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

Let MM be the Hardy–Littlewood maximal operator on Rn\mathbb{R}^n, and let u,vA1u,v\in A_1. Sawyer's mixed weak-type conjecture. There exists a dimensional constant cnc_n such that

M(fv)vL1,(uv)cn[u]A1[v]A1fL1(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.

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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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