Accessible parts of the boundary for domains with lower content regular complements

Abstract

We show that if 0 < t < s ≤ n − 1, Ω ⊆ R n with lower s-content regular complement, and z ∈ Ω, there is a chord-arc domainThis was originally shown by Koskela, Nandi, and Nicolau with John domains in place of chord-arc domains when n = 2, s = 1, and Ω is a simply connected planar domain.Domains satisfying the conclusion of this result support (p, β)-Hardy inequalities for β < p − n + t by a result of Koskela and Lehrbäck; Lehrbäck also showed that s-content regularity of the complement for some s > n − p + β was necessary. Thus, the combination of these results characterizes when a domain supports a pointwise (p, β)-Hardy inequality for β < p − 1 in terms of lower content regularity.

Assouad Type Dimensions in Geometric Analysis

“…This leads to the following theorem. We omit the details and refer to [19] and [24] for the definitions and proofs; see also [5] for recent progress concerning such accessibility conditions. We have seen in Theorems 5.3 and 5.6 that the "dual" conditions dim A (Ω c ) < q p (n − p + β) and dim A (Ω c ) > n − p + β, possibly together with some additional requirements, are sufficient for the (q, p, β)-Hardy-Sobolev inequality in Ω ⊂ R n .…”

Accessible parts of the boundary for domains in metric measure spaces

“…Shortly after, the result was generalized to R n . More precisely, Azzam proved in [2] that if for some 0 < s ≤ n − 1 and C 0 > 0, H s ∞ B(ω, λ) \ Ω ≥ C 0 λ s holds for all ω ∈ ∂Ω and 0 < λ < diam(Ω), a condition he calls having lower scontent regular complement, then (1) holds for some c ≥ 1 and for 0 < t < s. In fact, Azzam shows the stronger result with the visible boundary defined with chordarc subdomains playing the role of John subdomains. The basis of his proof is the construction of a subset of the visible boundary that makes use of projections, and so it features strong reliance on the linear structure of the space.…”

“…The case t = 0 is not interesting as condition ( 1) is trivially satisfied with c = 1 for any proper subdomain of R n . On the other hand, condition (1) can fail for t = n − 1 even when Ω is assumed to satisfy some nice geometric properties (see [2]), and so the focus is on the interval 0 < t < n − 1.…”

Accessible parts of the boundary for domains in metric measure spaces

“…Shortly after, the result was generalized to R n . More precisely, Azzam proved in [2] that if for some 0 < s ≤ n − 1 and C 0 > 0,…”

“…It was shown in [9], in the Euclidean setting, that if a domain has large visible boundary in the sense of (3), then it admits certain Hardy inequalities. This was generalised to metric measure spaces in [11], where the author also shows that it is enough for a domain to satisfy (2) to guarantee that it admits a Hardy inequality. The present work complements these results by showing, in fact, that (2) implies (3).…”