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 ₁. 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. We show that, given a Banach space E, there exists a continuous operator from a Lipschitz-free space F (M) onto E if and only if F (M) contains a subset homeomorphic to E if and only if d (M) d (E). We obtain a new characterization of the Schur property for spaces F (M): a space F (M) has the Schur property if and only if for every discrete metric space N with cardinality d (M) the spaces F (M) and F (N) are weakly sequentially homeomorphic. It is also showed that if a metric space M contains a bilipschitz copy of the unit sphere S₂䃐 of the space c₀, then Lip₀ (M) admits a continuous operator onto ₁ and hence onto c₀. We provide several conditions for a space M implying that Lip₀ (M) is not a Grothendieck space.