Solovay reduction and continuity

The objective of this study is a better understanding of the relationships between reduction and continuity. Solovay reduction is a variation of Turing reduction based on the distance of two real numbers. We characterize Solovay reduction by the existence of a certain real function that is computable (in the sense of computable analysis) and Lipschitz continuous. We ask whether there exists a reducibility concept that corresponds to H\"older continuity. The answer is affirmative. We introduce quasi Solovay reduction and characterize this new reduction via H\"older continuity. In addition, we separate it from Solovay reduction and Turing reduction and investigate the relationships between complete sets and partial randomness.