Singular integral characterization of nonisotropic generalized BMO spaces

We extend a result of Coifman and Dahlberg [Singular integral characterizations of nonisotropic H 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 spaces by singular integrals of R with a nonisotropic metric. Then we apply it to produce singular integral versions of generalized BMO spaces. More precisely, if Tλ is the family of dilations in R induced by a matrix with a nonnegative eigenvalue, then there exist 2n singular integral operators homogeneous with respect to the dilations Tλ that characterize BMOφ under a natural condition on φ.