Wavelet Decomposition Of Homogeneous Type Space Associated With Tricomi Operator

In this thesis,wavelet decomposition of L2(R+2)functions on space(R+2,?,?)associated with degenerated elliptic equations are constructed,where p is quasi-distance(see chapter 2),? is Lebesgte measure.The innovation is that the supports of wavelets are dyadic "squares" with different shapes in different positions of R+2,and the specific geometry of the dyadic "squares" will be very helpful to study the boundedness of the singular integral operators associated with the Tricomi operator.The following is the structure of this thesis:In chapter ? we review the classical wavelet decomposition of L2(Rn)function on Euclidean space ?,where the wavelets supported on dyadic cubes which have the same length at any direction.Wavelets in homogeneous space Rn supported on dyadic cubes which have the different length at any direction(see chapter 3).Under the same scale,the length of each edge of above two classes of "cubes" has nothing to do with center coordinate of cube.In chapter 2 we construct wavelet decomposition of L2(R+2)function on homogeneous space(R2+,?,?)associated with Tricomi operator,where the wavelets supported on dyadic "cubes(squares)" which has different length at x and y direction.Different from "cubes" in Euclidean space and homogeneous space,the side length of "cubes" in homogeneous type space under the same scale varies with the y coordinate of cube center.In chapter 3 we extend anisotropic wavelet bases and the anisotropic function spaces on Rn constructed by Triebel to homogeneous type space(R2,?,?).