On the weak maximizing properties

Quite recently, a new property related to norm-attaining operators has been introduced: the weak maximizing property (WMP). In this note, we define a generalised version of it considering other topologies than the weak one (mainly the $$\hbox {weak}^*$$ topology). We provide new sufficient conditions, based on the moduli of asymptotic uniform smoothness and convexity, which imply that a pair (X, Y) enjoys a certain maximizing property. This approach not only allows us to (re)obtain as a direct consequence that the pair $$(\ell _p,\ell _q)$$ has the WMP but also provides many more natural examples of pairs having a given maximizing property.