Does $\Delta u\in H^{-1}(M)$ imply $u\in H^{1}(M)$? Suppose that $M$ is a compact smooth Riemannian Manifold, Can we conclude that the above question? I just know there is an opposite question $u\in H^1(\Omega)$ implies $\Delta u\in H^{-1}(\Omega)$?. Thank you for your reading!!!
My attemption: $\forall\varphi\in H^{1}(M)$, then we have $\|\Delta u\|_{H^{-1}}\|\varphi\|_{H^{1}}\geq\langle\Delta u,\varphi\rangle_{H^{-1},H^{1}}=\langle\nabla u,\nabla\varphi\rangle$, what should we do next?