Linear functions and duality on the infinite polytorus

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 $p\geq 3.31138$. 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.