Let $g_n, \phi \in \dot{H}^1(\mathbb R^d)= \{f: \nabla f \in L^{2}(\mathbb R^d) \}.$
Assume that $\int_{\mathbb R^d} \nabla g_n \cdot \nabla h \ dx \to \int_{\mathbb R^d} \nabla \phi \cdot \nabla h \ dx$ as $n\to \infty$ for all $h\in \dot{H}^1.$ (In other words, $g_n$ converges to $\phi$ weakly in $\dot{H}^1$.)
Question: Can we expect to find $\psi \in \mathcal{S}(\mathbb R^d)$ (Schwartz Space) so that $$\lim_{n\to \infty} \int_{\mathbb R^d} g_{n}(x) \psi(x) dx = \int_{\mathbb R^d} \phi(x) \psi (x)dx?$$
Side Note: Let $\phi$ be a radial bump function supported on $\{ \xi: |\xi|\leq 2 \}$ which equals to 1 on $\{\xi: |\xi|\leq 1\}.$ Put $h(\xi)= \phi(\xi)- \phi(2\xi),$ and $h_{1}(\xi)= h(\xi/2).$ Take $\psi(x)= (h_{1})^{\vee}(x)$ (Inverse Fourier transform of $h_1$) (Is this the right choice?)