$A_1$ weights on $\R$, an alternative approach
We will prove that if $\phi$ belongs to the class $A_1(\R)$ with constant $c\ge1$ then the decreasing rearrangement of $\phi$, belongs to the same class with constant not more than $c$. We also find for such $\phi$ the exact best possible range of those $p>1$ for which $\phi\in L^p$. In this way we provide alternative proofs of the results that appear in \cite{1}.