We consider the following question: are there exponents 2 such that the Riesz projection is bounded from ?? to ?? 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 ?∞ to ?? if ?≥3.31138 . A similar result can be extracted for any ?>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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