Weak Slice Conditions and Hölder Imbeddings

Abstract

Abstract. We introduce weak slice conditions and investigate imbeddings of Sobolev spaces in various Lipschitz-type spaces. IntroductionBojarski [B] proved that Sobolev-Poincaré imbeddings are valid on all John domains; see also [Mto]. In [BK1], it is shown that John domains are essentially the right class for this imbedding, since a bounded domain G ⊂ R n is a John domain if and only if it supports a Sobolev-Poincaré imbedding and satisfies a certain separation condition. Corresponding results for the p = n (Trudinger) and p > n (Hölder) cases of the Sobolev Imbedding Theorem are given in [BK2], where it is shown that for domains satisfying a certain slice condition, each of these imbeddings is equivalent to a mean cigar condition dependent on p; see Section 1 for definitions of these concepts. For other results on Hölder imbeddings, we refer the reader to [A], [Mz], and [KR].In one way, the results of [BK2] are less satisfying than those of [BK1]. In [BK1], the strong geometric condition (John) is equivalent to the combination of the weak geometric condition (separation) and a Sobolev-Poincaré imbedding. However in [BK2], the weak geometric condition (slice) is not implied by the strong geometric condition (mean cigar) for any value of p ≥ n. For p = n, Buckley and O'Shea [BO] overcame this deficiency by showing that the strong geometric condition is equivalent to the combination of a so-called weak slice condition (which is implied by a slice condition) and the Trudinger imbedding. Here we prove the following analogous result for p > n; the terminology is explained in Sections 1 and 2.

Distinguishing properties of weak slice conditions

“…Obviously, efficient paths and rough geodesics exist, with C 1 , C 2 as close to zero as we wish (we can even take one of the two parameters to be zero), but α-geodesics might not exist. For instance in the Euclidean case, α-geodesics exist if α = 0, but might not if α > 0; see [GO] and [BS1,Example 1.2].…”

“…In the Euclidean context, some of the material presented below is to be found in [BS1] and [BS2], although this section is essentially self-contained. One notable new result is Theorem 2.14 which partially answers an open question in [BS2].…”

“…For instance, proper subdomains of R n are locally (1, 1)-nicely connected with respect to the Euclidean or inner Euclidean metric, or any metric intermediate between them (this is the class of metrics considered in [BS1] and [BS2]). More generally, it follows from Theorem 3.13 of [HK] that if Ω is an open connected set in a complete, Ahlfors upper regular, Loewner space (X, d X , µ) of Hausdorff dimension Q > 1, then Ω is locally nicely connected.…”

“…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.…”

“…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].…”