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Linear functions and duality on the infinite polytorus

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Abstract

We consider the following question: are there exponents \(2<p<q\) such that the Riesz projection is bounded from \(L^q\) to \(L^p\) on the infinite polytorus? We are unable to answer the question, but our counter-example improves a result of Marzo and Seip by demonstrating that the Riesz projection is unbounded from \(L^\infty \) to \(L^p\) if \(p\ge 3.31138\). A similar result can be extracted for any \(q>2\). Our approach is based on duality arguments and a detailed study of linear functions. Some related results are also presented.

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Acknowledgements

The author would like to extend his gratitude to A. Bondarenko, H. Hedenmalm, E. Saksman and K. Seip for an interesting discussion which culminated in the material presented in Sect. 3 and to the referee for a helpful suggestion.

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  1. Department of Mathematical Sciences, Norwegian University of Science and Technology (NTNU), 7491, Trondheim, Norway

    Ole Fredrik Brevig

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  1. Ole Fredrik Brevig
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Correspondence to Ole Fredrik Brevig.

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Brevig, O.F. Linear functions and duality on the infinite polytorus. Collect. Math. 70, 493–500 (2019). https://doi.org/10.1007/s13348-019-00243-8

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  • DOI: https://doi.org/10.1007/s13348-019-00243-8

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