Hausdorff--Choquet Angular Spaces and Weak-Type $(1,1)$ Bounds for Rough Maximal Operators

In the present paper, we consider the maximal operator \[ \mathcal M_Ωf(x) := \sup_{r>0}\frac{1}{r^n} \int_{|y|<r} |f(x-y)| \left| Ω\!\left(\frac{y}{|y|}\right) \right|\,dy. \] A longstanding open conjecture raised by E.~M.~Stein asks whether the maximal operator $\mathcal M_Ω$