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 ${\mathcal{M}}_{\Omega }$ 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 $$L{\log }^{+}L({\mathbb{S}}^{n-1})⊊\mathcal{X}({\mathbb{S}}^{n-1})\subset L^1({\mathbb{S}}^{n-1}).$$ Prior to our work, the condition $\Omega \in L{\log }^{+}L({\mathbb{S}}^{n-1})$ remained the best known size restriction guaranteeing the weak-type$(1,1)$ bounds for all dimensions $n>2$. To our knowledge, the present paper is the first to relax this threshold for $n>2$. 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.

24 pages