Unions of Lebesgue spaces and $A_1$ majorants

Greg Knese, John E. McCarthy and Kabe Moen

Vol. 280 (2016), No. 2, 411–432

DOI: 10.2140/pjm.2016.280.411

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 provide a systematic study of these questions and show that they are fundamentally related. We show that the union of $L_w^p\left({ℝ}^n\right)$ spaces with $w\in A_p$ is equal to the union of all Banach function spaces for which the Hardy–Littlewood maximal function is bounded on the space itself and its associate space.

Keywords

maximal functions, $L^p$ spaces, Hardy spaces, $A_p$ spaces, weighted $L^p$ spaces

Mathematical Subject Classification 2010

Primary: 42B25, 42B35, 46E30

Secondary: 30H10, 30H15

Milestones

Received: 22 August 2014

Revised: 9 January 2015

Accepted: 9 May 2015

Published: 28 January 2016