Sawyer's mixed weak-type conjecture for two A1A_1 weights

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Let MM be the Hardy–Littlewood maximal operator on Rn\mathbb{R}^n, and let u,v∈A1u,v\in A_1. Sawyer's mixed weak-type conjecture. There exists a dimensional constant cnc_n such that

∥M(fv)v∥L1,∞(uv)≤cn[u]A1[v]A1∥f∥L1(uv).\Big\|\frac{M(fv)}{v}\Big\|_{L^{1,\infty}(uv)}\le c_n [u]_{A_1}[v]_{A_1}\|f\|_{L^1(uv)}.

This is the A1A_1-weight case of the mixed weak-type estimate discussed in the source and is attributed there to O. P. and R. The parser supplies no evidence that it has been resolved.

References

Primary source

Kangwei Li, Sheldy Ombrosi and Carlos Pérez, “Proof of an extension of E. Sawyer's conjecture about weighted mixed weak-type estimates”, arXiv:1703.01530 (2017).

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