Weak slice conditions, product domains, and quasiconformal mappings

Abstract

Abstract. We investigate geometric conditions related to H older imbeddings, and show, among other things, that the only bounded Euclidean domains of the form U V that are quasiconformally equivalent to inner uniform domains are inner uniform domains. Inner uniform domains, as de ned by V ais al a V5], satisfy a uniformity condition with respect to the inner Euclidean metric. These domains form a class intermediate between uniform and John domains and, in particular, they include all Lipschitz domains see Section 1 for de nitions. We prove the following theorem which indicates that this class is well suited to the study of quasiconformal equivalence.

Distinguishing properties of weak slice conditions

“…The slice condition is a metric-geometric condition for domains in Euclidean spaces R n that was introduced by the first author and Koskela [BK2] to obtain a set of geometric classifications of a large class of domains in Euclidean spaces which support any of the Sobolev imbeddings, for integrability index p ≥ n. In later works ( [BO], [BS1], [BS2]), variations of the slice condition were used to refine these results and also to obtain unrelated results in other areas of analysis. There were many variants of these conditions such as (inner) α-wslice and (inner) α-wslice + conditions, all of which were defined for a variety of purposes; see Section 2 for definitions.…”

“…For instance, they are all satisfied by every simply connected planar domain. This follows by the Riemann mapping theorem from the more general fact that the quasiconformal images of inner uniform domains in R n satisfy the slice condition and all weak slice conditions (once a certain auxiliary parameter C is sufficiently large); see Theorem 3.1 of [BS2]. Inner uniform domains, introduced by Väisälä [V5], generalize the well-known class of uniform domains.…”

“…In [BS2], the authors used weak slice conditions to obtain results regarding quasiconformal equivalence of product domains. One pleasing aspect of the main results of this work was that the weak slice conditions did not appear at all in the statement of the results, but rather as a tool in the proof.…”

“…Many properties and examples of these conditions in the Euclidean context were obtained in [BS1] and [BS2], but some fundamental questions remain open even in this context. Several of these were presented in Section 6 of [BS2].…”

“…Many properties and examples of these conditions in the Euclidean context were obtained in [BS1] and [BS2], but some fundamental questions remain open even in this context. Several of these were presented in Section 6 of [BS2]. In this work we construct examples that fully answer one and partially answer another of these open problems, and we also answer another question that was not posed there, but is related to the results in [BS2].…”