Hardy Spaces and Heat Kernel Regularity

Abstract

In this paper, we show that, for doubling manifolds satisfiying the scaled Poincare inequalities and $p\in (2,\infty )$ , the boundedness of the Riesz transform dΔ−1/2 on L p , is essentially equivalent to the fact that $H_{1,d}^{p}$ is equal the L p closure of the set of L p exact harmonic 1-forms. Here, $H_{1,d}^{p}$ is a Hardy space of exact 1 −forms, naturally associated with the Riesz transform, as defined by Auscher, McIntosh and Russ.