On Hardy and Hardy-Littlewood transforms in classes of functions with given majorant of modulus of continuity

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