Abstract:We extend a result of Coifman and Dahlberg [{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 $.
Keywords: singular integral, nonisotropic generalized BMO
AMS Subject Classification: Primary 42B30; secondary 42B99