The Construction Of Two Kinds Of Doubling Weights On The Real Line

Doubling measure on a metric space extends Lebesgue measure on Euclidean s-pace.It is a fundamental concept to do analysis on metric space and,at the same time,it is an important object in the study of geometric measure theory(reference[1]).In this paper we discuss doubling measures on the line.As is known to all,there are doubling measures on the real line that are singular with respect to the Lebesgue measure(reference[19]).We only consider absolutely continuous doubling measures on the line.David Cruz-Mr Uribe Sfo proved the following theorem in[2].Theorem,If ω is a decreasing doubling weight on[0,∞),then there exists a constantα∈(0,1)such that αω(t)≤ω(2t)for all t ∈[0,+∞);conversely,if a decreasing nonnegative function ω on[0,oo)satisfies the inequality αω(t)≤ω(2t)for some α∈(1/2,1)and for all t ∈[0,+∞),then ω is a doubling weight.The theorem leaves us two problems.Question 1.Let α∈(0,1).Is there a decreasing doubling weight ω on[0,∞)such that αω(t)≤ω(2t)holds for all t ∈[0,+∞)?Question 2.Let ω be a decreasing nonnegative function on[0,∞)satisfying the in-equality αω(t)≤ω(2t)for some α ∈(0,1/2]and for all t∈[0,+∞).Is ω a doubling weight?We will answer negatively these two questions by comparing the convergent ve-locities of the general term and the tail term of positive term series and the properties of the maximum function(reference[17,20]).This paper mainly consists of five chapters.In chapter 1,we briefly describe the background of this thesis.In chapter 2,we introduce doubling weights and the rele-vant concepts on the line with an emphasis on the known results of doubling weights.Chapter 3 is devoted to comparing the convergent velocities of the general term and the tail term of positive term series.In chapter 4,we discuss questions 1 and 2.Using the results obtaining in Chapter 3,we answer negatively questions 1 and 2 by constructing two classes of weights.In Chapter 5,we shall present some further problems that are closely related to our results.