  In this paper we mainly discuss three things. First, there is no canonical
norm on the space $H^p_u(\mathbb{D})$. Second, we improve the weak-$*$
convergence of the measures $\mu_{u,r}$. Third, the dilations $f_t$ of the
function $f\in H^p_u(\mathbb{D})$ converge to $f$ in $H^p_u$-norm and hence the
polynomials are dense in $H^p_u(\mathbb{D})$.
