Medians, Oscillations, and Distance Functions

Vasin (for $n=1$) and Anderson, Lehrb\"ack, Mudarra, and V\"ah\"akangas (arXiv:2209.06284) (for $n>1$) provided a geometric characterization of the sets $E \subset \mathbb{R}^n$ so that $w = \text{dist}(\cdot, E)^{-\alpha}$ is a Muckenhoupt $A_1$ weight for some $\alpha > 0$. In this paper, we provide a geometric characterization of the sets $E \subset \mathbb{R}^n$ (which we call median porous sets) so that $w = \text{dist}(\cdot, E)^{-\alpha}$ is a Muckenhoupt $A_p$ weight for some $\alpha > 0$ (given any $1 < p \leq \infty$). Given $1 < p \leq \infty$, we also find the precise range of exponents $\alpha$ so that $w = \text{dist}(\cdot, E)^{-\alpha} \in A_p$, in analogy to the $p=1$ case done in arXiv:2209.06284. With our characterization we prove that $\mathbb{R}^n \setminus E$ supports a Hardy-Sobolev inequality if $E$ is an appropriate median porous set. All previous such results that we are aware of make the strictly stronger assumption that the set $E$ is porous, e.g. arXiv:1705.01360, arXiv:1502.01190. As far as we know, this is the first instance in the literature that the ``porosity barrier" is broken in this context. Examples of such appropriate median porous (but not porous) sets were known. We provide further such examples, additional applications to weighted Poincar\'e inequalities, and a geometric characterization of the nonnegative H\"older continuous functions $w$ such that $\log (w) \in BMO$. We prove that two of the methods we use ($A_p$ and Riesz potential methods) are sharp, i.e. they cannot be improved beyond the results we obtain. The proofs rely on a new median-value characterization of $BMO$: For a real-valued measurable function on $\mathbb{R}^n$ and constants $0 < s < t < 1$, $$\|f\|_{BMO} \approx_{s, t, n} \sup_{Q}[M_t(f, Q) - M_s(f, Q)]$$ where $M_s(f, Q)$ denotes the $s$-median value of $f$ on $Q$.