Gradient Estimates for Harmonic Functions on Regular Domains in Riemannian Manifolds

Abstract

Derivative formulae for heat semigroups are used to give gradient estimates for harmonic functions on regular domains in Riemannian manifolds. This probabilistic method provides an alternative to coupling techniques, as introduced by Cranston, and allows us to improve some known estimates. We discuss two slightly different ways to exploit derivative formulae where each one should be interesting by itself.1998 Academic Press

A Bismut type formula for the Hessian of heat semigroups

“…[7], Section 5, as well as [6]. ✷ In our last application we focus on the case of harmonic functions on rotationally symmetric manifolds.…”

“…x v)(ω, id M ); hence S τ (θ, ω) = sup v∈T x M, |v|=1 |S τ (ω, θ )(v)| is independent of θ . Writing the gradient formula as iterated integral with respect to the initial probability and the Haar measure, and exploiting the mean value property of u, we obtain |grad x u| u(x) E S τ , where the last expectation can be bounded as in [7], Section 5. The method immediately transfers to the Hessian case.…”

Untitled

“…We obtain the desired result by applying Theorem 3.2 to f n and letting n → ∞. In the case where cut(x) = ∅, we prove the same result by a trick used in part (2) of the proof to Corollary 5.1 in [20]. ✷…”

“…, where We shall apply Theorem 3.1 by constructing a proper finite energy process t as in [20]. Letting T (t) := t 0 f −2 (x s ) ds for t τ δ , we have…”

Some Remarks on the Heat Flow for Functions and Forms

“…with a constant c depending on t, dim M , dist(x, ∂D) and the lower bound of Ric on D. This follows from Theorem 2.3 with an explicit choice for h. See [8] for the details.…”