Gradient estimate for harmonic functions on Poincare disc

Abstract

The harmonic functionon the Poincare disc is a C[superscript 2] function satisfying [delta][g][u] = 0, where [delta][subscript g] is the Laplace-Beltrami operator. The gradient on the Poincare disc of u is [delta][g][u]. In this work, [delta][g][u] is to be estimated in terms of given qualitative data of u on a geodesic ball of radius r