room: A1-33, godz. 10:30–12:00
Andreas Defant (Carl von Ossietzky Universität, Niemcy)
Hp-Theory of Dirichlet series: A Few Aspects
Summary. For a finite Dirichlet polynomial D(s)=∑n≤xann−s and 1≤p<∞ the Hp-norm is given by ‖D‖Hp=limT→∞(12T∫T−T|∑n≤xann−it|pdt). 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 Hp-theory of Dirichlet series D=∑nann−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 Hp(T∞) of functions on the infinite dimensional torus T∞. We will focus on the following formula which for 1≤p<q<∞ compares the Hardy norms ‖D‖Hp and ‖D‖Hq of finite Dirichlet polynomials D of length x: supD=∑n≤xann−s‖D‖Hq‖D‖Hp=exp(logxloglogx(log√qp+O(logloglogxloglogx))), 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.