# Strong subdifferentiability and local Bishop-Phelps-Bollob\'as properties

## TL;DR

This paper investigates local Bishop-Phelps-Bollobás properties for multilinear mappings, highlighting differences from the uniform case and linking strong subdifferentiability to these properties in Banach spaces.

## Contribution

It extends local Bishop-Phelps-Bollobás properties to multilinear mappings, providing new examples and conditions related to strong subdifferentiability of Banach space norms.

## Key findings

- Differences between local and uniform Bishop-Phelps-Bollobás properties for multilinear mappings.
- Characterization of strong subdifferentiability of Banach space norms via bilinear mappings.
- Necessary and sufficient conditions for strong subdifferentiability in specific Banach spaces.

## Abstract

It has been recently presented some local versions of the Bishop-Phelps-Bollob\'as 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-Bollob\'as 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 $\ell_1^N \times Y$.