Sawyer-type inequalities for Lorentz spaces

Abstract

The Hardy-Littlewood maximal operator satisfies the classical Sawyer-type estimate $${\Vert \frac{Mf}{v}\Vert }_{L^{1,\infty }(uv)}\le C_{u,v}\Vert f{\Vert }_{L^1(u)},$$ where $u\in A_1$ and $uv\in A_{\infty }$. We prove a novel extension of this result to the general restricted weak type case. That is, for $p>1$, $u\in A_p^{\mathcal{R}}$, and $u{v}^p\in A_{\infty }$, $${\Vert \frac{Mf}{v}\Vert }_{L^{p,\infty }(u{v}^p)}\le C_{u,v}\Vert f{\Vert }_{L^{p,1}(u)}.$$ 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 $A_{\vec{P}}^{\mathcal{R}}$, 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 $A_{\vec{P}}^{\mathcal{R}}$ weights, and Lorentz spaces.

References in corpus (1)

• Weighted boundedness of the 2-fold product of Hardy-Littlewood maximal operators

Cited by in corpus (2)

• A class of multilinear bounded oscillation operators on measure spaces and applications
• Boundary Value Problems in graph Lipschitz domains in the plane with $A_{\infty }$-measures on the boundary