The tangential k-Cauchy-Fueter type operator and Penrose type integral formula on the generalized complex Heisenberg group

The tangential k-Cauchy-Fueter operator and k-CF functions are counterparts of the tangential Cauchy–Riemann operator and CR functions on the Heisenberg group in the theory of several complex variables,respectively.In this paper,we introduce a Lie group that the Heisenberg group can be imbedded into and call it generalized complex Heisenberg.We investigate quaternionic analysis on the generalized complex Heisenberg.We also give the Penrose integral formula for k-CF functions and construct the tangential k-Cauchy-Fueter complex.

一类Kirchhoff-Poisson系统在Heisenberg群上解的存在性

在Heisenberg群上研究了一类临界的Kirchhoff-Poisson系统。由于存在临界和非局部项,导致空间嵌入不紧,在非线性项适当的假设下,通过变分方法克服了空间的紧性并且得到该系统至少存在一个解。在此基础上,借助形变引理和拓扑度理论,证明了该解是一个变号解。

Heisenberg群中带奇异权的最优临界Hardy-Trudinger-Moser不等式

本文建立了Heisenberg群中有界域和一般无界域上带奇异权的最优临界Hardy-Trudinger-Moser不等式.克服临界Hardy不等式和奇异权函数带来的困难,利用带奇异权的Trudinger-Moser不等式和一些基本估计建立了有界域上一般的带奇异权的临界Hardy-Trudinger-Moser不等式,并通过选取适当的Moser函数得到了最佳常数.最后,利用分割积分区域的方法得到了一般无界域上带奇异权的最优临界Hardy-Trudinger-Moser不等式.

L^p BOUNDEDNESS OF COMMUTATOR OPERATOR ASSOCIATED WITH SCHRDINGER OPERATORS ON HEISENBERG GROUP

Let L = -△Hn ＋ V be a SchrSdinger operator on Heisenberg group Hn, where AHn is the sublaplacian and the nonnegative potential V belongs to the reverse HSlder class BQ/2 where Q is the homogeneous dimension of Hn. Let T1 = （--△Hn ＋V）-1V, T2 = （-△Hn ＋V）-1/2V1/2, and T3 = （--AHn ＋V）-I/2△Hn, then we verify that [b, Ti], i = 1, 2, 3 are bounded on some LP（Hn）, where b ∈ BMO（Hn）. Note that the kernel of Ti, i = 1, 2, 3 has no smoothness.

ENDPOINT ESTIMATES FOR FRACTIONAL INTEGRAL ASSOCIATED TO SCHRDINGER OPERATORS ON THE HEISENBERG GROUPS

Let ∠= -△Hn＋ V be the Schrdinger operator on the Heisenberg groups Hn,where V is a nonnegative function satisfying the reverse Hlder inequality. In this article, the author obtains the BMO_∠ and BLO_∠ estimates of the fractional integrals associated to ∠.

Characterizations of Sobolev spaces in Euclidean spaces and Heisenberg groups

Recently, many new features of Sobolev spaces W k,p ？RN ？ were studied in [4-6, 32]. This paper is devoted to giving a brief review of some known characterizations of Sobolev spaces in Euclidean spaces and describing our recent study of new characterizations of Sobolev spaces on both Heisenberg groups and Euclidean spaces obtained in [12] and [13] and outlining their proofs. Our results extend those characterizations of first order Sobolev spaces in [32] to the Heisenberg group setting. Moreover, our theorems also provide diff erent characterizations for the second order Sobolev spaces in Euclidean spaces from those in [4, 5].

MULTIRESOLUTION ANALYSIS, SELF-SIMILAR TILINGS AND HAAR WAVELETS ON THE HEISENBERG GROUP

In this article, the properties of multiresolution analysis and self-similar tilings on the Heisenberg group are studied. Moreover, we establish a theory to construct an orthonormal Haar wavelet base in L^2（H^d） by using self-similar tilings for the acceptable dilations on the Heisenberg group.

THE UNIFORMLY BOUNDEDNESS OF THE RIESZ TRANSFORMS ON THE CAYLEY HEISENBERG GROUPS

In this article,the authors estimate some functions by using the explicit expression of the heat kernels for the Cayley Heisenberg groups,and then prove the uniform boundedness of the Riesz transforms on these nilpotent Lie groups.

