Singular integral characterization of nonisotropic generalized BMO spaces

summary:We extend a result of Coifman and Dahlberg [{\it Singular integral characterizations of nonisotropic $H^p$ spaces and the F. and M. Riesz theorem\/}, Proc. Sympos. Pure Math., Vol. 35, pp.\,231--234; Amer. Math. Soc., Providence, 1979] on the characterization of $H^p$ spaces by singular integrals of $\Bbb R^n$ with a nonisotropic metric. Then we apply it to produce singular integral versions of generalized BMO spaces. More precisely, if $T_\lambda$ is the family of dilations in $\Bbb R^n$ induced by a matrix with a nonnegative eigenvalue, then there exist $2n$ singular integral operators homogeneous with respect to the dilations $T_\lambda$ that characterize BMO$_\varphi$ under a natural condition on $\varphi$