Let $\Omega \subseteq \mathbb{R}^n$ be open. For some $f \in L^2(\Omega)$ consider the continuous linear functional $$T \colon C^\infty_c(\Omega) \to \mathbb{R}, \qquad T(\varphi) := \int_\Omega f \varphi.$$ This functional can be uniquely extended to a continuous linear functional $$\overline{T} \colon L^2(\Omega) \to \mathbb{R}$$ with $\|\overline{T}\| = \|T\| = \|f\|_{L^2}$. This can be shown directly or by using the Riesz representation theorem. Now one can also extend $T$ to a continuous linear functional $$\overline{T} \colon H^1_0(\Omega) \to \mathbb{R}$$ with respect to the $H^1$-norm $$\|u\|_{H^1} := \|u\|_{L^2} + \|\nabla u\|_{L^2}.$$ Now it also holds that $\|\overline{T}\| = \|T\| \leq \|f\|_{L^2}$. But I think it is not necessarily true that $\|\overline{T}\| = \|T\| \geq \|f\|_{L^2}$ holds due to the inclusion of the first derivative in the $H^1$-norm. Is there a result about the operator norm of the functional $\overline{T}$ with respect to the $H^1$-norm? Of course, if $\Omega$ is bounded one might also ask for the operator norm of $\overline{T}$ with respect to the equivalent $H^1_0$-norm $$\|u\|_{H^1_0} := \|\nabla u\|_{L^2}$$ due to the Poincaré inequality.
Edit. Apparently, by the characterisation of $H^{-1}(\Omega) := H^1_0(\Omega)^*$ (see Evans: PDE p. 299), for every $\varphi \in H^{-1}(\Omega)$, there exist $f^0,\dots,f^n \in L^2(\Omega)$ with $$\varphi(u) = \int_\Omega f^0u dx + \sum_{i = 1}^n \int_\Omega f^i \partial_i u dx \qquad \forall u \in H^1_0(\Omega)$$ and $\|\varphi\|_{H^{-1}}$ is just the infimum over the sum of all the $L^2$-norms of the $f^i$. Thus in my case, $f^0 = f$ and $f^i = 0$ for the others does work, but only yields $\|T\| \leq \|f\|_{L^2}$ again.
