Composition operators on Hardy-Sobolev spaces with bounded reproducing kernels

For any real 8 let H82 be the Hardy-Sobolev space on the unit disc D. H82 is a reproducing kernel Hilbert space and its reproducing kernel is bounded when 8 > 1/2. In this paper, we prove Cw has dense range in H82 if and only if the polynomials are dense in a certain Dirichlet space domain w(D) for 1/2 < 8 < 1. It follows that if the range of Cw is dense in H82, then w is a weak generator of H03, although the conclusion is false for the classical Dirichlet space D. Moreover, study the relation between the density of the range of Cw and the cyclic vector of the multiplier M8w.