El mètode de Phragmén-Lindelöf i aplicacions: teorema de Riesz-Thorin i d’incertesa de Hardy

Abstract

[en] The Maximum Modulus Principle, which is one of the most important results in complex analysis, states that a holomorphic function defined on a bounded domain of $\mathbb{C}$, takes its maximum value at some point from the domain's boundary. Hence, the objective of this work is to introduce and apply the Phragmén-Lindelöf method in order to extend the conclusions given by the Maximum Modulus Principle to unbounded domains. Furthermore, this method will be used to see some applications such as: the Hadamard Three Lines Theorem, which provides good enough bounds for holomorphic functions on vertical strips; the Riesz-Thorin Interpolation Theorem, which establishes that a linear operator between measurable function spaces is bound in certain Lebesgue spaces $L^p$; and the Hardy's Uncertainty Principle, which claims that a measurable function and its Fourier transform cannot simultaneously have compact support, unless they both are identically zero.