Hausdorff--Choquet Angular Spaces and Weak-Type $(1,1)$ Bounds for Rough Maximal Operators
In the present paper, we consider the maximal operator \[ \mathcal M_{\Omega}f(x) := \sup_{r>0}\frac{1}{r^n} \int_{|y|<r} |f(x-y)| \left| \Omega\!\left(\frac{y}{|y|}\right) \right|\,dy. \] A longstanding open conjecture raised by E.~M.~Stein asks whether the maximal operator $\mathcal M_{\Omega}$ is of weak type $(1,1)$ when $\Omega$ is merely in $L^1(\mathbb S^{n-1})$. We partially settle this problem by proving weak type $(1,1)$ bounds of $\mathcal M_{\Omega}$ with kernel $\Omega\in\mathcal X(\mathbb S^{n-1})$, 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 $$ L\log^+\!L(\mathbb S^{n-1}) \subsetneq \mathcal X(\mathbb S^{n-1})\subset L^1(\mathbb S^{n-1}). $$ Finally, for the general Hausdorff--Choquet scale $\mathcal H\mathcal C_{\alpha}$, we show that $\alpha=(n-1)/2$ is the sharp exponent for uniform weak type $(1,1)$ estimates.