BOUNDEDNESS OF FRACTIONAL MAXIMAL OPERATOR AND THEIR HIGHER ORDER COMMUTATORS IN GENERALIZED MORREY SPACES ON CARNOT GROUPS

In the article we consider the fractional maximal operator Mα, 0 ≤α 〈 Q on any Carnot group G （i.e., nilpotent stratified Lie group） in the generalized Morrey spaces Mp,φ（G）, where Q is the homogeneous dimension of G. We find the conditions on the pair （φ1, φ2） which ensures the boundedness of the operator Ms from one generalized Morrey space Mp,φ1 （G） to another Mq,φ2 （G）, 1. 〈 p ≤q 〈 ∞. 1/p - 1/q = α/Q, and from the space M1,φ1 （G） to the weak space Wq,φ2 （G）, 1 〈 q 〈 ∞, 1 - 1/q = α/Q. Also find conditions on the φ which ensure the Adams type boundedness of the Ms from M α （G） from Mp,φ^1/p（G）to Mq,φ^1/q（G） for 1 〈p〈q〈∞ and fromM1,φ（G） toWMq,φ^1/q（G）for 1〈q〈∞. In the case b ∈ BMO（G） and 1 〈 p 〈 q 〈 ∞, find the sufficient conditions on the pair （φ1, φ2） which ensures the boundedness of the kth-order commutator operator Mb,α,k from Mp,φ1 （G） to Mq,φ2（G） with 1/p - 1/q = α/Q. Also find the sufficient conditions on the φ which ensures the boundedness of the operator Mb,α,k from Mp,φ^1/p（G） tom Mp,φ^1/p （G） for 1 〈p〈q〈∞. In all the cases the conditions for the boundedness of Mα are given it terms of supremaltype inequalities on （φ1, φ2） and φ , which do not assume any assumption on monotonicity of （φ1, φ2） and φ in r. As applications we consider the SchrSdinger operator -△G ＋ V on G, where the nonnegative potential V belongs to the reverse Holder class B∞（G）. The MB,φ1 - Mq,φ2 estimates for the operators V^γ（-△G ＋ V）^-β and V^γ△↓G（-△G ＋ V）^-β are obtained.

UNIVERSAL INEQUALITIES FOR A HORIZONTALLAPLACIAN VERSION OF THE CLAMPED PLATE PROBLEM ON CARNOT GROUP

In this paper, we investigate a horizontal Laplacian version of the clamped plate problem on Carnot groups and obtain some universal inequalities. Furthermore, for the lower order eigenvalues of this eigenvalue problem on carnot groups, we also give some universal inequalities.

Horizontal Connection and Horizontal Mean Curvature in Carnot Groups

In this paper we give a geometric interpretation of the notion of the horizontal mean curvature which is introduced by Danielli Garofalo-Nhieu and Pauls who recently introduced sub- Riemannian minimal surfaces in Carnot groups. This will be done by introducing a natural nonholonomic connection which is the restriction （projection） of the natural Riemannian connection on the horizontal bundle. For this nonholonomic connection and （intrinsic） regular hypersurfaces we introduce the notions of the horizontal second fundamental form and the horizontal shape operator. It turns out that the horizontal mean curvature is the trace of the horizontal shape operator.

Extremals in some classes of Carnot groups

Let G be a Carnot group and D={e 1,e 2 } be a bracket generating left invariant distribution on G.In this paper,we obtain two main results.We first prove that there only exist normal minimizers in G if the type of D is (2,1,...,1) or (2,1,...,1,2).This immediately leads to the fact that there are only normal minimizers in the Goursat manifolds.As one corollary,we also obtain that there are only normal minimizers when dim G 5.We construct a class of Carnot groups such as that of type (2,1,...,1,2,n 0,...,n a) with n 0 1,n i 0,i=1,...,a,in which there exist strictly abnormal extremals.This implies that,for any given manifold of dimension n 6,we can find a class of n-dimensional Carnot groups having strictly abnormal minimizers.We conclude that the dimension n=5 is the border line for the existence and nonexistence of strictly abnormal extremals.Our main technique is based on the equations for the normal and abnormal extremals.

Quasi-convex Functions in Carnot Groups

在这篇论文，作者在 Carnot 组 G 上介绍 h-quasiconvex 功能的概念。h-quasiconvex 功能和 h 凸的集合的观点是相等的， L~ ∞第一估计 h-quasiconvex 功能的衍生物被给，这被显示出。为步二的 aCarnot 组 G， h-quasiconvex 功能局部地从上面被围住，这被证明。而且，作者获得那 h 凸的功能是局部地连续的 Lipschitz 和那 h 凸的功能到处几乎是两次可辨的。

Rearrangements in Carnot Groups

In this paper we extend the notion of rearrangement of nonnegative functions to the setting of Carnot groups. We define rearrangement with respect to a given family of anisotropic balls Br , or equivalently with respect to a gauge‖x‖, and prove basic regularity properties of this construction. If u is a bounded nonnegative real function with compact support, we denote by u*its rearrangement. Then, the radial function u* is of bounded variation. In addition, if u is continuous then u* is continuous, and if u belongs to the horizontal Sobolev space W 1,ph , then Dhu*(x)/Dh( ‖x‖ )| is in Lp. Moreover, we found a generalization of the inequality of P(o)lya and Szeg(o) ∫|Dhu*|p/Dh(‖x‖)|pdx≤C ∫|Dhu|pdx,where p ≥ 1.

