Boundary values of vector-valued Hardy spaces on nonsmooth domains and the Radon–Nikodym property

Abstract

We define Hardy spaces of functions taking values on a Banach space $\mathcal{X}$ over nonsmooth domains. The types of functions we consider are harmonic functions on a starlike Lipschitz domain and solutions to the heat equation on a time-varying domain. Our purpose is twofold: (a) to characterize the Radon–Nikodym property of the Banach space $\mathcal{X}$ in terms of the existence of nontangential limits of $\mathcal{X}$-valued functions $u$ in the corresponding Hardy space with index $p\ge 1$, (b) to identify the function of the boundary values of $u$ in the Hardy space with index $p>1$ with an element in the space ${\mathcal{V}}_{\mathcal{X}}^p$ of measures of $p$-bounded variation in the absence of the Radon–Nikodym property of $\mathcal{X}$. This extends similar results already known on the unit disk of $\mathbb{C}$ and the semispace ${\mathbb{R}}^n\times (0,\infty )$.