Resumen de Aproximació en la classe de zygmund i distorsió per funcions internes
Odà Soler i Gibert
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 n1. 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 0s=1 (taking the Zygmund class when s=1) defined on the euclidean space of dimension n=1. For 0s=1, the image of BMO under the Riesz potential I_s in the sense of tempered distributions modulo polynomials, denoted by I_s(BMO), is a subspace of the Hölder class of parameter s. In particular, in the case n=s=1 this definition coincides with that of I_1(BMO) in Chapter 1. Then, given a function f in the Hölder class of parameter 0s=1 with compact support, we find an estimate of its distance to I_s(BMO). We take here the distance defined by the corresponding Hölder semi-norm. As before, this result is also expressed in terms of the second differences of f. Moreover, we also show two more equivalent estimates. One in terms of the coefficients of the wavelet series of f and the other in terms of the second hyperbolic derivative of the harmonic extension of f on the upper half-space.
In Chapter 3, we study a problem about inner functions in the unit disk. Löwner's Lemma states that the Lebesgue measure in the unit circle is invariant under any inner function f fixing the origin. In other words, every measurable set E in the unit circle has the same Lebesgue measure as its preimage under f. In the case that an inner function has no fixed point in the unit disk, C. I. Doering and R. Mañé studied an infinite measure in the unit circle that depends on the Denjoy-Wolff fixed point of f and that has the property of being quasi-invariant. Here we generalise the result of Doering and Mañé taking any point of the unit circle to define their measure. In a similar way, J. L. Fernández and D. Pestana studied the distortion of Hausdorff contents in the unit circle under inner functions fixing the origin. Using a Hausdorff content based in the measure defined by Doering and Mañé, we present a generalisation of the result of Fernandez and Pestana for inner functions without fixed points in the unit disk.
