Let $(M,g)$ be a Riemannian manifold. Consider $A : W^{1,r}(M,\mathbb{R}) \rightarrow W^{-1,r'}(M,\mathbb{R}), k \mapsto Ak$, where $Ak$ is defined by $(Ak)(\varphi) = \int_{M}g(\nabla k, \nabla \varphi) \text{vol}, \varphi \in W^{1,r'}(M,\mathbb{R})$ and $\frac{1}{r} + \frac{1}{r'} = 1$. Does there exist an inequality of the form: $\exists C>0$ such that $||Au||_{W^{-1,r'}} \geq C||u||_{W^{1,r}}, \forall u \in W^{1,r}(M,\mathbb{R})$ ?
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