Let X be a pointed metric space and let E be a Banach space. It is known that the Lipschitz space Lip 0 (X, E * ) is isometrically isomorphic to (F(X) ⊗ π E) * , the dual of the projective tensor product of the Lipschitz-free space F(X) and E. Since the injective norm ε on F(X) ⊗ E is smaller than the projective norm π, we study Lipschitz Grothendieck-integral operators which are exactly those elements in Lip 0 (X, E * ) which correspond to the elements of (F(X) ⊗ ε E) * , the dual of the injective tensor product of F(X) and E.

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There are known results showing a canonical association between Lipschitz cross-norms (norms on the Lipschitz tensor product of a metric space and a Banach space) and ideals of Lipschitz maps from a metric space to a dual Banach space. We extend this association, relating Lipschitz cross-norms to ideals of Lipschitz maps taking values in general Banach spaces. To do that, we prove a Lipschitz version of the representation theorem for maximal operator ideals. As a consequence, we obtain linear characterizations of some ideals of (nonlinear) Lipschitz maps between metric spaces.

We develop a systematic approach to the study of duality for ideals of Lipschitz maps from a metric space to a Banach space, inspired by the classical theory that relates ideals of operators and tensor norms for Banach spaces, by using the Lipschitz tensor products previously introduced by the same authors. We first study spaces of Lipschitz maps, from a metric space to a dual Banach space, that can be represented canonically as the dual of a Lipschitz tensor product endowed with a Lipschitz cross-norm. We show that several known examples of ideals of Lipschitz maps (Lipschitz maps, Lipschitz $p$-summing maps and maps admitting a Lipschitz factorization through a subset of an $L_p$ space) admit such a representation, and more generally we characterize when a space of Lipschitz maps from a metric space to a dual Banach space is in canonical duality with a Lipschitz cross-norm. Furthermore, we give conditions on the Lipschitz cross-norm that are almost equivalent to the space of Lipschitz maps having an ideal property. We introduce a concept of operators which are approximable with respect to one of these ideals of Lipschitz maps, and identify them in terms of tensor-product notions. Finally, we also prove a Lipschitz version of the representation theorem for maximal operator ideals. This allows us to relate Lipschitz cross-norms to ideals of Lipschitz maps taking values in general Banach spaces, and not just dual spaces.Comment: 26 page