IUMJ

Title: A new characterization of the Muckenhoupt $A_p$ weights through an extension of the Lorentz-Shimogaki Theorem

Authors: Andrei K. Lerner and Carlos Perez

Issue: Volume 56 (2007), Issue 6, 2697-2722

Abstract:

Given any quasi-Banach function space $X$ over $\mathbb{R}^n$ an index $\alpha_X$ is defined that coincides with the upper Boyd index $\bar{\alpha}_X$ when the space $X$ is rearrangement-invariant. This new index is defined by means of the local maximal operator $m_{\lambda}f$. It is shown then that the Hardy-Littlewood maximal operator $M$ is bounded on $X$ if and only if $\alpha_X < 1$ providing an extension of the classical theorem of Lorentz and Shimogaki for rearrangement-invariant $X$.\par As an application, a new characterization of the Muckenhoupt $A_p$ class of weights is shown, $u \in A_p$, if and only if for any $\varepsilon > 0$ there is a constant $c$ such that for any cube $Q$ and any measurable subset $E \subset Q$, \[ \frac{|E|}{|Q|}\log^{\varepsilon} \left( \frac{|Q|}{|E|} \right) \le c \left( \frac{u(E)}{u(Q)} \right)^{1/p}.\] The case $\varepsilon = 0$ is false corresponding to the class $A_{p,1}$.\par Other applications are given, in particular within the context of the variable $L^p$ spaces.