Continuous operators from spaces of Lipschitz functions

We study the existence of continuous (linear) operators from the Banach spaces Lip0(M) of Lipschitz functions on infinite metric spaces M vanishing at a distinguished point and from their predual spaces F(M) onto certain Banach spaces, including C(K)-spaces and the spaces c0 and ℓ1. For pairs of spaces Lip0(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. It is also showed that if a metric space M contains a bilipschitz copy of the unit sphere Sc0 of the space c0, then Lip0(M) admits a continuous operator onto ℓ1 and hence onto c0. Using this, we provide several conditions for a space M implying that Lip0(M) is not a Grothendieck space. Finally, we obtain a new characterization of the Schur property for Lipschitz-free spaces: a space F(M) has the Schur property if and only if for every complete discrete metric space N with cardinality d(M) the spaces F(M) and F(N) are weakly sequentially homeomorphic.