Poincaré inequalities in quasihyperbolic boundary condition domains

We study the validity of (q,p)-Poincaré 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)-Poincaré 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.

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