room: A1-33, godz. 10:30–12:00

Andreas Defant (Carl von Ossietzky Universität, Niemcy)
\(\mathcal{H}_p\)-Theory of Dirichlet series: A Few Aspects

Summary. For a finite Dirichlet polynomial \(D(s) = \sum_{n\leq x}a_nn^{-s}\) and \(1\leq p<\infty\) the \(\mathcal{H}_p\)-norm is given by \[ \|D\|_{\mathcal{H}_p}=\lim_{T\to\infty} \bigg(\frac{1}{2T} \int_{-T}^T \bigg| \sum_{n\leq x}a_n n^{-it} \bigg|^pdt\bigg). \] This definition goes back to the early work of A.S. Besicovitch on almost periodic functions, and it in recent years motivated a sort of \(H_p\)-theory of Dirichlet series \(D = \sum_{n} a_n n^{-s}\). The aim of this talk is to recall some of the basic definitions, understanding that this theory can be equivalently seen as the study of Hardy spaces \(H_p (\mathbb{T}^\infty)\) of functions on the infinite dimensional torus \(\mathbb{T}^\infty\). We will focus on the following formula which for \(1 \leq p < q < \infty\) compares the Hardy norms \(\|D\|_{\mathcal{H}_p}\) and \(\|D\|_{\mathcal{H}_q}\) of finite Dirichlet polynomials \(D\) of length \(x\): \[ \sup_{D=\sum_{n\leq x} a_n n^{-s}} \frac{\|D\|_{\mathcal{H}_q}}{\|D\|_{\mathcal{H}_p}} = \text{exp} \bigg( \frac{\log x}{\log \log x} \bigg( \log \sqrt{\frac{q}{p}}+O\bigg( \frac{\log \log \log x}{\log \log x} \bigg) \bigg) \biggr), \] and we try to illustrate how this equality is linked with various interesting topics in the field. Joint work with Antonio Pérez, Murcia.

Tea and coffee at 10:00.