A Fractional Orlicz-Sobolev Poincaré Inequality in John Domains

Abstract

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 (ϕn/(n−β), ϕ)β-Poincare 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 (ϕn/(n−β),ϕ)β-Poincare inequality, then we show that it is a John domain.