The Generalization And Applications Of The Rising Sun Lemma

This paper mainly revolves around Riesz’s Rising Sun Lemma. Firstly it introduced the Lebesgue theorem and some basic knowledges about Hardy-Littlewood maximal function, And then it proved the Lebesgue theorem, introducted some definitions about Mximal Function. In order to describe a property of maximal function later, it introduced an important theorem of functional analysis-Marcinkiewisz theorem.In chapter two, firstly it made a further elaboration for the Rising Sun Lemma. Nextly it introduced the Calderon-Zygmund decomposition and a result about n dimensional generalization of the Rising Sun Lemma. Finally the good results of the Rising Sun Lemma will be generalized to two-dimensional and three-dimensional, and it gives the detailed proofs.This paper got two conclusions about the employments of the Rising Sun Lemma, they are best constants for uncentered maximal function and the classic problem in calculus-The differentiability of the monotone continuous function.