2025/07/28 by Pasquariello, Marcus, Uriarte-Tuero, Ignacio
#42B37 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics
paper · doi:10.48550/arxiv.2507.21020
Vasin (for n=1) and Anderson, Lehrbäck, Mudarra, and Vähäkangas (arXiv:2209.06284) (for n>1) provided a geometric characterization of the sets E ⊂ ℝn so that w = dist(⋅, E)-α is a Muckenhoupt A1 weight for some α> 0. In this paper, we provide a geometric characterization of the sets E ⊂ ℝn (which we call median porous sets) so that w = dist(⋅, E)-α is a Muckenhoupt Ap weight for some α> 0 (given any 1 < p ≤ ∞). Given 1 < p ≤ ∞, we also find the precise range of exponents α so that w = dist(⋅, E)-α ∈ Ap, in analogy to the p=1 case done in arXiv:2209.06284. With our characterization we prove that ℝn ∖ 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 Poincaré inequalities, and a geometric characterization of the nonnegative Hölder continuous functions w such that log (w) ∈ BMO. We prove that two of the methods we use (Ap 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 ℝn and constants 0 < s < t < 1, ‖f‖BMO ≈s, t, n supQ[Mt(f, Q) - Ms(f, Q)] where Ms(f, Q) denotes the s-median value of f on Q.