ARMONIC ANALYSIS

Author: INFANTE LINARES ADRIAN RAMON.
Year: 2004.
University: AUTÓNOMA DE MADRID [www.uam.es].
Place of defense: UNIVERSIDAD AUTONOMA DE MADRID.
Place of preparation: AUTONOMA DE MADRID.

Summary: One of the great achievements of Analysis Armónico in the past century was the introduction of the role of maximal Hardy allowing clarify certain phenomena of convergence, including theorem differentiation Lebesgue. Its importance was even more important with the development of the theory of singular integrals Calderon and Zygmund, since that operadorb controls in a way the singularities of them. Since then, there has been considerable interest in knowing what other ways and contexts it is possible to define this operator maximal maintaining their properties regularly and marking, for example replacing the balls Euclídeas by other bodies geometric or changing the underlying Lebesgue measure by other measures anisotrópicas. The aim of the thesis is to deepen the study of the following problem: To determine conditions under which the operator of maximal Hardy associated with a measure of Borel, an operator is bounded on the area of Lebesgue or some space Orlicz . The answer to this question in the space Euclídeo dimensional is easy to find using a simple slogan geomérico coating. In this case it appears that the maximal operator associated with any measure, is a weak (1.1). In higher dimensions, the same result is true if we assume that the measure is doblante, because in this case you can use the slogan Vitali. Also known results in the other direction. The example of P. Sjà ¶ gren demonstrates that the operator associated with the maximal extent Gaussiana, a measure that radio is clearly not doblante, is not a weak (1.1). This result was widespread by A. Vargas where presents conditions that characterize the measures invariant porrotaciones supported on the entire space, so that the operator maximal associated with this type of measure is weak (1.1). Sededuce of this work that these measures are precisely those that are doblantes "away" from home. Subsequently, Forzani, Scotto, Sjà ¶ gren and Urbina show that the operator associated with the maximal extent Gaussiana is a strong (p, p), where p> 1. Recently P. Sjà ¶ gren and F. Soria investigated the properties of integrabilidad operator maximal associated with certain types of non doblantes radio and decreasing among which the Gaussiana and showed that the operator is a strong (p, p), p> 1. The result is obtained by showing that in fact these operators meet certain inequality modular described in Section 1.2.5) on an area of Orlicz near L1. The idea of the test is to replace the balls for other geometric figures of comparable measures and maximal operator whose partner has a simple description. This result is the starting point of this work. In order to substantiate the content of this memory, dedicate Chapter 1 describes some of these results known. We started this chapter mentioning some of the results that can be obtained from geometric slogans coating and then focuses on the operator associated with maximal measures Borel defined radial ball where coating techniques are not valid and which will play a the geométria of role so far. The property doblante, or how it degenerates, it is crucial to get this property, as described in Section 1.2.5, where the results compiled by P. Sjà ¶ gren and F. Soria. In addition to these results is presented in this chapter a counterexample demonstrates the necessity of the conditions imposed on the measures considered in the theorem P. Sjà ¶ gren and F. Soria. Inspired by the techniques introduced by P. Sjà ¶ gren and F. Soria, and to explore the natural extensions of estosresultados, is presented in Chapter 2 a study of maximal operator associated with some kind of measures radial and growing. We show here that also verifies the same kind of modular inequality. This result can be calculated marking strong type (p, p), for any p & g 8 t; 1. Res 849 ulta natural explore below the maximal operator defined buckets and associated with a class action radial and decreasing. We extend these studies to cubes whose sides are not parallel to the axes. Here is also apparent that the maximal operator is a strong partner (p, p), p> 1. To compare this with the extent of the child's cube that contains cone. This study suggests the operator defined maximal external and regular cones. These definitions and results are described in Chapter 3. An important part of the memory estádedicada the explicit calculation of the extent of the figures geoméricas used to define operators maximal under study, cones, cubes, balls, hiperplanos, etc.. Without wishing to minimize complications ténicas that this results, we preferred to be thorough when submitting these calculations to facilitate the task of the reader, at the risk of being repetitive. Chapter 4 is devoted to other problems related to the size of the constants that appear on the type of inequalities weak. Other test results that the maximal operator associated with a class action growing, and functions defined for radio, is bounded with constant independent of the size. The report closes with a few appendages in order to influence various aspects mentioned along the same.