Asymptotics, trace, and density results for weighted Dirichlet spaces defined on the halfline

We give analytic description for the completion of $C_0^{\mathrm{\infty }}({\mathbf{R}}_{+})$ in Dirichlet space $D^{1,p}({\mathbf{R}}_{+},\omega ):=\{u:{\mathbf{R}}_{+}\to \mathbf{R}:u\text{ is locally absolutely continuous on}{\mathbf{R}}_{+}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{\parallel }{u}^{{}^{\mathrm{\prime }}}{\mathrm{\parallel }}_{L^p({\mathbf{R}}_{+},\omega )}<\mathrm{\infty }\}$, for given continuous weight $\omega$, in terms of the local $B_p$ conditions due to Kufner and Opic, where $1<p<\mathrm{\infty }$. 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.