MODULUS OF CONTINUITY OF p-DIRICHLET SOLUTIONS 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-Dirichlet solution of f over Omega. It is well known that if Omega is p-regular and f is an element of C(partial derivative Omega), then P(Omega)f is p-harmonic in Omega and continuous in (Omega) over bar. We characterize the family of domains Omega such that improved continuity of boundary functions f ensures improved continuity of P(Omega)f. We specify such improved continuity if X is Ahlfors regular and X \ Omega is uniformly p-fat.