$A_1$ weights on $\mathbb{R}$, an alternative approach

Abstract

We will prove that if $\varphi$ belongs to the class $A_1(\mathbb{R})$ with constant $c\ge 1$ then the decreasing rearrangement of $\varphi$, belongs to the same class with constant not more than $c$. We also find for such $\varphi$ the exact best possible range of those $p>1$ for which $\varphi \in L^p$. In this way we provide alternative proofs of the results that appear in \cite{1}.

arXiv comment8 pages