Sharpness of uniform continuity of quasiconformal mappings onto s-John domains

Abstract

Abstract. We show that a prediction in [8] is inaccurate by constructing quasiconformal mappings onto s-John domains so that the mappings fail to be uniformly continuous between natural distances. These examples also exhibit the sharpeness of the assumptions in [5].

Sharp Capacity Estimates in S-John Domains

“…The restriction becomes 1 ≥ (s − 1)(n − 1) + ε, which is equivalent to s ≤ 1 + 1−ε n−1 . The range for s is essentially sharp, see [6]. If q = n, then (1.5) reduces to the estimate…”

“…The recent studies [1,5,7] on mappings of finite distortion have generated new interest in the class of s-John domains. In particular, uniform continuity of quasiconformal mappings onto s-John domains was studied in [4,6].…”