Linear functions and duality on the infinite polytorus

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.Comment: This paper has been accepted for publication in Collectanea Mathematic

Sharp norm estimates for composition operators and Hilbert-type inequalities

Let $\mathscr{H}^2$ denote the Hardy space of Dirichlet series $f(s) = \sum_{n\geq1} a_n n^{-s}$ with square summable coefficients and suppose that $\varphi$ is a symbol generating a composition operator on $\mathscr{H}^2$ by $\mathscr{C}_\varphi(f) = f \circ \varphi$. Let $\zeta$ denote the Riemann zeta function and $\alpha_0=1.48\ldots$ the unique positive solution of the equation $\alpha\zeta(1+\alpha)=2$. We obtain sharp upper bounds for the norm of $\mathscr{C}_\varphi$ on $\mathscr{H}^2$ when $0<\operatorname{Re}\varphi(+\infty)-1/2 \leq \alpha_0$, by relating such sharp upper bounds to the best constant in a family of discrete Hilbert-type inequalities.Comment: This paper has been accepted for publication in Bulletin of the LM

High pseudomoments of the Riemann zeta function

The pseudomoments of the Riemann zeta function, denoted $\mathcal{M}_k(N)$, are defined as the $2k$th integral moments of the $N$th partial sum of $\zeta(s)$ on the critical line. We improve the upper and lower bounds for the constants in the estimate $\mathcal{M}_k(N) \asymp_k (\log{N})^{k^2}$ as $N\to\infty$ for fixed $k\geq1$, thereby determining the two first terms of the asymptotic expansion. We also investigate uniform ranges of $k$ where this improved estimate holds and when $\mathcal{M}_k(N)$ may be lower bounded by the $2k$th power of the $L^\infty$ norm of the $N$th partial sum of $\zeta(s)$ on the critical line.Comment: This paper has been accepted for publication in Journal of Number Theor

Minimal norm Hankel operators

Let $\varphi$ be a function in the Hardy space $H^2(\mathbb{T}^d)$. The associated (small) Hankel operator $\mathbf{H}_\varphi$ is said to have minimal norm if the general lower norm bound $\|\mathbf{H}_\varphi\| \geq \|\varphi\|_{H^2(\mathbb{T}^d)}$ is attained. Minimal norm Hankel operators are natural extremal candidates for the Nehari problem. If $d=1$, then $\mathbf{H}_\varphi$ has minimal norm if and only if $\varphi$ is a constant multiple of an inner function. Constant multiples of inner functions generate minimal norm Hankel operators also when $d\geq2$, but in this case there are other possibilities as well. We investigate two different classes of symbols generating minimal norm Hankel operators and obtain two different refinements of a counter-example due to Ortega-Cerd\`{a} and Seip