Abstract. From the works of D.$ $V. Giang and F. Móricz (see [5]) and B.$ $I. Golubov (see [7]) it follows that the Hardy--Littlewood operator ${\cal B}(f)(x)=x^{-1}\int ^x_0f(t) dt$, $x\not=0$, is bounded on $BMO({\msbm R})$. We prove that ${\cal B}$ is also bounded on $VMO({\msbm R})$ and that the generalized Lipschitz classes $H^{\omega }_X({\msbm R})$ under additional conditions are invariant with respect to the operator ${\cal B}$. A direct approximation theorem for $VMO({\msbm R})$ is also obtained.

AMS Subject Classification (1991): 44A15, 47B38, 41A17

Keyword(s): Hardy--Littlewood operator, generalized Lipschitz classes, real Hardy space, functions of vanishing mean oscillation, direct approximation theorem

Received May 19, 2008, and in revised form October 29, 2008. (Registered under 6073/2009.)