Harnack Inequalities For Weak Solutions Of Quasilinear Degenerate Elliptic Equations

Quasilinear degenerate elliptic equations have an important theoretical and practical significance in sub-Riemann geometry,harmonic analysis and geometry analysis and so on.This thesis mainly studied the Harnack inequalities of weak solutions to quasilinear degenerate elliptic equations constructed by H?rmander vector fields and homogeneous spaces,and the Green estimates of weighted subelliptic p-Laplace operator constructed by H?rmander vector fields,these results extend and improve the corresponding results in Euclidean setting.This work consists of three parts:In the first part(Chapter 2),we study the maximum principle and local boundedness of weak solutions to weighted subelliptic p-Laplace equation,where the coefficient satisfies the weighted uniformly elliptic conditions,and the non-homogeneous function belongs to weighted Lebesgue spaces.Different from the non-weighted cases,we first modified the Moser iteration technique and use the weighted Sobolev inequality to obtain maximum principle and local boundedness.and then proved Harnack inequality by John-Nirenberg inequality.By applications,we establish the H?lder continuity.The second part(Chapter 3) is concerned with the Green estimates of weight-ed subelliptic p-Laplace operator.Unlike the linear operators,we first prove the existence of the modified Green function by virtue of Minty-Browder theorem.and then the existence of the Green function by checking the convergence of se-quence of modified Green function.Next,we derive upper bounds of the modified Green function by establishing the interpolation inequality in the weighted weak Lebesgue spaces.Finally,the bounds of the Green function are also obtained by maximum principle and Harnack inequality.The last part(Chapter 4) is devoted to the local boundedness and Harnack inequality for the weak solutions to the following quasilinear degenerate elliptic equation with rough and singular coefficients in homogeneous spaces,under some structural conditions and the coefficients belong to some Stummel-Kato classes We establish the Fefferman-Phong inequality related to the Stummel-Kato classes firstly and derive an embedding inequality subsequently.Based on these inequalities,then we obtain the local boundedness,Harnack inequality,continuity and H?lder continuity for the weak solutions.