Unions of Lebesgue spaces and $A_1$ majorants

TL;DR

This paper explores the relationship between functions belonging to unions of Lebesgue spaces and those having an $A_1$ majorant, providing characterizations and equivalences on fixed cubes and the entire space.

Contribution

It establishes fundamental equivalences between membership in Lebesgue spaces, $A_1$ majorants, and weighted Lebesgue spaces with $A_p$ weights, on fixed cubes and on ${ m I extbf{R}}^n$.

Findings

Functions in $L^p$ for some $p>1$ have an $A_1$ majorant.

Characterizations of unions of weighted Lebesgue spaces with $A_p$ weights.

Equivalent conditions for functions on ${ m I extbf{R}}^n$ having $A_1$ majorants.

Abstract

We study two questions. When does a function belong to the union of Lebesgue spaces and when does a function have an $A_1$ majorant? We show these questions are fundamentally related. For functions restricted to a fixed cube we prove that the following are equivalent: a function belongs to $L^p$ for some $p>1$; the function has an $A_1$ majorant; for any $p>1$ the function belongs to $L^p_w$ for some $A_p$ weight $w$. We also examine the case of functions defined on ${\mathbb R}^n$ and give characterizations of the union of $L^p_w$ over $w$ in $A_p$ and when a function has an $A_1$ majorant on all of ${\mathbb R}^n$.