This is an announcement for the paper “On strongly norm attaining Lipschitz operators” by Bernardo Cascaleshttps://arxiv.org/search/math?searchtype=author&query=Cascales%2C+B, Rafa Chiclanahttps://arxiv.org/search/math?searchtype=author&query=Chiclana%2C+R, Luis García-Lirolahttps://arxiv.org/search/math?searchtype=author&query=Garc%C3%ADa-Lirola%2C+L, Miguel Martínhttps://arxiv.org/search/math?searchtype=author&query=Mart%C3%ADn%2C+M, Abraham Rueda Zocahttps://arxiv.org/search/math?searchtype=author&query=Zoca%2C+A+R.
Abstract: We study the set $\SA(M,Y)$ of those Lipschitz operators from a (complete pointed) metric space $M$ to a Banach space $Y$ which (strongly) attain their Lipschitz norm (i.e.\ the supremum defining the Lipschitz norm is a maximum). Extending previous results, we prove that this set is not norm dense when $M$ is length (or local) or when $M$ is a closed subset of $\R$ with positive Lebesgue measure, providing new example which have very different topological properties than the previously known ones. On the other hand, we study the linear properties which are sufficient to get Lindenstrauss property A for the Lipschitz-free space $\mathcal{F}(M)$ over $M$, and show that all of them actually provide the norm density of $\SA(M,Y)$ in the space of all Lipschitz operators from $M$ to any Banach space $Y$. Next, we prove that $\SA(M,\R)$ is weak sequentially dense in the space of all Lipschitz functions for all metric spaces $M$. Finally, we show that the norm of the bidual space to $\mathcal{F}(M)$ is octahedral provided the metric space $M$ is discrete but not uniformly discrete or $M'$ is infinite.
The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1807.03363