  We give analytic description for the completion of $C_0^\infty (
\mathbf{R}_+)$ in Dirichlet space $D^{1,p}(\mathbf{R}_+, \omega):= \{
u:\mathbf{R}_+\rightarrow \mathbf{R}: u\ \hbox{ is locally absolutely
continuous on} \ \mathbf{R}_+ \ {\rm and}\
  \| u^{'}\|_{L^p(\mathbf{R}_+, \omega)}<\infty \}$, for given continuous
weight $\omega$, in terms of the local $B_p$ conditions due to Kufner and Opic,
where $1<p<\infty$. Moreover, we propose applications of our results to:
analysis of Hardy inequalities, boundary value problems, complex interpolation
theory, and to derivation of new Morrey type inequalities.
