Hardy spaces and heat kernel regularity

TL;DR

This paper establishes a fundamental link between the boundedness of the Riesz transform and the equality of Hardy and Lebesgue spaces on certain manifolds, advancing understanding of heat kernel regularity.

Contribution

It proves the equivalence between Riesz transform boundedness and Hardy space equality on manifolds with doubling measure and Poincaré inequalities.

Findings

Riesz transform boundedness on L^p is equivalent to Hardy space L^p equality.

Results apply to manifolds with doubling measure and scaled Poincaré inequalities.

Provides a characterization of heat kernel regularity via Hardy space theory.

Abstract

In this paper, we show the equivalence between the boundedness of the Riesz transform $d\Delta^{-1/2}$ on $L^p$, $p\in (2,p_0)$, and the equality $H^p=L^p$, $p\in(2,p_0)$, in the class of manifold whose measure is doubling and for which the scaled Poincar\'{e} inequalities hold. Here, $H^p$ is a Hardy space of exact $1-$forms, naturally associated with the Riesz transform.