Unions of Lebesgue spaces and $A_1$ 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 $L_w^p$ 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 $L_w^p$ over $w$ in $A_p$ and when a function has an $A_1$ majorant on all of ${\mathbb{R}}^n$.

arXiv commentv2: typos corrected