Dyadic weights on $R^n$ and reverse Holder inequalities

We prove that for any weight $\varphi$ defined on $[0,1{]}^n$ that satisfies a reverse Holder inequality with exponent p > 1 and constant $c\ge 1$ upon all dyadic subcubes of $[0,1{]}^n$, it’s non increasing rearrangement satisfies a reverse Holder inequality with the same exponent and constant not more than $2^{n}c-2^n+1$, upon all subintervals of $[0;1]$ of the form $[0;t]$. This gives as a consequence, according to the results in [8], an interval $[p; p_0(p; c)) = I{p,c}$,suchthatforany$q \in I{p,c}$,wehavethat$\phi$ is in $L^q$.