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

TL;DR

This paper demonstrates that weights satisfying a dyadic reverse Holder inequality on the unit cube have a non-increasing rearrangement that also satisfies a reverse Holder inequality on intervals, leading to integrability results.

Contribution

It establishes a link between dyadic reverse Holder inequalities and classical inequalities for rearranged weights, providing explicit bounds and integrability intervals.

Findings

Rearranged weights satisfy reverse Holder inequalities with controlled constants.

Identifies intervals of q where weights belong to L^q spaces.

Provides bounds on the reverse Holder constant for rearranged weights.

Abstract

We prove that for any weight $\phi$ defined on $[0,1]^n$ that satisfies a reverse Holder inequality with exponent p > 1 and constant $c\ge1$ 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 $2^nc-2^n + 1$, upon all subintervals of $[0; 1]$ of the form $[0; t]$. This gives as a consequence, according to the results in [8], an interval $[p; p_0(p; c)) = I{p,c}$, such that for any $q \in I{p,c}$, we have that $\phi$ is in $L^q$.