Proof of an extension of E. Sawyer's conjecture about weighted mixed weak-type estimates

Abstract

We show that if $v\in A_{\infty }$ and $u\in A_1$, then there is a constant $c$ depending on the $A_1$ constant of $u$ and the $A_{\infty }$ constant of $v$ such that $$\Vert \frac{T(fv)}{v}{\Vert }_{L^{1,\infty }(uv)}\le c\Vert f{\Vert }_{L^1(uv)},$$ where $T$ can be the Hardy-Littlewood maximal function or any Calderón-Zygmund operator. This result was conjectured in [IMRN, (30)2005, 1849--1871] and constitutes the most singular case of some extensions of several problems proposed by E. Sawyer and Muckenhoupt and Wheeden. We also improve and extends several quantitative estimates.

19 pages, submitted

References in corpus (3)

• Weighted norm inequalities for rough singular integral operators
• Weighted mixed weak-type inequalities for multilinear operators
• Three observations on commutators of Singular Integral Operators with BMO functions

Cited by in corpus (6)

• A class of multilinear bounded oscillation operators on measure spaces and applications
• A different approach to endpoint weak-type estimates for Calderón-Zygmund operators
• Sawyer-type inequalities for Lorentz spaces
• Weighted Lorentz spaces: sharp mixed $A_p-A_{\infty }$ estimate for maximal functions
• Two-weight extrapolation on function spaces and applications
• Boundary Value Problems in graph Lipschitz domains in the plane with $A_{\infty }$-measures on the boundary