Alptekin Can Goksan | Searcharxiv

arXiv2026

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 $d_E^{-α}$ to belong to a one-sided Muckenhoupt $A_p$ class for some $α>0$ and $1 0$ if and only if $E$ is one-sided weakly porous. In this paper, we find the precise range of exponents $α>0$ such that $d_E^{-α}$ 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.

math.CA↗

arXiv2025

Sharpness of the side condition in a characterization of Békollé-Bonami weights

We study the sharpness of the side condition in a recent characterization of a limiting class $B_\infty$ of Békollé-Bonami weights by Aleman, Pott and Reguera. This side condition bounds the oscillation of a weight on the top halves of Carleson squares and allows for the development of a rich theory for Békollé-Bonami weights, analogous to that of Muckenhoupt weights. First, we prove that the side condition can essentially be dropped when the weight is radial and monotonic. Then, by means of counterexamples, we show that the side condition is sharp for non-monotonic weights. In addition, we extend the characterization of the $B_\infty$ class so that it includes all twelve $A_\infty$ conditions recently studied by Duoandikoetxea, Martín-Reyes and Ombrosi, and we present a complete picture of the relationships between these twelve conditions for arbitrary weights on the unit disc. Finally, we use our results to prove an analogue of the self-improvement property of Muckenhoupt weights for monotonic Békollé-Bonami weights.

math.CA↗