## Banach Journal of Mathematical Analysis

### 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 $p\geq1$, (b) to identify the function of the boundary values of $u$ in the Hardy space with index $p\gt 1$ 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 $\mathbb{R}^{n}\times(0,\infty)$.

#### Article information

Source
Banach J. Math. Anal., Volume 10, Number 3 (2016), 523-546.

Dates
Accepted: 7 November 2015
First available in Project Euclid: 6 June 2016

https://projecteuclid.org/euclid.bjma/1465230962

Digital Object Identifier
doi:10.1215/17358787-3607222

Mathematical Reviews number (MathSciNet)
MR3509883

Zentralblatt MATH identifier
1357.46034

#### Citation

Ocampo-Salgado, Hugo; Rivera-Noriega, Jorge; San Martin, Luis. Boundary values of vector-valued Hardy spaces on nonsmooth domains and the Radon–Nikodym property. Banach J. Math. Anal. 10 (2016), no. 3, 523--546. doi:10.1215/17358787-3607222. https://projecteuclid.org/euclid.bjma/1465230962

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