Asymptotics, Trace, and Density Results for Weighted Dirichlet Spaces Defined on the Half-line

Abstract

AbstractWe give an analytic description for the completion of $$C_0^\infty (\mathbb {R}_+)$$ C 0 ∞ ( R + ) , where $$\mathbb {R}_+= (0,\infty )$$ R + = ( 0 , ∞ ) , in Dirichlet space $$D^{1,p}(\mathbb {R}_+, \omega ):= \{ u \, :\mathbb {R}_+\rightarrow {{\mathbb {R}}}: u\ $$ D 1 , p ( R + , ω ) : = { u : R + → R : u is locally absolutely continuous on $$\mathbb {R}_+\, and \, \Vert u^{'}\Vert _{L^p(\mathbb {R}_+, \omega )}<\infty \}$$ R + a n d ‖ u ′ ‖ L p ( R + , ω ) < ∞ } , for given continuous positive weight $$\omega $$ ω defined on $$\mathbb {R}_+$$ R + , where $$1<p<\infty $$ 1 < p < ∞ . The conditions are described in terms of the modified variants of the $$B_p$$ B p conditions due to Kufner and Opic from 1984, which in our approach are focusing on the integrability of $$\omega ^{-p/(p-1)}$$ ω - p / ( p - 1 ) near zero or near infinity. Moreover, we propose applications of our results to: obtaining new variants of Hardy inequality, interpretation of boundary value problems in ODE’s defined on the half-line with solutions in $$D^{1,p}(\mathbb {R}_+, \omega )$$ D 1 , p ( R + , ω ) , new results from complex interpolation theory dealing with interpolation spaces between weighted Dirichlet spaces, and for deriving new Morrey type embedding theorems for our Dirichlet space.