Dimension free properties of strong Muckenhoupt and Reverse H\"older weights for Radon measures

In this paper we prove self-improvement properties of strong Muckenhoupt and Reverse H\"older weights with respect to a general Radon measure on $\mathbb{R}^n$. We derive our result via a Bellman function argument. An important feature of our proof is that it uses only the Bellman function for the one-dimensional problem for Lebesgue measure; with this function in hand, we derive dimension free results for general measures and dimensions.