Teoria de diferenciació d’integrals

Abstract

The theory of differentiation of integrals comes from the widely known theorem of Lebesgue. One could think that taking on this theorem euclidean balls instead of other type of sets might well be irrelevant. But it’s not true. It became a difficult problem to find out whether the replacement of euclidean balls by other type of sets in the Lebesgue theorem would lead to a true statement or not. The aime of this work is to present the theory of differentiation of integrals as an interaction between covering properties of families of sets in R n , estimations for an adequate extension of the maximal operator of Hardy and Littlewood and differentiation properties. First chapter is devoted to the main covering theorems that are used in the subject. The second one introduces the notions of a differentiation basis and the maximal operator associated to it. Third chapter hows how closely related are the properties of the maximal operator and the differentiation properties of a basis. Finally, in the fourth chapter we solve some classical problems: the Perron tree, the Kakeya problem and the Nikodym set.