PDOs Related To Carnot Groups And Generalized Greiner Vector Field

The thesis consists of two parts.一. PDOs on Carnot groupsPDEs on H-type groups which are a special class of Carnot groups of step two are considered. First, we construct the polar coordinates for H-type groups. Using this we explicitly compute the volume of balls in the sense of the distance and the constant in the fundamental solution of the p-sub-Laplacian on the N-type group. Then we discuss nonexistence of weak solutions for some partial differential inequalities on H-type groups. We prove nonexistence results of weak solutions for a degenerate elliptic inequality, a degenerate first order evolution inequality and a degenerate second order evolution inequality on the H-type group by the method of integral relations. Roughly speaking, this approach is based on the derivation of suitable a priori bounds on the possible weak solutions of the problem under consideration by carefully choosing special nonnegative test functions and scaling argument. It avoids use of comparison or maximum principle arguments and is useful in the study of some general problems.In succession nonexistence of "general positive local solutions" for a doubly nonlinear degenerate parabolic equation on Carnot group G is studied. This equation has very strong real background and important application value. In particular we take into account the singular potential V=γ|â–½_GN|^p／N^p(γ＞0). Our results indicate that nonexistence of positive solutions to this equation is intimately related to Hardy's inequality on polarizable Carnot groups.二. PDOs related to generalized Greiner vector fieldGeneralized Greiner vector field is k≥1, which is not the basis for any nilpotent Lie gronp, does not possesses the translation invariance and does not satisfies Hormander's hypoellipticity condition. Generalized Greiner operatorΔ_L related to it is no longer the translation invariante differential operator on any nilpotent Lie group, but is a quasihomogeneous PDO.It is still an open problem whether the result similar to the geometric maximum principle for elliptic equations holds in the situation of generalized Greiner vector field. This open problem can be stated as follows: LetΩ(?)R²ⁿ⁺¹ be a connected, bounded open set and f∈L^Q(Ω). Suppose that u∈L_loc^{2, Q} (Ω)∩C((?)) satisfies Lu=sum from i,j=1 to 2n a_iju,_ij≥f inΩ. There exists a constant C=C (Q, v,Ω)＞0 such that (?) u≤(?) u⁺ + C‖f‖_L^Q(Ω). Through constructing solutions to nondivergence form equations associated to generalized Greiner vector field, we point out that in this case the L^Q norm of f in the right-hand side is the best possible character.Hardy type inequalities and Rellich type inequalities for generalized Greiner vector field are established. They play important roles in the study of PDEs. We yield some Hardy type inequalities on the domain in R²ⁿ⁺¹ by constructing suitable auxiliary functions; we give generalized Picone type identities for p—subLaplacianΔ_{L, p} related to generalized Greiner vector field and applying it get a generalized Hardy type inequality; we prove that the constant is sharp in the Hardy type inequality of generalized Greiner operatorΔ_L. A representation formula for smooth functions with compact support is found. Then a Hardy type inequality involving part variables is established and the optimality of the constant is proved. Finally we devote to researching improved Hardy type inequalities and Rellich type inequalities associated to generalized Greiner vector field. Some Hardy type inequalities and Rellich type inequalities with remainder terms are gained by improving inequalities established.