On a Class of Functionals on a Weighted First-Order Sobolev Space on the Real Line

Let $g$ be a Lebesgue measurable function on an interval $I\subset \mathbb R$. We find conditions on $g$ under which the mapping $f\mapsto \int _I g(x)(Df)(x)\,dx$ is a continuous linear functional on a weighted first-order Sobolev space $W_{p,p}^1(I)$; we also obtain estimates for the norm of this functional in $[W_{p,p}^1(I)]^*$.