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
We calculate the Hormander index in the finite-dimensional case. Then we use the result to give some iteration inequalities, and prove almost existence of mean indices for given complete autonomous Hamiltonian system ...We calculate the Hormander index in the finite-dimensional case. Then we use the result to give some iteration inequalities, and prove almost existence of mean indices for given complete autonomous Hamiltonian system on compact symplectic manifold with symplectic trivial tangent bundle and given autonomous Hamiltonian system on regular compact energy hypersurface of symplectic manifold with symplectic trivial tangent bundle.展开更多
This paper is devoted to investigating the weighted LP-mapping properties of oscillation and variation operators related to the families of singular integrals and their commutators in higher dimension. We establish th...This paper is devoted to investigating the weighted LP-mapping properties of oscillation and variation operators related to the families of singular integrals and their commutators in higher dimension. We establish the weighted type （p, p） estimates for 1 〈 p 〈 ∞ and the weighted weak type （1, 1） estimate for the oscillation and variation operators of singular integrals with kernels satisfying certain HSrmander type conditions, which contain the Riesz transforms, singular integrals with more general homogeneous kernels satisfying the Lipschitz conditions and the classical Dini＇s conditions as model examples. Meanwhile, we also obtain the weighted LP-boundeness for such operators associated to the family of commutators generated by the singular integrals above with BMO（Rd）-functions.展开更多
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.
基金Acknowledgements The authors would like to thank the referees for their critical reading and very helpful comments and suggestions. This work was partially supported by the National Natural Science Foundation of China （Grant Nos. 11221091, 11471169） and LPMC of MOE of China.
文摘We calculate the Hormander index in the finite-dimensional case. Then we use the result to give some iteration inequalities, and prove almost existence of mean indices for given complete autonomous Hamiltonian system on compact symplectic manifold with symplectic trivial tangent bundle and given autonomous Hamiltonian system on regular compact energy hypersurface of symplectic manifold with symplectic trivial tangent bundle.
基金Supported by the NNSF of China(Grant Nos.11371295 and 11471041)the NSF of Fujian Province of China(Grant No.2015J01025)Foundation for Doctors of Yili Normal College(Grant No.2017YSBS09)
文摘This paper is devoted to investigating the weighted LP-mapping properties of oscillation and variation operators related to the families of singular integrals and their commutators in higher dimension. We establish the weighted type （p, p） estimates for 1 〈 p 〈 ∞ and the weighted weak type （1, 1） estimate for the oscillation and variation operators of singular integrals with kernels satisfying certain HSrmander type conditions, which contain the Riesz transforms, singular integrals with more general homogeneous kernels satisfying the Lipschitz conditions and the classical Dini＇s conditions as model examples. Meanwhile, we also obtain the weighted LP-boundeness for such operators associated to the family of commutators generated by the singular integrals above with BMO（Rd）-functions.
基金supported by the Youth Scientific Research Fund of Hunan Normal.University(No.Math140650)the Scientific Research Foundation for Ph.D Hunan Normal University(No.Math140675)
