Weak type (1,1) for the rough maximal operator with merely integrable angular kernel

Determine whether the rough maximal operator \(\mathcal M_{\Omega}f(x)=\sup_{r>0}r^{-n}\int_{|y|<r}|f(x-y)|\,|\Omega(y/|y|)|\,dy\) is of weak type \((1,1)\) for every angular kernel \(\Omega\in L^1(\mathbb S^{n-1})\).

Background

The paper studies the rough maximal operator associated with a degree-zero homogeneous angular kernel $\Omega$ on the sphere. Earlier work of Christ and Rubio de Francia established weak type $(1,1)$ bounds under the stronger assumption $\Omega\in L\log^+L(\mathbb S^{n-1})$, while the paper proves such bounds for a larger Hausdorff–Choquet angular space that remains contained in $L^1(\mathbb S^{n-1})$.

The unresolved issue is whether the endpoint weak type estimate holds under the minimal size condition $\Omega\in L^1(\mathbb S^{n-1})$, without additional regularity, monotonicity, entropy, or Orlicz-integrability assumptions. The paper provides only a partial answer and does not settle the full $L^1$ problem.