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

Abstract: 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 $\mathcal M_Ω$ is of weak type $(1,1)$ when $Ω$ is merely in $L^1(\mathbb S<sup>{n-1})$. We partially settle this problem by proving weak type $(1,1)$ bounds of $\mathcal M_Ω$ with kernel $Ω\in\mathcal X(\mathbb S<sup>{n-1})$, yielding a significant improvement over the work of M.~Christ and Rubio de Francia. Here $\mathcal X(\mathbb S<sup>{n-1})$ is the space related to the Hausdorff--Choquet angular space and $$L\log^+!L(\mathbb S^{n-1}) \subsetneq \mathcal X(\mathbb S^{{n-1})\subset} L^1(\mathbb S^{n-1}).$$ Finally, for the general Hausdorff--Choquet scale $\mathcal H\mathcal C_α$, we show that $α=(n-1)/2$ is the sharp exponent for uniform weak type $(1,1)$ estimates.