Some essential properties of Q(p)(partial derivative Delta)-spaces

For p epsilon (-infinity, infinity), let Q(p)(partial derivative Delta) be the space of all complex-valued functions f on the unit circle partial deri vative Delta satisfying sup I subset of partial derivative Delta \I\-P integral I integral I \f(z) - f)w)\2 / \z - w\2-p \dz\\dw\ < infinity, where the supremum is taken over all subarcs I subset of partial derivative Delta with the arclength \I\. In this paper we consider some essential pro perties of Q(p)(partial derivative Delta). We first show that if p > 1, the n Q(p)(partial derivative Delta) = BMO(partial derivative Delta), the space of complex-valued functions with bounded mean oscillation on aa. Second, w e prove that a function belongs to Qp(partial derivative Delta) if and only if ii is Mobius bounded in the Sobolev space L-p(2)(partial derivative Del ta). Finally, a characterization of Qp(partial derivative Delta) is given v ia wavelets.