Strong subdifferentiability and local Bishop-Phelps-Bollobás properties

Abstract

It has been recently presented some local versions of the Bishop-Phelps-Bollobás type property for operators. In the present article, we continue studying these properties for multilinear mappings. We show some differences between the local and uniform versions of the Bishop-Phelps-Bollobás type results for multilinear mappings, and also provide some interesting examples which shows that this study is not just a mere generalization of the linear case. We study those properties for bilinear forms on ${\ell }_p\times {\ell }_q$ using the strong subdifferentiability of the norm of the Banach space ${\ell }_p{\hat{\otimes }}_{\pi }{\ell }_q$. Moreover, we present necessary and sufficient conditions for the norm of a Banach space $Y$ to be strongly subdifferentiable through the study of these properties for bilinear mappings on ${\ell }_1^N\times Y$.