Dyadic weights on $\mathbb {R}^n$ and reverse Hölder inequalities
Volume 234 / 2016
Studia Mathematica 234 (2016), 281-290
MSC: Primary 42B25.
DOI: 10.4064/sm8621-6-2016
Published online: 5 August 2016
Abstract
We prove that for any weight $\phi $ defined on $[0,1]^n$ that satisfies a reverse Hölder inequality with exponent $p \gt 1$ and constant $c\ge 1$ on all dyadic subcubes of $[0,1]^n$, its non-increasing rearrangement $\phi ^\ast $ satisfies a reverse Hölder inequality with the same exponent and constant not more than $2^nc-2^n+1$ on all subintervals of the form $[0,t]$, $0 \lt t\le 1$. As a consequence, there is an interval $[p,p_0(p,c))=I_{p,c}$ such that $\phi \in L^q$ for any $q\in I_{p,c}$.