Carnot群上一类半线性方程的Liouville型定理(英文)

本文研究了Carnot群上一类具有超线性非齐次项的半线性次Laplace方程非负解的存在性问题.结合Birindelli等[4]在Heisenberg群上利用积分不等式研究解的方法和拟齐性分析技巧,给出了此类方程在Carnot群上的一类Liouville型定理.

SOME SHARP RELLICH TYPE INEQUALITIES ON NILPOTENT GROUPS AND APPLICATION

We prove some Rellich type inequalities for the sub-Laplacian on Carnot nilpotent groups. Using the same method, we obtain some analogous inequalities for the Heisenberg-Greiner operators. In most cases, the constants we obtained are optimal.

Carnot群上Schrodinger方程解的全局Orlicz正则性

研究Carnot群上一类退化椭圆型Schr?dinger方程,在位势项属于某反向H?lder类的条件下,利用迭代-覆盖引理得到该类方程解的全局Orlicz估计,从而将Carnot群上退化椭圆型Schr?dinger方程解的经典L^p估计推广到Orlicz估计.

Carnot群上凸函数的比较原理

Carnot群上凸函数的单调性质对研究完全非线性次椭圆方程的正则性理论起关键作用.通过在Carnot群上引入(Hr)-凸函数类,利用辅助函数方法并结合基于群结构的散度定理,建立了关于(H2)-凸函数的比较原理.此外,作为该结论的应用,得到了高维Heisenberg群上关于凸函数的比较原理.这些结果有望为进一步研究Carnot群上凸函数的性质和完全非线性方程的正则性提供理论基础.

Carnot群上退化半线性抛物型不等方程的Liouville型定理

本文研究Carnot群上一类退化半线性抛物型不等方程的Liouville型定理,将Fujita和Kartsatos- Kurta经典的关于欧氏空间上相应方程的非平凡解的不存在性结果推广到Carnot群上。通过建立一个先验不等式,本文还证明了无界子域上的Liouville型定理。

Carnot群上的r-凸函数与广义加权平均值

该文建立了Carnot群上的r-凸函数与广义加权平均值间的关系,它们是经典凸函数的Hadamard不等式的推广.

Carnot群上散度型退化椭圆方程组的内Morrey估计(英文)

设X1,…,X q是Carnot群G=(Rn,o)(q<n)上的一组水平向量场的基.主要考虑下面的散度型退化方程组N∑β=1q∑i,j=1Xi(aijαβ(x)Xjuβ)=q∑i=1Xifαi,α=1,2,…,N,其中,系数aijαβ是ΩG上的实的有界VMO loc(Ω)函数并且满足强Legendre条件.得到该方程组弱解的内Morrey估计.

关于齐次Carnot群上广义Morrey空间中一些性质（英文）

本文研究了关于Heisenberg群上的广义Morrey空间和Carnot群上的Lebesgue空间中Riesz位势算子或者分数阶极大算子的行为.根据Heisenberg群中抽象调和分析方法以及sub Laplacian算子的Dirichlet问题解的表示公式,本文主要给出了关于齐次Carnot群G上消失的广义Morrey空间V L^(p,?)(G)中的加权Hardy算子、分数阶极大算子和分数阶位势算子的有界性刻画.进而也得到无消失模的广义Morrey空间上Morrey位势的浸入不等式.所有这些结果推广了关于Heisenberg群上的广义Morrey空间和Carnot群上的Lebesgue空间中的相关结论.

齐次Carnot群上次拉普拉斯算子的奇点可去性定理

研究了齐次Carnot群上次拉普拉斯算子的奇点可去性理论,应用次调和函数的最大最小值原理得到了齐次Carnot群上次拉普拉斯算子的奇点可去性定理,并可以推广到一般的分层群H上的次拉普拉斯算子.

Carnot群上次椭圆方程组的正则性:可控增长条件

本文考虑Carnot群上具有VMO系数的拟线性次椭圆方程组,在可控增长条件下,利用改进的A-调和逼近技巧建立其弱解的H lder连续性.

Carnot群上水平Laplace算子的二次多项式算子的特征值不等式

本文研究了Carnot群上水平Laplace算子的二次多项式算子的Dirichlet特征值问题,并建立了一些特征值不等式.特别地,我们的结果涵盖了文献[10]对双调和水平Laplace算子所获得的结果.

Carnot群上一类拟线性次椭圆方程组的正则性

本文考虑Carnot群上具有VMO不连续系数的拟线性次椭圆方程组,利用改进的A-调和逼近技巧建立其弱解的H?lder连续性.