  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.
