Dirichlet Series and Holomorphic Functions in High Dimensions

Andreas Defant is Professor of Mathematics at Carl V. Ossietzky Universität Oldenburg, Germany. Domingo García is Professor of Mathematics at Universitat de València, Spain. Manuel Maestre is Full Professor of Mathematics at Universitat de València, Spain. Pablo Sevilla-Peris is Associate Professor of Mathematics at Universitat Politècnica de València, Spain.

• Introduction; Part I. Bohr's Problem and Complex Analysis on Polydiscs: 1. The absolute convergence problem; 2. Holomorphic functions on polydiscs; 3. Bohr's vision; 4. Solution to the problem; 5. The Fourier analysis point of view; 6. Inequalities I; 7. Probabilistic tools I; 8. Multidimensional Bohr radii; 9. Strips under the microscope; 10. Monomial convergence of holomorphic functions; 11. Hardy spaces of Dirichlet series; 12. Bohr's problem in Hardy spaces; 13. Hardy spaces and holomorphy; Part II. Advanced Toolbox: 14. Selected topics on Banach space theory; 15. Infinite dimensional holomorphy; 16. Tensor products; 17. Probabilistic tools II; Part III. Replacing Polydiscs by Other Balls: 18. Hardy–Littlewood inequality; 19. Bohr radii in lp spaces and unconditionality; 20. Monomial convergence in Banach sequence spaces; 21. Dineen's problem; 22. Back to Bohr radii; Part IV. Vector-Valued Aspects: 23. Functions of one variable; 24. Vector-valued Hardy spaces; 25. Inequalities IV; 26. Bohr's problem for vector-valued Dirichlet series; References; List of symbols; Subject index.

'Dirichlet series have been studied for well over a century and still form an integral part of analytic number theory … The purpose of this text is to illustrate the connections between the Dirichlet series per se and the fields just mentioned, e.g., both functional and harmonic analysis … The authors succeed in transferring important concepts and theorems of analytic function theory, in finitely many variables, to the theory in infinitely many variables.' J. T. Zerger, Choice