Improved Buckley's theorem on LCA groups

Abstract

We present sharp quantitative weighted norm inequalities for the Hardy-Littlewood maximal function in the context of Locally Compact Abelian Groups, obtaining an improved version of the so-called Buckley's Theorem. On the way, we prove a precise reverse Hölder inequality for Muckenhoupt $A_{\infty }$ weights and provide a valid version of the "open property" for Muckenhoupt $A_p$ weights.