Poincar\'e inequalities in quasihyperbolic boundary condition domains
  We study the validity of (q,p)-Poincar\'e inequalities, q<p, on domains in R^n which satisfy a quasihyperbolic boundary condition, i.e. domains whose quasihyperbolic metric satisfies a logarithmic growth condition. In the present paper, we show that the quasihyperbolic boundary condition domains support a (q,p)-Poincar\'e inequality whenever p>p_0, where p_0 is an explicit constant depending on q, on the logarithmic growth condition, and on the boundary of the domain. 
Ritva Hurri-Syrj\"anen, Niko Marola, and Antti V. V\"ah\"akangas
math.CA math.AP