Extension of Lipschitz p-Compact Operators in Metric Spaces, New Results on Finite-Dimensional Extensions

Resumo

This paper presents a comprehensive study of extension properties for Lipschitz \(p\)-compact operators acting between metric spaces. The linearization technique via Lipschitz-free spaces \(\F(X)\) allows the transfer of problems from the nonlinear metric setting to the well-developed theory of \(p\)-compact linear operators on Banach spaces. We establish several fundamental results concerning the extendability of such operators to larger domains while preserving \(p\)-compactness properties. Our main contributions include: (i) a corrected and fully rigorous proof of the finite-dimensional extension theorem for Lipschitz \(p\)-compact operators, addressing critical gaps in previous arguments; (ii) extension theorems for operators with values in \(P_\lambda^L\)-spaces under optimal hypotheses; (iii) preservation of dual \(p\)-compactness under extensions via pre-adjoint techniques, with complete proofs using Pietsch factorization; (iv) a characterization of \(L_1\)-preduals through local extension properties of Lipschitz \(p\)-compact operators; and (v) uniform boundedness principles and the equivalence of approximate and genuine extensions. The theory is developed using the machinery of Lipschitz-free spaces and builds upon recent advances in the theory of \(p\)-compact operators and their Lipschitz counterparts.