Asymptotic structure, ℓp-estimates of sequences, and compactness of multilinear mappings

We relate the moduli of asymptotic uniform smoothness and convexity of a Banach space with the existence of upper and lower ℓp-estimates of sequences in the space. To this end, we introduce two properties which are related to the (mp)-property defined by Kalton and Werner. In this way we obtain a connection between the moduli of asymptotic uniform smoothness and convexity, and compactness or weak-sequential continuity of multilinear mappings. Finally, we give some applications to the existence of analytic and asymptotically flat norms on a Banach space.