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24 - Vector-Valued Hardy Spaces

from Part 4 - Vector-Valued Aspects

Published online by Cambridge University Press:  19 July 2019

Andreas Defant
Affiliation:
Carl V. Ossietzky Universität Oldenburg, Germany
Domingo García
Affiliation:
Universitat de València, Spain
Manuel Maestre
Affiliation:
Universitat de València, Spain
Pablo Sevilla-Peris
Affiliation:
Universitat Politècnica de València, Spain
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Summary

Given a Banach space X, we consider Hardy spaces of X-valued functions on the infinite polytorus, Hardy spaces of X-valued Dirichlet series (defined as the image of the previous ones by the Bohr transform), and Hardy spaces of X-valued holomorphic functions on l_2 ∩ B_{c0}. The chapter is dedicated to study the interplay between these spaces. It is shown that the space of functions on the polytorus always forms a subspace of the one of holomorphic functions, and these two are isometrically isomorphic if and only if X has ARNP. Then the question arises of what do we find in the side of Dirichlet series when we look at the image of the Hardy space of holomorphic functions. This is also answered, showing that this consists of Dirichlet series for which all horizontal translations (those whose coefficients are (a_n/n^ε)) are in \mathcal{H}_p with uniformly bounded norms. Also, a version of the brothers Riesz theorem for vector-valued functions is given.

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Publisher: Cambridge University Press
Print publication year: 2019

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