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

We give analytic description for the completion of C?0 (R+) in Dirichletspace D1,p(R+, ?) := {u : R+ -> R : u is locally absolutely continuous on R+ and ||u? ||_Lp(R+,?) < ?}, for given continuouspositive weight ? defined on R+, where 1 < p < ?. The conditions are described in terms of the modified variants of the Bpconditions due to Kufner and Opic from 1984, which in our approach are focusing on integrability of ?^-p/(p-1) near zero or near infinity. Moreover, we propose applications of our results to: obtaining newvariants of Hardy inequality, interpretation of boundary value problems in ODE's defined on the halpfline with solutions in D1,p(R+, ?),new results from complex interpolation theory dealing with interpolation spaces between weighted Dirichlet spaces, and to derivationof new Morrey type embedding theorems for our Dirichlet space.