Lipschitz measures and vector-valued Hardy spaces

Abstract

We define certain spaces of Banach-valued measures called Lipschitz measures. When the Banach space is a dual space X*, these spaces can be identified with the duals of the atomic vector-valued Hardy spaces HXp(ℝn), 0<p<1. We also prove that all these measures have Lipschitz densities. This implies that for every real Banach space X and 0<p<1, the dual HXp(ℝn)∗ can be identified with a space of Lipschitz functions with values in X*.