On the norm of the Riesz projection from $L^{\infty}$ to $L^p$

We will consider $2\pi$-periodic functions of countably many variables. Let $\mathbb{T} = \mathbb{R}/(2\pi\mathbb{Z})$ and $\mu_\infty$ denote the Haar measure on $\mathbb{T}^\infty$ normalized so that $\mu_\infty(\mathbb T^\infty)=1$. Any function $f\in L(\mathbb{T}^\infty)$ has the Fourier expansion
$$ f\sim \sum_{\mathbf{k}} \hat f(\mathbf{k}) e^{i \mathbf{k} \mathbf{x}},$$
where now the sum is taken over all $\mathbf{k} = (k_1,k_2,\dots,)$ with integers $k_1,k_2,\dots,$ such that all these numbers but finitely many are equal to $0$. We consider the Riesz operator $R$ defined on the space $L^2(\mathbb{T}^\infty)$:
$$ Rf\sim \sum_{\mathbf{k} \ge 0} \hat f(\mathbf{k}) e^{i \mathbf{k} \mathbf{x}}.$$
We prove that for any $p>2, q>2$ the Riesz operator is not a bounded operator from $L^p$ to $L^q$.
The talk is based on a joint paper with Herve Queffélec, Eero Saksman,and Kristian Seip.