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

Abstract

We give analytic description for the completion of $C_0^{\infty }({\mathbf{R}}_{+})$ in Dirichlet space $D^{1,p}({\mathbf{R}}_{+},\omega ):=\{u:{\mathbf{R}}_{+}\to \mathbf{R}:uis\; locally\; absolutely\; continuous\; on{\mathbf{R}}_{+}\mathrm{and}\Vert u^{{}^{\prime }}{\Vert }_{L^p({\mathbf{R}}_{+},\omega )}<\infty \}$, 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.