In this article, we deal with weak solutions to non-degenerate sub-elliptic equations in the Heisenberg group, and study the regularities of solutions. We establish horizontal Calderón-Zygmund type estimate in Be...In this article, we deal with weak solutions to non-degenerate sub-elliptic equations in the Heisenberg group, and study the regularities of solutions. We establish horizontal Calderón-Zygmund type estimate in Besov spaces with more general assumptions on coefficients for both homogeneous equations and non-homogeneous equations. This study of regularity estimates expands the Calderón-Zygmund theory in the Heisenberg group.展开更多
In the 1970’s, Folland and Stein studied a family of subelliptic scalar operators $ \mathcal{L}_\lambda $ which arise naturally in the $ \bar \partial _b $ -complex. They introduced weighted Sobolev spaces as the nat...In the 1970’s, Folland and Stein studied a family of subelliptic scalar operators $ \mathcal{L}_\lambda $ which arise naturally in the $ \bar \partial _b $ -complex. They introduced weighted Sobolev spaces as the natural spaces for this complex, and then obtained sharp estimates for $ \bar \partial _b $ in these spaces using integral kernels and approximate inverses. In the 1990’s, Rumin introduced a differential complex for compact contact manifolds, showed that the Folland-Stein operators are central to the analysis for the corresponding Laplace operator, and derived the necessary estimates for the Laplacian from the Folland Stein analysis. In this paper, we give a self-contained derivation of sharp estimates in the anisotropic Folland-Stein spaces for the operators studied by Rumin using integration by parts and a modified approach to bootstrapping.展开更多
Let Ω be a bounded open domain in Rn with smooth boundary Ω, X =（X1,X2,... ,Xm） be a system of real smooth vector fields defined on Ω and the bound-ary Ω is non-characteristic for X. If X satisfies the HSrmande...Let Ω be a bounded open domain in Rn with smooth boundary Ω, X =（X1,X2,... ,Xm） be a system of real smooth vector fields defined on Ω and the bound-ary Ω is non-characteristic for X. If X satisfies the HSrmander＇s condition, then the vectorfield is finitely degenerate and the sum of square operator △X =m∑j=1 X2 j is a finitely de-generate elliptic operator. In this paper, we shall study the sharp estimate of the Dirichlet eigenvalue for a class of general Grushin type degenerate elliptic operators △x on Ω.展开更多
In this paper, we study the Dirichlet problems for the following quasilinear secondorder sub-elliptic equation, sum from i,j=1 to m(X~*(A(x,u)Xu)+sum from j=1 to m(B(x,u)Xu+C(x,u)=0 in Ω, u=φ on Ω,where X={X, …, ...In this paper, we study the Dirichlet problems for the following quasilinear secondorder sub-elliptic equation, sum from i,j=1 to m(X~*(A(x,u)Xu)+sum from j=1 to m(B(x,u)Xu+C(x,u)=0 in Ω, u=φ on Ω,where X={X, …, X} is a system of real smooth vector fields which satisfies the Hrmander’scondition, A(i,j), B, C∈C~∞(■×R) and (A(x, z)) is a positive definite matris. We have provedthe existence and the maximal regularity of solutions in the "non-isotropic" Hlder space associatedwith the system of vector fields X.展开更多
Let Q be a bounded open domain in R^n with smooth boundaryаΩ.Let X=(X1,X2…,Xm)be a system of general Grushin type vector fields defined onΩand the boundaryаΩis non-characteristic for X.For△x=∑j=1^mXj^2,we den...Let Q be a bounded open domain in R^n with smooth boundaryаΩ.Let X=(X1,X2…,Xm)be a system of general Grushin type vector fields defined onΩand the boundaryаΩis non-characteristic for X.For△x=∑j=1^mXj^2,we denoteλk as the k-th eigenvalue for the bi-subelliptic operator△X2^2 onΩ.In this paper,by using the sharp sub-elliptic estimates and maximally hypoeliptic estimates,we give the optimal lower bound estimates ofλk for the operatork△X^2.展开更多
文摘In this article, we deal with weak solutions to non-degenerate sub-elliptic equations in the Heisenberg group, and study the regularities of solutions. We establish horizontal Calderón-Zygmund type estimate in Besov spaces with more general assumptions on coefficients for both homogeneous equations and non-homogeneous equations. This study of regularity estimates expands the Calderón-Zygmund theory in the Heisenberg group.
基金This work was supported by NSERC(Grant No.RGPIN/9319-2005)
文摘In the 1970’s, Folland and Stein studied a family of subelliptic scalar operators $ \mathcal{L}_\lambda $ which arise naturally in the $ \bar \partial _b $ -complex. They introduced weighted Sobolev spaces as the natural spaces for this complex, and then obtained sharp estimates for $ \bar \partial _b $ in these spaces using integral kernels and approximate inverses. In the 1990’s, Rumin introduced a differential complex for compact contact manifolds, showed that the Folland-Stein operators are central to the analysis for the corresponding Laplace operator, and derived the necessary estimates for the Laplacian from the Folland Stein analysis. In this paper, we give a self-contained derivation of sharp estimates in the anisotropic Folland-Stein spaces for the operators studied by Rumin using integration by parts and a modified approach to bootstrapping.
基金partially supported by the NSFC(11631011,11626251)
文摘Let Ω be a bounded open domain in Rn with smooth boundary Ω, X =（X1,X2,... ,Xm） be a system of real smooth vector fields defined on Ω and the bound-ary Ω is non-characteristic for X. If X satisfies the HSrmander＇s condition, then the vectorfield is finitely degenerate and the sum of square operator △X =m∑j=1 X2 j is a finitely de-generate elliptic operator. In this paper, we shall study the sharp estimate of the Dirichlet eigenvalue for a class of general Grushin type degenerate elliptic operators △x on Ω.
文摘In this paper, we study the Dirichlet problems for the following quasilinear secondorder sub-elliptic equation, sum from i,j=1 to m(X~*(A(x,u)Xu)+sum from j=1 to m(B(x,u)Xu+C(x,u)=0 in Ω, u=φ on Ω,where X={X, …, X} is a system of real smooth vector fields which satisfies the Hrmander’scondition, A(i,j), B, C∈C~∞(■×R) and (A(x, z)) is a positive definite matris. We have provedthe existence and the maximal regularity of solutions in the "non-isotropic" Hlder space associatedwith the system of vector fields X.
基金supported by National Natural Science Foundation of China (Grants Nos. 11631011 and 11626251)
文摘Let Q be a bounded open domain in R^n with smooth boundaryаΩ.Let X=(X1,X2…,Xm)be a system of general Grushin type vector fields defined onΩand the boundaryаΩis non-characteristic for X.For△x=∑j=1^mXj^2,we denoteλk as the k-th eigenvalue for the bi-subelliptic operator△X2^2 onΩ.In this paper,by using the sharp sub-elliptic estimates and maximally hypoeliptic estimates,we give the optimal lower bound estimates ofλk for the operatork△X^2.
