On Piecewise Monotonic Doubling Weights On The Real Line

Let ? be a locally integrable function on R.We say that ? is a weight function on R if ?(t)>0 for almost all t ? R.Moreover,if ?(A)=? A ?(t)dt is a doubling measure on R,we say that ? is a doubling weight on R.Similarly,we may define the double weight on an interval I of R.In the present thesis,we discuss piecewise monotonic weights and piecewise dou-bling weights.The questions are as follows:1.Under what conditions a piecewise monotonic weight is doubling?2.Under what conditions a piecewise doubling weights is doubling on the whole line?We obtained various sufficient and necessary conditions for a weight in the above two classes.The paper consists of four parts.In Part 1,we shall present some backgrounds and known results on doubling measures and doubling weights.Using an equivalent condition of doubling measure.In Part 2,we obtain various conditions for a piecewise monotonic weight to be a doubling weight.In Part 3,we obtain various conditions for a piecewise doubling weight to be a doubling weight on the whole line.In Part 4,we present several problems that are closely related to our study,for example,what are the.conditions for a weight on higher dimensional Euclidean space to be doubling;what are the difference between doubling weights and A? weights,etc.