Weak reverse Hölder inequalities for doubling measures

Abstract

Let $\mu$ be a doubling Radon measure in ${\mathbb{R}}^d$ with support $X$. We prove a local and quantitative form of the Bennewitz--Lewis theorem in this setting: if a Radon measure $\nu$ satisfies the Bennewitz--Lewis condition on a bounded relatively open set $V\subset X$ with a small enough constant, then $\nu {|}_V\ll \mu {|}_V$ and its density satisfies a reverse Hölder inequality on every compact subset $K\subset V$. Along the way we develop the dyadic toolkit these arguments run on --- triple cubes defined through neighbours, the basis $\mathcal{D}\cup 3\mathcal{D}$, and a Whitney decomposition --- and we collect the Muckenhoupt-type classes of weights over that basis. The same machinery yields a local Gehring lemma for enlarged balls: both theorems are deduced from a single bootstrapping lemma, stated once for $p=1$ and $p>1$.

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