GRADIENT ESTIMATES FOR HARMONIC-FUNCTIONS ON REGULAR DOMAINS IN RIEMANNIAN-MANIFOLDS

Derivative formulae for heat semigroups are used to give gradient esti mates for harmonic functions on regular domains in Riemannian manifold s. This probabilistic method provides an alternative to coupling techn iques, as introduced by Cranston, and allows us to improve some known estimates. We discuss two slightly different ways to exploit derivativ e formulae where each one should be interesting by itself. (C) 1998 Ac ademic Press.