THE HEAT KERNEL ON THE CAYLEY HEISENBERG GROUP

The authors obtain an explicit expression of the heat kernel for the Cayley Heisenberg group of order n by using the stochastic integral method of Gaveau. Apart from the standard Heisenberg group and the quaternionic Heisenberg group, this is the only nilpotent Lie group on which an explicit formula for the heat kernel has been obtained.

On fundamental solution for powers of the sub-Laplacian on the Heisenberg group

We discuss the fundamental solution for m-th powers of the sub-Laplacian on the Heisenberg group. We use the representation theory of the Heisenberg group to analyze the associated m-th powers of the sub-Laplacian and to construct its fundamental solution. Besides, the series representation of the fundamental solution for square of the sub-Laplacian on the Heisenberg group is given and we also get the closed form of the fundamental solution for square of the sub-Laplacian on the Heisenberg group with dimension n = 2, 3, 4.

REGULARITIES AND SINGULARITIES OF THE ENERGY MINIMIZERS OF THE HEISENBERG GROUP TARGETS

In this paper, the properties of the maps for the Heisenberg group targets are studied. For u e∈W1,α(Ω, Hm), some Poincare type inequalities are proved. For the energy minimizers, the ∈-regularity theorems and the singularity theorems are obtained.

EXISTENCE OF MINIMISERS FOR A CLASS OF FREE DISCONTINUITY PROBLEMS IN THE HEISENBERG GROUP H^n

The purpose of this paper is to prove existence of minimisers of the functional where Ω is an open set of the Heisenberg group Hn, K runs over all closed sets of Hn, u varies in C_H^1(Ω\ K), α,β> 0,q≥1, g ∈ Lq(Ω) ∩ L∞(Ω) and f : R2n→R is a convex function satisfying some structure conditions (H1)(H2)(H3) (see below).

IMPROVED GAGLIARDO-NIRENBERG INEQUALITIES ON HEISENBERG TYPE GROUPS

Motivated by the idea of M. Ledoux who brings out the connection between Sobolev embeddings and heat kernel bounds, we prove an analogous result for Kohn’s sub-Laplacian on the Heisenberg type groups. The main result includes features of an inequality of either Sobolev or Galiardo-Nirenberg type.

A WEIGHTED ESTIMATE OF THE HORMANDER MULTIPLIER ON THE HEISENBERG GROUP

In this article, the anthors prove the weighted boundedness of Hoermander-type multiplier on the Heisenberg group.

ON CRITICAL CASES OF SOBOLEV'S INEQUALITIES FOR HEISENBERG GROUPS

We prove some Trudinger-type inequalities and Brezis-Gallouet-Wainger inequality on the Heisenberg group, extending to this context the Euclidean results by T. Ozawa.

A restriction theorem for the quaternion Heisenberg group

We prove that the restriction operator for the sublaplacian on the quaternion Heisenberg group is bounded from L^p to L^p＇ if 1 ≤ p ≤4/3. This is different from the Heisenberg group, on which the restriction operator is not bounded from Lp to Lp＇ unless p = 1.

Nonlinear Liouville Theorem in the Quaternionic Heisenberg Group

This paper deals with the problem of the type triangle open_H f+ f^p =O inquaternionic Heisenberg group, where triangle open_H is the quaternionic Heisenberg Laplacian. Itis proved that, under suitable conditions on p and /, the only solution of triangle open_H f+ f^p=O.

BILINEAR SPECTRAL MULTIPLIERS ON HEISENBERG GROUPS

As we know,thus far,there has appeared no definition of bilinear spectral multipliers on Heisenberg groups.In this article,we present one reasonable definition of bilinear spectral multipliers on Heisenberg groups and investigate its boundedness.We find some restrained conditions to separately ensure its boundedness from C0(H^(n))×L^(2)(H^(n))to L^(2)(H^(n)),from L2(H^(n))×C0(H^(n))to L^(2)(H^(n)),and from L^(p)×L^(q) to L^(r) with 2<p,q<∞,2≤r≤∞.

SEMILINEAR DEGENERATE HEAT INEQUALITIES WITH SINGULAR POTENTIAL ON THE HEISENBERG GROUP

This article deals with the global existence and nonexistence of solutions to the degenerate heat inequalities with singular potential on the Heisenberg group. To prove the existence results, the authors adjust the method of supersolutions to their setting. The nonexistence results are obtained by means of the test function method.

SOLVABILITY OF THE FOURTH ORDER NONLINEAR SUBELLIPTIC EQUATIONS ON THE HEISENBERG GROUP

In this paper,some existence results for the fourth order nonlinear subelliptic equations on the Heisenberg group are given by means of variational methods.