Dirichletreihen und komplexe Analysis für Funktionen von unendlich vielen Variablen
Zusammenfassung der Projektergebnisse
A series ∑ an e−λn s , where an ’s are complex coefficients, s a complex variable, and λ = (λn ) a frequency (i.e. a strictly increasing unbounded sequence of non-negative real numbers) is called a general Dirichlet series, shortly a λ -Dirichlet series. The study of such series, and the analytic functions they generate, lived a golden moment in the first third of the 20th century, with deep contributions of Bohr, Bohnenblust, Hardy, Hille, Riesz and Landau among others. Since then ordinary Dirichlet series ∑ an n−s , so ((log n))-Dirichlet series, play a fundamental role in analytic number theory. But after the 1930s the area of Dirichlet series as a subject of independent interest lost its influence (ordinary or general), until the mid 1990s when several important contributions again called the attention. A new field emerged, intertwining the classical work in novel ways with modern methods. So far a growing number of researchers with different backgrounds have become engaged in the area. As a result of this work, a number of challenging research problems have been solved which required unconventional combinations of expertise from functional analysis, harmonic analysis, infinite dimensional holomorphy, probability theory, as well as analytic number theory. This ’modern theory of Dirichlet series’ mainly focuses on the study of ordinary series ∑ an n−s , and the major goal of our project was to extend relevant parts to general Dirichlet series ∑ an e−λn s . Making the jump from the ordinary case λ = (log n) to arbitrary frequencies λ reveals serious difficulties, and in fact many of the fundamental ideas and techniques ruling the ordinary theory fail for general Dirichlet series. The main conceptional contribution is the novel invention of Hardy spaces H p (λ ) of λ - Dirichlet series. For 1 ≤ p ≤ ∞ these spaces may be defined as follows: Given a λ - Dirichlet polynomial D = ∑N an e−λn s define its norm by n=1 1 T D p = lim |D(it)| p dt. T →∞ 2T −T Then H p (λ ) is the completion of the linear space of all these λ - Dirichlet polynomials with this norm. For the ordinary case λ = (log n) this construction is due to Bayart. But already in the ordinary case, and here even for the Hilbert case p = 2, this point of view does not capture all relevant information on these Hardy spaces. Thanks to the so-called Bohr transform, most of the interesting results on them come by identifying H p ((log n)) with the corresponding Hardy space of functions on the infinite polytorus. But this transformation depends heavily on the factorization of n into a product of prime numbers, namely on the multiplicative structure of the sequence (log n). So at first glance it seems that there is no hope to use such a tool in the context of a general frequency λ . The main success of our work is to overcome this difficulty. We show that without making any assumption on the sequence λ , the spaces H p (λ ) may be identified with some Hardy space λ H p (G) of functions which are defined on (what we call) λ -Dirichlet groups G. This initiates a new abstract theory of Hardy spaces of general Dirichlet series, which allows to transport fundamental tools from functional analysis, harmonic analysis on compact abelian groups, Fourier analysis on R, and the theory of holomorphic almost periodic functions on half-planes, to the environment of general Dirichlet series.
Projektbezogene Publikationen (Auswahl)
- Dirichlet series and holomorphic functions in high dimensions; New Mathematical Monographs, 37. Cambridge University Press, Cambridge, 2019
A. Defant, D. García, M. Maestre, and P. Sevilla-Peris
- Hardy spaces of general Dirichlet series – a survey; Banach Center Publications 119 (2019) 123-149
A. Defant and I. Schoolmann
(Siehe online unter https://doi.org/10.4064/bc119-6) - Henry Helson meets other big shots – a brief survey; Math. Proc. Roy. Irish Acad. 119 A(2) (2019) 111-132
A. Defant and I. Schoolmann
(Siehe online unter https://doi.org/10.1353/mpr.2019.0002) - Hardy spaces of general Dirichlet series and their maximal inequalities, PhD thesis, Carl von Ossietzky University Oldenburg (2020), 216 pages
I. Schoolmann
