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

TL;DR

This paper characterizes the completion of smooth functions in weighted Dirichlet spaces on the halfline, providing insights into boundary behavior and applications to inequalities and boundary value problems.

Contribution

It offers an analytic description of the completion of smooth functions in weighted Dirichlet spaces using local Bp conditions, with applications to various analysis problems.

Findings

Characterization of the completion of $C_0^ abla$ in weighted Dirichlet spaces.

Application to Hardy inequalities and boundary value problems.

Derivation of new Morrey type inequalities.

Abstract

We give analytic description for the completion of $C_0^\infty ( \mathbf{R}_+)$ in Dirichlet space $D^{1,p}(\mathbf{R}_+, \omega):= \{ u:\mathbf{R}_+\rightarrow \mathbf{R}: u\ \hbox{ is locally absolutely continuous on} \ \mathbf{R}_+ \ {\rm and}\ \| u^{'}\|_{L^p(\mathbf{R}_+, \omega)}<\infty \}$, 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.