Homogeneous Measures And A₁ Weights On Ahlfors Regular Spaces And Quasisymmetrically Minimality And Dimension Of Moran Sets

In this dissertation, we first study the d-homogeneous measures on an Ahlfors d-regular space; Then we are concerned with Quasisymmetrically minimality and Hausdorff dimen-sion of a great deal of Moran sets on the line; At last, we discuss some properties and applications of A1 weight on doubling metric spaces. The paper contains three parts.The first part, let X be a compact metric space, Kaufman and Wu showed all doubling measures on X are mutually absolutely continuous if and only if X carries only purely atomic doubling measures. Jonsson investigated the absolute continuity of measures with a refined doubling condition on a d-set in Euclidean space. In the first part of this dissertation, we study the d-homogeneous measures on an Ahlfors d-regular space (X, m). It is proved that any two d-homogeneous measures on X are mutually absolutely continuous. Moreover, we prove that ifμis a measure on X which is continuous with respect to m, thenμis d-homogeneous if and only if the Radon-Nikodym derivative Dmμ(x) is an A1 weight. Finally, we show there exists a d-homogeneous but not d-regular measure on X.The second part, let f:Rn→Rn is a quasi symmetric mapping, E(?)Rn. How can f charge or change the dimensions of E is an important question of quasisymmetric mapping. Bishop, Tyson, Kovalev and Hakohyan have gotten some interesting conclusions on this aspect. In the second part of this dissertation, we mainly discuss the quasisym-metrically minimality of Moran class. Suppose E (?) R is a Moran set associated with (J,｛nk｝k≥1,{ck,j}1>j≤nk1,k≥1), we give conditions on which E satisfies We also obtain a general theorem on the Hausdorff dimension of Moran sets on the line. The main tool is some Gibbs-like measures. In chapter 4, we will give the construction and properties of these measures. Together with the mass distribution principle we estimate the infimum of Hausdorff dimension.The third part, we know that A∞weights gave a general maximal function theorem in Euclidean space. In fact, Semmes and Heinonen guessed A1 weight have associations with bilipchitz embedding problem and quasiconformal Jacobian problem. Semmes, Bishop, Bonk and Heinonen have gotten some interesting conclusions on this aspects. In the last part of this dissertation, we study some properties and applications of A1 weight on doubling metric spaces. The basic thought is "dyadic" division of a ball in a doubling metric space which is analogues of the Euclidean dyadic cubes. So we also have Calderon-Zygmund theorem and reverse Holder inequality.