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

In the present paper, we consider the maximal operator $${\mathcal{M}}_{\mathrm{\Omega }}f(x):=\underset{r>0}{\sup }\frac{1}{r^n}{\int }_{\mathrm{\mid }y\mathrm{\mid }<r}\mathrm{\mid }f(x-y)\mathrm{\mid }\mid \mathrm{\Omega }\text{\hspace{0.17em}⁣}\left(\frac{y}{\mathrm{\mid }y\mathrm{\mid }}\right)\mid dy\mathrm{.}$$ A longstanding open conjecture raised by E.~M.~Stein asks whether the maximal operator ${\mathcal{M}}_{\mathrm{\Omega }}$ is of weak type $(1,1)$ when $\mathrm{\Omega }$ is merely in $L^1({\mathbb{S}}^{n-1})$. We partially settle this problem by proving weak type $(1,1)$ bounds of ${\mathcal{M}}_{\mathrm{\Omega }}$ with kernel $\mathrm{\Omega }\in \mathcal{X}({\mathbb{S}}^{n-1})$, yielding a significant improvement over the work of M.~Christ and Rubio de Francia. Here $\mathcal{X}({\mathbb{S}}^{n-1})$ is the space related to the Hausdorff–Choquet angular space and $$L{\log }^{+}\text{\hspace{0.17em}⁣}L({\mathbb{S}}^{n-1})⊊\mathcal{X}({\mathbb{S}}^{n-1})\subset L^1({\mathbb{S}}^{n-1})\mathrm{.}$$ Finally, for the general Hausdorff–Choquet scale $\mathcal{H}{\mathcal{C}}_{\alpha }$, we show that $\alpha =(n-1)\mathrm{/}2$ is the sharp exponent for uniform weak type $(1,1)$ estimates.