Composition operators on Hardy-Sobolev spaces with bounded reproducing kernels

For any real $\beta$ let $H_{\beta }^2$ be the Hardy-Sobolev space on the unit disc $\mathbb{D}$. $H_{\beta }^2$ is a reproducing kernel Hilbert space and its reproducing kernel is bounded when $\beta >1/2$. In this paper, we prove that $C_{\phi }$ has dense range in $H_{\beta }^2$ if and only if the polynomials are dense in a certain Dirichlet space of the domain $\phi (\mathbb{D})$ for $1/2<\beta <1$. It follows that if the range of $C_{\phi }$ is dense in $H_{\beta }^2$, then $\phi$ is a weak-star generator of $H^{\mathrm{\infty }}$, although the conclusion is false for the classical Dirichlet space $\mathfrak{D}$. Moreover, we study the relation between the density of the range of $C_{\phi }$ and the cyclic vector of the multiplier $M_{\phi }^{\beta }.$