Abstract
Let \(n_j\) be a lacunary sequence of integers, such that \(n_{j+1}/n_j\ge r\). We are interested in linear combinations of the sequence of finite Riesz products \(\prod _{j=1}^N(1+\cos (n_j t))\). We prove that, whenever the Riesz products are normalized in \(L^p\) norm (\(p\ge 1\)) and when r is large enough, the \(L^p\) norm of such a linear combination is equivalent to the \(\ell ^p\) norm of the sequence of coefficients. In other words, one can describe many ways of embedding \(\ell ^p\) into \(L^p\) based on Fourier coefficients. This generalizes to vector valued \(L^p\) spaces.
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Acknowledgements
We would like to thank F. Nazarov for stimulating correspondence which encouraged us to continue working on this project. We are also indebted to P. Ohrysko for a helpful discussion, and to anonymous referees for very helpful reports significantly improving the paper. This material is partially based upon work supported by the NSF Grant DMS-1440140, while the authors were in residence at the MSRI in Berkeley, California, during the fall semester of 2017. P. N. and T. T. were also partially supported by the Simons Foundation. R. L. and P. N. were partially supported by the National Science Centre Poland Grant 2015/18/A/ST1/00553 and T. T. by NSF Grant DMS-1955175. The research leading to these results is part of a project that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 Research and Innovation Programme (Grant Agreement No. 637851).
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Communicated by Krzysztof Stempak.
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Bonami, A., Latała, R., Nayar, P. et al. Bounds on Moments of Weighted Sums of Finite Riesz Products. J Fourier Anal Appl 26, 84 (2020). https://doi.org/10.1007/s00041-020-09800-3
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DOI: https://doi.org/10.1007/s00041-020-09800-3