We will discuss a conjecture of Zygmund concerning maximal operators defined on a family of axis parallel rectangles in the Euclidean space. If the historical version of the problem has been disproved by Soria, we will see that the idea behind Zygmund's conjecture may still be true.

In particular, a certain reformulation of the problem has been solved in the Euclidean plane by Stokolos but it remains open in higher dimensions. In the past fews years, different authors (among which D'Aniello, Hagelstei, Oniani, Moonens, Rey, Stokolos etc.) have established sharp weak type estimates in specific settings and their work lend weight to a certain reformulation of Zygmund's conjecture.

We will discuss this problem and in particular, I would like to focus on a specific family of rectangles that exhibits a product structure.