Logarithmic Holder Estimates of p-Harmonic Extension Operators in a Metric Measure Space

Let 1 < p < infinity and let X be a metric measure space with a doubling measure and a (1, p)-Poincare inequality. Let Omega be a bounded domain in X. For a function f on partial derivative Omega we denote by P(Omega)f the p-harmonic extension of f over Omega. It is well known that if Omega is p-regular and f is an element of (partial derivative Omega), then P(Omega)f is continuous in (Omega) over bar. We characterize the family of domains such that logarithmic Holder continuity of boundary functions f ensures logarithmic Holder continuity of P(Omega)f.