Continuous operators from spaces of Lipschitz functions

We study the existence of continuous (linear) operators from the Banach spaces $\mathrm{Lip}_0(M)$ of Lipschitz functions on infinite metric spaces $M$ vanishing at a distinguished point and from their predual spaces $\mathcal{F}(M)$ onto certain Banach spaces, including $C(K)$-spaces and the spaces $c_0$ and $\ell_1$. For pairs of spaces $\mathrm{Lip}_0(M)$ and $C(K)$ we prove that if they are endowed with topologies weaker than the norm topology, then usually no continuous (linear or not) surjection exists between those spaces. We show that, given a Banach space $E$, there exists a continuous operator from a Lipschitz-free space $\mathcal{F}(M)$ onto $E$ if and only if $\mathcal{F}(M)$ contains a subset homeomorphic to $E$ if and only if $d(M)\ge d(E)$. We obtain a new characterization of the Schur property for spaces $\mathcal{F}(M)$: a space $\mathcal{F}(M)$ has the Schur property if and only if for every discrete metric space $N$ with cardinality $d(M)$ the spaces $\mathcal{F}(M)$ and $\mathcal{F}(N)$ are weakly sequentially homeomorphic. It is also showed that if a metric space $M$ contains a bilipschitz copy of the unit sphere $S_{c_0}$ of the space $c_0$, then $\mathrm{Lip}_0(M)$ admits a continuous operator onto $\ell_1$ and hence onto $c_0$. We provide several conditions for a space $M$ implying that $\mathrm{Lip}_0(M)$ is not a Grothendieck space.