Let $g \in H^1(\mathbb{R}^n)$, we may define $\Delta g$ as an element of $H^{-1}$ by \begin{equation} \langle \Delta g, f \rangle_{H^{-1} \times H^1}:= -\langle \nabla g, \nabla f \rangle_{L^2} \end{equation}
Then, I see that \begin{equation} \lvert \langle \Delta g, f \rangle_{H^{-1} \times H^1} \rvert \leq \lVert \nabla g \rVert_{L^2} \lVert \nabla f \rVert_{L^2} \leq \lVert \nabla g \rVert_{L^2} \lVert f \rVert_{H^1} \end{equation} implying that \begin{equation} \lVert \Delta g \rVert_{H^{-1}} = \lVert \nabla g \rVert_{L^2} \leq \lVert g \rVert_{H^1} \end{equation}
Could anyone please confirm that this is indeed correct? It is kind of surprising that a distribution can be bounded by the function from which it is derived..