Some fundamental topics in the theory of real-valued functions and their applications in harmonic analysis

Author: Giorgi Gognadze
Keywords: Maximal function, distribution functions, singular integrals
Annotation:

Many topics in the real-variable function theory are related to the operations of integration and differentiation. One of the main results of this theory is Lebesgue’s theorem on differentiability of integrals. Usually, it is more convenient to differentiate integrals in terms of maximal function. The term “Maximal Function” emerged in the process of investigation of these topics and its properties in most cases can be written by means of weak inequalities. Other important topics are theorems about covers. The main idea here is that any open set can be covered by pair-wise disjoint cubes or balls (the choice of the type of a cover depends on the topic being analyzed). One of the main topics in the real-variable function theory represents Marcinkiewicz’s interpolation theorem, which is widely used in proofs of many other statements. Singular integrals which commute with dilations and translations are also discussed.

Lecture files:

ნამდვილი ცვლადის ფუნქციათა თეორიის ზოგიერთი ფუნდამენტური საკითხი და მათი გამოყენება ჰარმონიულ ანალიზში [ka]