Abstract
The Hardy-Littlewood maximal operator M satisfies the classical Sawyer-type estimate MfvL1,∞(uv)≤Cu,v‖f‖L1(u),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left\\| \\frac{Mf}{v}\\right\\| _{L^{1,\\infty }(uv)} \\le C_{u,v} \\Vert f \\Vert _{L^{1}(u)}, \\end{aligned}$$\\end{document}where uin A_1 and uvin A_{infty }. We prove a novel extension of this result to the general restricted weak type case. That is, for p>1, uin A_p^{{mathcal {R}}}, and uv^p in A_infty , MfvLp,∞(uvp)≤Cu,v‖f‖Lp,1(u).\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left\\| \\frac{Mf}{v}\\right\\| _{L^{p,\\infty }(uv^p)} \\le C_{u,v} \\Vert f \\Vert _{L^{p,1}(u)}. \\end{aligned}$$\\end{document}From these estimates, we deduce new weighted restricted weak type bounds and Sawyer-type inequalities for the m-fold product of Hardy-Littlewood maximal operators. We also present an innovative technique that allows us to transfer such estimates to a large class of multi-variable operators, including m-linear Calderón-Zygmund operators, avoiding the A_infty extrapolation theorem and producing many estimates that have not appeared in the literature before. In particular, we obtain a new characterization of A_p^{{mathcal {R}}}. Furthermore, we introduce the class of weights that characterizes the restricted weak type bounds for the multi(sub)linear maximal operator {mathcal {M}}, denoted by A_{mathbf {P}}^{{mathcal {R}}}, establish analogous bounds for sparse operators and m-linear Calderón-Zygmund operators, and study the corresponding multi-variable Sawyer-type inequalities for such operators and weights. Our results combine mixed restricted weak type norm inequalities, A_p^{{mathcal {R}}} and A_{mathbf {P}}^{{mathcal {R}}} weights, and Lorentz spaces.
Highlights
From these estimates, we deduce new weighted restricted weak type bounds and Sawyer-type inequalities for the m-fold product of Hardy-Littlewood maximal operators
“Sawyer-type inequalities” is a terminology coined in the paper [19], where the authors prove that if u ∈ A1, and v ∈ A1 or uv ∈ A∞, uv x
Muckenhoupt in [47], where the terminology “mixed type norm inequalities” was introduced and was used since in other papers like [2] or [44]. This terminology refers to certain weighted estimates for some classical operators T, where a weight v is included in their level sets; that is, x ∈ Rn : |T f (x)| > t, t > 0
Summary
“Sawyer-type inequalities” is a terminology coined in the paper [19], where the authors prove that if u ∈ A1, and v ∈ A1 or uv ∈ A∞, uv x. Observe that this result is an extension of (1.4) To our knowledge, this multi-variable mixed restricted weak type inequalities for maximal operators involving the ARp condition on the weights have not been previously studied, and we found no record of them being conjectured in the literature. It is worth mentioning that we couldn’t find in the literature any trace of results like (1.10) involving M or multi-linear Calderón-Zygmund operators, ARp or ARP weights, and mixed restricted weak type inequalities. P. has shown in [53] that Sawyer-type inequalities for Lorentz spaces play a fundamental role in the extension to the multi-variable setting of the restricted weak type Rubio de Francia’s extrapolation presented in [13,15] His approach suggests that Conjecture 1 will be crucial for proving multi-variable extrapolation theorems involving weights in ARP
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