Minicourse Introduction to Tauberian theory. A distributional approach

Tauberian theory provides striking methods to attack hard problems in analysis. The study of Tauberian type theorems has been historically stimulated by their potential applications in diverse fields of mathematics such as number theory, combinatorics, complex analysis, probability theory, and the analysis of differential operators [1, 3, 7, 8, 17, 19]. Even mathematical physics has pushed forward developments of the subject; indeed, theoretical questions in quantum field theory motivated the incorporation of generalized functions into the theory [17]. The aim of this minicourse is to give a modern introduction to Tauberian theory via distributional methods. The central topic is Tauberian theorems for the Laplace transform of (Schwartz) distributions with applications to prime number theory and PDE with constant coefficients. It will also be shown how to recover the classical Tauberian theorems from their distributional versions. The minicourse consists of four parts. The first part is dedicated to explain the nature of Tauberian theory through classical examples and its formulation from a functional analysis perspective. Basics from distribution theory will be also recalled. In the second part, we study one-dimensional Hardy–Littlewood–Karamata type Tauberian theorems. The third part is devoted to complex Tauberian theorems and applications to the theory of (Beurling) generalized prime numbers. The final part deals with multidimensional theory; the results will be applied to asymptotics of solutions to convolution equations, in particular, hyperbolic PDE. The minicourse aims to give a modern introduction to Tauberian theory for generalized functions and some of its applications. We will focus in Tauberian theorems for the Laplace transform and applications to number theory and PDE. It is intended for a general audience ranging from (advanced) undergraduate and graduate students to experienced researchers. It only requires knowledge of basics from functional analysis (specifically, the Hahn–Banach and Banach–Steinhaus theorems). Some familiarity with the Laplace transform and distribution theory would be helpful, though not a requirement. The course consists of five lectures and, conceptually, it may be divided into four parts: Tauberian theory and functional analysis methods We will introduce the main problem in Tauberian theory through classical examples. We start with various summability procedures for divergent series and integrals and state their corresponding Abelian and Tauberian theorems; special attention will be paid to the summability methods by Abel, Cesàro, and Lambert means. We then reinterpret the particular cases within the language of functional analysis, and explain a general scheme to attack problems in Tauberian theory; the