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

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