Actions of groups of diffeomorphisms on one-manifolds by $C^1$ diffeomorphisms

Denote by $\text{DC}(M{)}_0$ the identity component of the group of the compactly supported $C^r$ diffeomorphisms of a connected $C^{\mathrm{\infty }}$ manifold $M$. We show that if $\dim (M)\ge 2$ and $r\mathrm{\ne }\dim (M)+1$, then any homomorphism from $\text{DC}(M{)}_0$ to ${\text{Diff}}^1(\mathbb{R})$ or ${\text{Diff}}^1(S^1)$ is trivial.