A Note on Capacity Muckenhoupt Weights with respect to the Hausdorff Content

Abstract

Let $p\in[1,\infty)$ and $δ\in(0,n]$. Recently, to characterize the weighted boundedness of maximal operator on Choquet integrals based on Hausdorff content $\mathcal H_\infty^δ$, a class of Capacity Muckenhoupt weights is introduced, denoted by $\mathcal A_{p,δ}$, and, for any fixed $δ\in(0,n]$, this new class of weights is proved to satisfy the self-improving property with respect to the index $p\in(1,\infty)$. In this note, we show that, for every fixed $p\in[1,\infty)$, the class of capacity Muckenhoupt weights $\mathcal A_{p,δ}$ fails to satisfy the self-improving property with respect to the index $δ\in(0,n]$.