Can $f\in L^2(\Omega)$ imply $\nabla f\in [(H^1(\Omega))^*]^n$?

Question

This is closely related to a previous question of mine. The only difference is the definition of $H^{-1}(\Omega).$

Suppose $f\in L^2(G_R)$ where $$ G_R=\{x\in\mathbb{R}^n\mid x_n>0,|x|<R\}\quad\textrm{or }\quad G_R=\{x\in\mathbb{R}^n\mid |x|<R\}. $$ Then $\nabla f\in H^{-1}(G_R)^n$ (component-wise), where $H^{-1}(G_R)$ is defined as the dual of $\color{red}{H^1(G_R)}$.

Would anybody show me whether this is true or not?