The Range of the Cesaro Operator Acting on H∞

In 1993, N. Danikas and A. G. Siskakis showed that the Cesaro operator C is not bounded on Ham; that is, C(H-infinity) not subset of H-infinity, but C(H-infinity) is a subset of BMOA. In 1997, M. Essen and J. Xiao gave that C(H-infinity) not subset of Q(p) for every 0 < p < 1. In this paper, we characterize positive Borel measures such that C(H-infinity') g M( DO and show that C(H-infinity) not subset of M( D mu(0)) not subset of boolean AND(0<p<,infinity) Qp by constructing some measures mu(0). Here, M(D-mu) denotes the Mobius invariant function space generated by D-mu, where D-mu is a Dirichlet space with superharmonic weight induced by a positive Borel measure mu on the open unit disk. Our conclusions improve results mentioned above.