatica, Università di Catania

Viale A. Doria 6 - I, 95125, Catania, Italy; fanciullo 'at' dmi.unict.it

**Abstract.**
In the general setting of a space of
homogeneous type endowed with an Ahlfors regular
measure, we introduce the Banach spaces *BMO* and *VMO*
defined through suitable cubes, and
we prove that these spaces are topologically equivalent to the standard ones
usually defined by means of balls. Through this fact we extend
a known result of Sarason showing that *C*^{\infty} is
locally dense in *VMO* in the setting of Carnot-Carathéodory metric spaces
related to
a family of free Hörmander vector fields
*X*_{1},...,*X*_{q}.

**2000 Mathematics Subject Classification:**
Primary 43A80, 46E30, 46E35, 54E35.

**Key words:**
VMO, spaces of homogeneous type,
Carnot-Carathéodory metric, Ahlfors regular measures.

**Reference to this article:** A.O. Caruso and M.S. Fanciullo:
*BMO* on spaces of homogeneous type: A density result on
*C-C* spaces.
Ann. Acad. Sci. Fenn. Math. 32 (2007), 13-26.

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