A Fractional Orlicz–Sobolev Poincaré Inequality in John Domains

Let n≥2,β∈(0,n)andΩ■R^(n) be a bounded domain.Support thatφ:[0,∞)→[0,∞)is a Young function which is doubling and satisfies sup x>0∫^(1)_(0)φ(tx)/φ(x)dt/t^(β+1)<∞.If Ω is a John domain,then we show that it supports a(φ^(n/(n-β)),φ)_(β)-Poincaréinequality.Conversely,assume thatΩis simply connected domain when n=2 or a bounded domain which is quasiconformally equivalent to some uniform domain when n≥3.IfΩsupports a((φ^(n/(n-β)),φ)β-Poincaréinequality,then we show that it is a John domain.