For every 1<p<∞ and every integer d≥ 3, does the natural two-weight Muckenhoupt condition A_p,d^+(w,v)<∞ characterize the weak-type estimate for the one-sided Hardy--Littlewood maximal operator M_d^+ on R^d, namely |M_d^+|_L^p(v)→ L^p,∞(w)<∞? Equivalently, is A_p,d^+(w,v)<∞ sufficient for this weak-type bound for all weights w and v? The cited source claims that the answer is negative: for every 1<p<∞ and d≥ 3, there exist weights w and v such that A_p,d^+(w,v)<∞ but |M_d^+|_L^p(v)→ L^p,∞(w)=∞.

Two-weight problem for one-sided Hardy–Littlewood maximal operators

For every $1<p<\infty$ and every integer $d\ge 3$ , does the natural two-weight Muckenhoupt condition $A_{p,d}^+(w,v)<\infty$ characterize the weak-type estimate for the one-sided Hardy--Littlewood maximal operator $M_d^+$ on $\mathbb{R}^d$ , namely $\|M_d^+\|_{L^p(v)\to L^{p,\infty}(w)}<\infty$ ? Equivalently, is $A_{p,d}^+(w,v)<\infty$ sufficient for this weak-type bound for all weights $w$ and $v$ ? The cited source claims that the answer is negative: for every $1<p<\infty$ and $d\ge 3$ , there exist weights $w$ and $v$ such that $A_{p,d}^+(w,v)<\infty$ but $\|M_d^+\|_{L^p(v)\to L^{p,\infty}(w)}=\infty$ .

References

Primary source

arXiv

Additional references

On the Two-Weight Problem for One-Sided Maximal Operators in Higher Dimensions — arXiv

harmonic-analysis weighted-inequalities

Posted 4 weeks ago

Progress summary

Refreshed 4 weeks ago

Claimed solved

A new report claims the proposed rule fails in three or more dimensions, while the two-dimensional case is already understood.

The problem asks for a characterization of the weight pairs governing weak-type bounds for the non-dyadic one-sided maximal operator. The two-dimensional theory is known; the corresponding question in dimensions $n \ge 3$ was previously open.

Known results

Forzani, Martín-Reyes, and Ombrosi characterized the two-weight weak-type problem in $\mathbb{R}^2$ ; their geometric argument was not known to extend to $n \ge 3$ .

September 2026 claimed disproof

A report dated September 8, 2026, says the proposed characterization is disproved in every dimension $n \ge 3$ , obstructing a direct extension of the one- and two-dimensional theory. This claim is not independently verified in the retrieved material.

Current status (as of September 2026): The two-dimensional characterization is established, while the proposed characterization for $n \ge 3$ is claimed to be false but remains unverified.

Sources

No solutions have been posted yet.