One-sided median porous sets and one-sided Muckenhoupt distance functions

We introduce the notion of one-sided median porosity for subsets $E$ of $\mathbb{R}$. We prove that this condition is necessary and sufficient for the distance weight $d_E^{-\alpha}$ to belong to a one-sided Muckenhoupt $A_p$ class for some $\alpha>0$ and $1<p<\infty$. As part of the proof, we obtain new characterizations of one-sided $A_p$ weights and one-sided $\mathrm{BMO}$ functions, in terms of medians. It was recently shown that $d_E^{-\alpha}$ is a one-sided Muckenhoupt $A_1$ weight for some $\alpha>0$ if and only if $E$ is one-sided weakly porous. In this paper, we find the precise range of exponents $\alpha>0$ such that $d_E^{-\alpha}$ belongs to a one-sided $A_p$ class, both for $p=1$ and for $1<p<\infty$. In addition, we show that $E$ is median porous if and only if it is both left and right median porous, and we give an example of a one-sided median porous set which is neither median porous nor one-sided weakly porous.