Banach Journal of Mathematical Analysis

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

Hugo Ocampo-Salgado, Jorge Rivera-Noriega, and Luis San Martin

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Abstract

We define Hardy spaces of functions taking values on a Banach space 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 X in terms of the existence of nontangential limits of X-valued functions u in the corresponding Hardy space with index p1, (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 VXp of measures of p-bounded variation in the absence of the Radon–Nikodym property of X. This extends similar results already known on the unit disk of C and the semispace Rn×(0,).

Article information

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

Dates
Received: 18 March 2015
Accepted: 7 November 2015
First available in Project Euclid: 6 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1465230962

Digital Object Identifier
doi:10.1215/17358787-3607222

Mathematical Reviews number (MathSciNet)
MR3509883

Zentralblatt MATH identifier
1357.46034

Subjects
Primary: 46B22: Radon-Nikodým, Kreĭn-Milman and related properties [See also 46G10]
Secondary: 31A20: Boundary behavior (theorems of Fatou type, etc.) 46G10: Vector-valued measures and integration [See also 28Bxx, 46B22] 31A15: Potentials and capacity, harmonic measure, extremal length [See also 30C85]

Keywords
vector-valued Hardy spaces Radon–Nikodym property Fatou-type theorems Lipschitz domains noncylindrical domains

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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