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Fatou and brothers Riesz theorems in the infinite-dimensional polydisc

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Abstract

We study the boundary behavior of functions in the Hardy spaces on the infinite-dimensional polydisc. These spaces are intimately related to the Hardy spaces of Dirichlet series. We exhibit several Fatou and Marcinkiewicz- Zygmund type theorems for radial convergence of functions with Fourier spectrum supported on \(\mathbb{N}_0^\infty\cup(-\mathbb{N}_0^\infty)\). As a consequence one obtains easy new proofs of the brothers F. and M. Riesz Theorems in infinite dimensions, as well as being able to extend a result of Rudin concerning which functions are equal to the modulus of an H1 function almost everywhere to \(\mathbb{T}^\infty\). Finally, we provide counterexamples showing that the pointwise Fatou theorem is not true in infinite dimensions without restrictions to the mode of radial convergence even for bounded analytic functions.

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Correspondence to Jan-Fredrik Olsen.

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The third author was supported by the Finnish Academy CoE in Analysis and Dynamics Research and by a Knut and Alice Wallenberg Grant.

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Aleman, A., Olsen, JF. & Saksman, E. Fatou and brothers Riesz theorems in the infinite-dimensional polydisc. JAMA 137, 429–447 (2019). https://doi.org/10.1007/s11854-019-0006-x

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  • DOI: https://doi.org/10.1007/s11854-019-0006-x

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