Let n ≥ 2, β ∈ (0, n) and Ω ⊂ ℝⁿ be a bounded domain. Support that ϕ:[0, ∞) → [0, ∞) is a Young function which is doubling and satisfies

$$\mathop {\sup }\limits_{x > 0} \int_0^1 {{{\phi (tx)} \over {\phi (x)}}{{dt} \over {{t^{\beta + 1}}}} < \infty .} $$

If Ω is a John domain, then we show that it supports a (ϕ^n/(n−β), ϕ)_β-Poincaré inequality. Conversely, assume that Ω is simply connected domain when n = 2 or a bounded domain which is quasiconformally equivalent to some uniform domain when n ≥ 3. If Ω supports a (ϕ^n/(n−β),ϕ)_β-Poincaré inequality, then we show that it is a John domain.

中文翻译：

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John Domains中的分数Orlicz-SobolevPoincaré不等式

让Ñ ≥2，β＆Element;（0，Ñ）和Ω⊂ℝ ^Ñ是有界域。支持ϕ：[0，∞）→[0，∞）是一个加倍且满足的Young函数

$$ \ mathop {\ sup} \ limits_ {x> 0} \ int_0 ^ 1 {{{\ phi（tx）} \ over {\ phi（x）}} {{dt} \ over {{t ^ {\ beta + 1}}}} <\ infty。} $$

如果Ω是John域，则表明它支持（ϕ ^{n /（n - β）}，ϕ）_β-庞加莱不等式。相反地，假设Ω是简单地连接域时Ñ = 2或这quasiconformally相当于一些均匀域的有界区域时Ñ ≥3.如果Ω支持（φ ^{ñ /（ñ - β）}，φ）_β -Poincaré不等式，那么我们证明它是一个John域。