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arXiv · 1302.7315

Unions of Lebesgue spaces and $A_1$ majorants

Greg Knese·John E. McCarthy·Kabe Moen
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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 ${\mathbb R}^n$.

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Greg Knese, John E. McCarthy, Kabe Moen. 2013-02-28. Unions of Lebesgue spaces and $A_1$ majorants. https://arxiv.org/abs/1302.7315

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