Hardy Spaces, Carleson Measures and a Gradient Estimate for Harmonic Functions on Negatively Curved Manifolds

Abstract

In this paper we study Hardy spaces, BMO, Carleson measures, Green potential and Bloch functions on a Cartan-Hadamard manifold $M$ of pinched negative curvature. Further, using our results on Carleson measure and BMO, we give a gradient estimate for harmonic functions on $M$. It is different from Yau's gradient estimates, and is applied to the existence problem of harmonic Bloch functions described in §10. We deal also with boundary behavior of harmonic Bloch functions on $M$.