For n≥ 2 and every measurable angular kernel Omega∈ L^1( S^n-1), define M_Omegaf(x)=sup_r>0frac1r^nint_|y| 0, |x∈ R^n: M_Omegaf(x)>lambda|≤ fracC_n|Omega|_L^1( S^n-1)|f|_L^1( R^n)lambda?

Stein’s weak-type (1,1) conjecture for rough maximal operators

For $n\ge 2$ and every measurable angular kernel $\Omega\in L^1(\mathbb S^{n-1})$ , define $\mathcal M_{\Omega}f(x)=\sup_{r>0}\frac{1}{r^n}\int_{|y|<r}|f(x-y)|\left|\Omega\!\left(\frac{y}{|y|}\right)\right|\,dy$ . Does there exist a constant $C_n$ , depending only on $n$ , such that for every $f\in L^1(\mathbb R^n)$ and every $\lambda>0$ , $\left|\left\{x\in\mathbb R^n:\mathcal M_{\Omega}f(x)>\lambda\right\}\right|\le \frac{C_n\|\Omega\|_{L^1(\mathbb S^{n-1})}\|f\|_{L^1(\mathbb R^n)}}{\lambda}$ ?

References

Primary source

arXiv

Additional references

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

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed progress

A September 2026 paper reportedly extends the known estimate to a broader class of rough kernels, but the full conjecture remains open.

Stein’s conjecture concerns a weak-type $(1,1)$ estimate for rough maximal operators. The case of every $L^1$ angular kernel remains unresolved.

Known results

Weak-type $(1,1)$ bounds were known for angular kernels in $L\log L(\mathbb{S}^{n-1})$ .
A 2021 result established limiting weak-type behavior and improved bounds in that $L\log L$ setting, without proving the full $L^1$ conjecture.
Related maximal truncated rough singular integrals have endpoint results weaker than weak type $(1,1)$ , including an $L\log\log L$ bound.

September 2026 claimed extension

A paper dated September 2026 claims weak-type $(1,1)$ bounds for angular kernels in the larger space $X(\mathbb{S}^{n-1})$ and identifies $(n-1)/2$ as the sharp exponent for uniform bounds on the associated Hausdorff–Choquet scale. This is substantial claimed progress, but the full $L^1$ conjecture is not claimed to be solved and the development is unverified here.

Current status (as of September 2026): The conjecture is claimed to hold for the larger class $X(\mathbb{S}^{n-1})$ , while the general $L^1$ angular-kernel case remains open.

Sources

No solutions have been posted yet.