Linear functions and duality on the infinite polytorus

TL;DR

This paper investigates the boundedness of the Riesz projection on the infinite polytorus, providing new bounds and counterexamples using duality and linear function analysis.

Contribution

It offers improved counterexamples showing unboundedness of the Riesz projection for certain $p$ and $q$, advancing understanding of its behavior on the infinite polytorus.

Findings

Riesz projection unbounded from $L^ Infty$ to $L^p$ for $p \,\geq\, 3.31138$

Counterexamples improve previous results by Marzo and Seip

Analysis based on duality and linear functions

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.