Criteria for optimal global integrability of Hajłasz–Sobolev functions

Abstract

The author establishes some geometric criteria for a domain of R n with n ≥ 2 to support a (pn/(n − ps), p) s -Haj lasz-Sobolev-Poincaré imbedding with s ∈ (0, 1] and p ∈ (n/(n + s), n/s) or an s-Haj lasz-Trudinger imbedding with s ∈ (0, 1]. g L p (Ω) < ∞. 2000 Mathematics Subject Classification: 46E35 Key words and phases: John domain, weak carrot domain, local linear connectivity, Haj lasz-Sobolev space, Haj lasz-Sobolev-Poincaré imbedding, Haj lasz-Trudinger imbedding, Haj lasz-Sobolev extension

An Orlicz-Besov Poincaré Inequality via John Domains

“…We see that (6) coincides with (2). Thus, Theorem 1 extends above criteria for bounded domains supporting fractional Poincaré inequality given in [2,3] (see also [4]).…”

“…whenever ∈ 1 (Ω). Moreover, as proved by Zhou [4], when 0 < < 1 and ∈ [1, / ), a John domain Ω ⊂ R always supports the following Hajłasz-Sobolev Poincaré inequality: there exists a constant ≥ 1 such that…”

“…We prove Theorem 1 in Section 3. To prove Theorem 1 (i), we use Boman's chain rule for John domains and an Orlicz-Besov imbedding in cubes under (7) and (8) as proved by [7] (see Lemma 8) and also borrow some ideas from [1,4]. To prove Theorem 1 (ii), we use (7) and (8) to calculate theḂ , * (Ω)-norms of some cut-off functions in a precise way; see Lemmas 6 and 7.…”

“…This allows us to prove the LLC(2) property of domains supporting inequality (6); see Proposition 9. By borrowing some ideas from [4,5,8], we then obtain Theorem 1 (ii).…”

Fractional Sobolev extension and imbedding

“…The proofs of Theorems 1.1 and 1.2 will be given in Section 2. We borrow some ideas from [13,14,18,24,32].…”