Maximal theorems and Calderón-Zygmund type decompositions forthe fractional maximal function

Sammanfattning: A very significant role in the estimation of different operators in analysis is played by the Hardy-Littlewood maximal function. There are a lot of papers dedicated to the study of properties of it, its variants, and their applications. One of the important variants of the Hardy-Littlewood maximal function is the so-called fractional maximal function, which is deeply connected to the Riesz potential operator. The main goal of the thesis is to establish analogues of some important properties of the Hardy-Littlewood maximal function for the fractional maximal function. In 1930 Hardy and Littlewood proved a remarkable result, known as the Hardy-Littlewood maximal theorem. Therefore its naturally arose a problem: what is an analogue of the Hardy-Littlewood maximal theorem for the fractional maximal function? In the thesis we will give an answer for this problem. Particularly, we will show that the so-called Hausdorff capacity and Morrey spaces, introduced by C. Morrey in 1938 in connection with some problems in elliptic partial differential equations and the theory of variations, naturally appears here. Moreover, recently Morrey spaces found important applications in connection with the Navier-Stokes and Schrödinger equations, elliptic problems with discontinuous coefficients and potential theory. The Hardy-Littlewood maximal theorem is deeply connected with the Stein-Wiener and Riesz-Herz equivalences. Analogues of these equivalences for the fractional maximal function are also given. In 1971 C. Fefferman and E. Stein, by using the Calderón-Zygmund decomposition, obtained the generalization of the maximal theorem of Hardy-Littlewood for a sequence of functions. This result of Fefferman and Stein found many important applications in Harmonic Analysis and its applications, e.g. in Signal Processing. In the thesis we will give an analogue of one part of the Fefferman-Stein maximal theorem for the fractional maximal operator. In 1952 A. Calderón and A. Zygmund published the paper "On Existence of Certain Singular Integrals", which has made a significant influence on the Analysis of the last 50 years. One of the main new tools used by A. Calderón and A. Zygmund was a special family of the decomposition of a given function in its "good" and "bad" parts. This decomposition provides a multidimensional substitution of the famous "sunrise" lemma by F. Riesz and it was used for proving a weak-type estimate for singular integrals. Furthermore, we want to emphasize that Calderón-Zygmund type decompositions have played an important and sometimes crucial role in the proofs of many fundamental results, such as the John-Nirenberg inequality, the theory of Ap-weights, Fefferman-Stein maximal theorem, etc. In the thesis it is showed that it is possible to construct an analogue of the Calderón-Zygmund decomposition for the Morrey spaces.