Hausdorff operators of special kind in $BMO$-type spaces and Hölder–Lipschitz spaces

We show that some $BMO$-type spaces are invariant with respect to the Hausdorff operators of special kind (weighted Hardy-Cesaro operators). Also we obtain a sufficient and necessary conditions for such operators to be bounded in spaces of functions of generalized bounded variation. Finally, we study the invariance of Hölder–Lipschitz spaces with respect to these operators.