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

The authors thank the anonymous reviewers for their constructive comments and suggestions, which help to improve the expression of this paper. This project is partly supported by the Natural Science Foundation of Hunan province (2024JJ3023).