Discussion on the Complex Structure of Nilpotent Lie Groups Gk

Consider the real, simply-connected, connected, s-step nilpotent Lie group G endowed with a left-invariant, integrable almost complex structure J, which is nilpotent. Consider the simply-connected, connected nilpotent Lie group Gk, defined by the nilpotent Lie algebra g/ak, where g is the Lie algebra of G, and ak is an ideal of g. Then, J gives rise to an almost complex structure Jk on Gk. The main conclusion obtained is as follows: if the almost complex structure J of a nilpotent Lie group G is nilpotent, then J can give rise to a left-invariant integrable almost complex structure Jk on the nilpotent Lie group Gk, and Jk is also nilpotent.

Herz Spaces on Nilpotent Lie Groups and Its Applications

In this paper, we define Herz type spaces on nilpotent Lie groups and give the boundedess of heat kernel and Riesz transform on Herz spaces.

A Note on the Heat Kernel for the Rescaled Harmonic Oscillator from Two Step Nilpotent Lie Groups

In this note,we use Schr?dinger representations and the Fourier transform on two step nilpotent Lie groups to compute the explicit formula of the sub-Laplacian operator and its symbol,which is associated with the rescaled harmonic oscillator.Then we can give an explicit formula for the heat kernel of the rescaled harmonic oscillator for the singularity at the origin.Our results are useful for the general two step nilpotent Lie groups,including the Heisenberg group and H-type group.

The Weighted Estimates of the Schrodinger Operators on the Nilpotent Lie Group

In this paper we consider the SchrSdinger operator -△G ＋ W on the nilpotent Lie group G where the nonnegative potential W belongs to the reverse H51der class Bq1 for some q1 ≥ D and D is the dimension at infinity of G. The weighted L^p -L6q estimates for the operators W^a（-△G ＋ W）^-β and W^a△G（-△G ＋ W）^-β are obtained.

Isoperimetry of nilpotent groups

This survey gives an overview of the isoperimetric properties of nilpotent groups and Lie groups. It discusses results for Dehn functions and filling functions as well as the techniques used to retrieve them. The content reaches from long standing results up to the most recent development.

Regularity of b complex on nondegenerate Cauchy-Riemann manifolds

For each point ξ in a CR manifold M of codimension greater than 1, the CR structure of M can be approximated by the CR structure of a nilpotent Lie group Gξ of step two near ξ. Gξ varies with ξ. □b and b on M can be approximated by □4 and b on the nilpotent Lie group Gξ. We can construct the parametrix of □b on M by using the parametrix of □b on nilpotent group of step two, and define a quasidistance on M by the approximation. The regularity of □b and b follows from the Harmonic analysis on M.

Name Change Announcement

We are pleased to announce that from 2012 the name of the Journal of Mathematical Re- search and Exposition （JMRE） will be changed to Journal of Mathematical Research with Applications （JMRA） with ISSN changed from 1000-341X to 2095-2651, and CN from 21-1208/O1 to 21-1579/O1.