Over 100 years ago Harald Bohr identified a deep problem about the convergence of Dirichlet series, and introduced an ingenious idea relating Dirichlet series and holomorphic functions in high dimensions. Elaborating on this work, almost twnety years later Bohnenblust and Hille solved the problem posed by Bohr. In recent years there has been a substantial revival of interest in the research area opened up by these early contributions. This involves the intertwining of the classical work with modern functional analysis, harmonic analysis, infinite dimensional holomorphy and probability theory as well as analytic number theory. New challenging research problems have crystallized and been solved in recent decades. The goal of this book is to describe in detail some of the key elements of this new research area to a wide audience. The approach is based on three pillars: Dirichlet series, infinite dimensional holomorphy and harmonic analysis.
Author(s): Andreas Defant, Domingo García, Manuel Maestre, Pablo Sevilla-Peris
Publisher: Cambridge University Press
Year: 2019
Language: English
Pages: 709
Cover......Page 1
Front Matter
......Page 2
NEW MATHEMATICAL MONOGRAPHS......Page 3
Dirichlet Series and Holomorphic
Functions in High Dimensions......Page 4
Copyright
......Page 5
Contents
......Page 6
Introduction
......Page 12
PART ONE:
BOHR’S PROBLEM AND COMPLEX
ANALYSIS ON POLYDISCS......Page 30
1 The Absolute Convergence Problem......Page 32
2 Holomorphic Functions on Polydiscs......Page 66
3 Bohr’s Vision......Page 105
4 Solution to the Problem......Page 122
5 The Fourier Analysis Point of View......Page 140
6 Inequalities I......Page 158
7 Probabilistic Tools I......Page 182
8 Multidimensional Bohr Radii......Page 210
9 Strips under the Microscope......Page 234
10 Monomial Convergence of Holomorphic
Functions......Page 259
11 Hardy Spaces of Dirichlet Series......Page 297
12 Bohr’s Problem in Hardy Spaces......Page 318
13 Hardy Spaces and Holomorphy......Page 344
PART TWO:
ADVANCED TOOLBOX......Page 366
14 Selected Topics on Banach Space Theory......Page 368
15 Infinite Dimensional Holomorphy......Page 380
16 Tensor Products......Page 439
17 Probabilistic Tools II......Page 464
PART THREE:
REPLACING POLYDISCS BY OTHER
BALLS......Page 502
18 Hardy–Littlewood Inequality......Page 504
19 Bohr Radii in lp Spaces and Unconditionality......Page 515
20 Monomial Convergence in Banach Sequence
Spaces......Page 535
21 Dineen’s Problem......Page 560
22 Back to Bohr Radii......Page 584
PART FOUR:
VECTOR-VALUED ASPECTS......Page 594
23 Functions of One Variable......Page 596
24 Vector-Valued Hardy Spaces......Page 613
25 Inequalities IV......Page 641
26 Bohr’s Problem for Vector-Valued Dirichlet
Series......Page 674
References......Page 693
Symbol Index......Page 706
Subject Index......Page 707
