Approximation in the Zygmund Class and Distortion under Inner Functions

Student thesis: Doctoral thesis


In this work we deal with two different problems. The first one is an approximation problem in the Zygmund class by functions in the subspace I_1(BMO), which is the space of continuous functions with derivative in BMO in the sense of distributions. We consider the distance defined by the Zygmund semi-norm. In Chapter 1, given a function f in the Zygmund class in the real line with compact support, we find an estimate of its distance to the subspace I_1(BMO). In addition, this result is expressed in terms of the second differences of f, which define its Zygmund semi-norm. As a corollary, we obtain a characterisation of the closure of I_1(BMO) in this semi-norm. The methods presented in this first part are not applicable to the Zygmund class in the euclidean space of dimension n>1. However, we present an analogous result for Zygmund measures in dimension n>=1. In this case, the subspace that we consider is the space of absolutely continuous measures with Radon-Nykodim derivative in BMO. In Chapter 2, we consider the space of Hölder continuous functions with parameter 0=1. For 0
Date of Award15 Jul 2020
Original languageEnglish
SupervisorArturo Nicolau Nos (Director)

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