Jingzhi Tie

1.   The Inverse of Some Differential Operators on the Heisenberg Group,

Comm. in PDEs. Vol.20 (7&8)(1995), 1275-1302. 

Abstract: By using the pseudo-differential operator methods introduced by Beals and Greiner in studying the fundamental solution for the Heisenberg Laplacian, we find the symbol of the inverse of  a very interesting PDO and the corresponding kernel.  It is interesting to observe that the kernel of the fundamental solution depends very strongly on the dimension n.

2.  Embedding $\text{\bf C}^1$ into $\text{\bf H}_1$ 

Canad.J. Math. Vol. 47 (6)(1996), 1317-1328.

Abstract: In this paper, we proved directly a theorem of Greiner in $n=1$. This result implies that the classical Mikhlin-Calderon-Zygmund calculus for the Principal value convolution operators on $C^1$ is, in a natural way, the limit of Laguerre calculus for principal value convolution operators on the Heisenberg group.

3.   (Joint with Der-Chen Chang)) Estimates for the spectral projection operators of the sub-Laplacian on the Heisenberg group 

J. Analyse. Math Vol.71 (6)(1997), 313-347. 

Abstract: We use Laguerre calculus to find the $L^p$ spectrum of the pair $({\mathcal L}, T)$. Here ${\mathcal L}$ is the Kohn's sub-Laplacian on the Heisenberg group. We find the kernels for the spectrum projection operators and show that they are Mikhlin-Calderon-Zygmundoperators.

Estimates for the projection operators in various spaces were deduced.

4.  The explicit solution of the $\bar\partial$-Neumann problem in the non-isotropic Siegel Domain. 

Canad. J. Math. Vol.49  (6) (1997),1299-132.

Abstract: In this paper, we solve the $\bar\partial$-Neumann problem on (0,q) forms, 0≤q≤n, in the strictly pseudoconvex non-isotropic Siegel domain. The metric we use is invariant under the action of the Heisenberg group on the domain. The fundamental solution of the related differential equation is derived via the Laguerre calculus. We obtain an explicit formula for the kernel of the Neumann operator. We also construct the solution of the corresponding heat equation and the fundamental solution of the Laplacian operator on the Heisenberg group.

      

5.  (Joint with Der-Chen Chang) Applications of Laguerre calculus to Dirichlet problem of the Heisenberg Laplacian.

Finite or infinite dimensional complex analysis, (Fukuoka, 1999), 47--53, Lecture Notes in Pure and Appl. Math., 214, Dekker, New York, 2000.

Abstract: The purpose of this paper is to describe research we have been doing recently in an area of analysis lies at the confluence of two directions of research: the study of Laguerre calculus on the Heisenberg group, and the theory of subelliptic boundaryvalue problems. Here we intend only to outline part of the results we have obtained.

6.  (Joint with Der-Chen Chang) Estimates for the powers of the sub-Laplacian on the non-isotropic Heisenberg group.

J. Geom. Anal Vol.10(4) (2000), 653-678.      

Abstract: We apply the Laguerre calculus to obtain the explicit kernel for the fundamental solution of the powers and heat kernels of the Kohn's sub-Laplacian on the non-isotropic Heisenberg group. We then proved some estimates of these kernels in various spaces.

7.  (Joint with Der-Chen Chang) An identity related to the Riesz transforms on the Heisenberg group.

Complex Variables Theory Appl. Vol.40(6) (2000),  395--421. 

Abstract: Let  be a real basis for the Heisenberg Lie algebra  . In this paper, we calculate the explicit kernels of the Riesz transforms  on the Heisenberg group H_n . Here  is the sub-Laplacian operator. We show that  for j = 1,2, … 2n where R_j 's are the regular Riesz transforms on  . We also construct the “missing transform”  such that  in analogy of the classical result 

8. (Joint with Der-Chen Chang) Some differential operators related to the sub-Laplacian on the non-isotropic Heisenberg group 

Math. Nachr. Vol. 221 (2001) 19-29. 

Abstract: Let ℒ_{α =}  + iαT be the sub–Laplacian on the non–isotropic Heisenberg group H_n where Z_j for j = 1, 2, …, n and T are a basis of the Lie algebra. We apply the Laguerre calculus to obtain the fundamental solution of the heat kernel exp{—sℒ_α}, the Schrödinger operator exp{—isℒ_α} and the operator  + iαT. We also discuss some basic properties of the wave operator.

           

9. (Joint with  Der-Chen Chang and Robert Gilbert), Bergman Projection and Weighted Holomorphic Functions,

Operator Theory: Adavances and Applications, Vol. 143, 147-169,  2003.

Abstract: In this paper, we prove the boundedness of the generalized Cesaro operators on Hardy and the generalized Bergman spaces and present a class of invariant subspace under the action of this operator.

 

10.  (Joint with  Der-Chen Chang and Peter C. Greiner) Sub-Riemannian Geometry and Subelliptic PDEs,

Function Theory in Several Complex Variables, Editors: Carl H FitaGerald and  Sheng Gong,

Proceedings of a Satellite Conference to the ICM in Beijing 2002, 1-36,  2004.

Abstract: the article is based on the lectures given by Der-Chen Chang and Peter C. Greiner a the Satellite Conferences of ICM 2002 on Geometric Function Theory in Several Complex Variables. We have surveyed the basic notion of the sub-Riemannian geometry on the Heisenberg group and the Martinet case.

 

11. (Joint with Der-Chen Chang) Hermit operator and Subelliptic Operators,

Acta Math. Sin. (Engl. Ser.) Vol 21, no. 4, 803--818,  (2005).

12.  (Joint with Ovidiu Calin and Der-Chen Chang) Hermite Operator on the Heisenberg Group,

Harmonic Analysis, Signal Processing and Complexity: Festschrift in Honor of the 60th Birthday of Carlos A.

Berenstein, 37-54,  2005.

 

13. The fundamental solution and heat kernel of the twisted Laplacian on $\mathbb{R}^{2n}$.

Communication in PDEs, Vol 31, no.7-9,1047--1069, {\bf 2006}.

 

14. (Joint with  Ovidiu Calin and Der-Chen Chang) Fundamental Solutions for Hermite and Subelliptic Operators,

J. Analyse. Math.,  Vol 100, 223--248,  2006.

 

15. (Joint with Der-Chen Chang and Peter C. Greiner) Laguerre Calculus on the Heisenberg group and Bessel-Fourier transform on ${\mathbb C}^n$,

Sciences in China, Series A, Vol 49, no. 11, 1722--1739,  2006.

16. (Joint with M.W. Wong) The wave kernel of the twisted Laplacian on ${\mathbb C}^n$,

Modern Trends in Pseudo-Differential Operators, 107--115, Oper. Theory Adv. Appl., Vol 172,  2007.

17. (Joint with  M.W. Wong) The Heat Kernel and Green Functions of Sub-Laplacians on the Quaternion Heisenberg Group,

 Journal of Geometric Analysis, Vol 19, 191-210,  2009.

18. (Joint with Der-Chen Chang and Shu-Cheng Chang) Laguerre Calculus and Paneitz Operator on the Heisenberg group,

 Sci. China Ser. A , Vol 52 , No. 12, 2549-2966,  2009.

19. (Joint with Shu-Cheng Chang and  Chin-Tung Wu) Subgradient Estimate and Liouville-type Theorems for the CR Heat Equation on Heisenberg groups,

Asian Journal of Mathematics, Vol 14, Number 1, 41-72, 2010.

 

20. (Joint with  Malcolm R. Adams) On Sub-Riemannian Geodesics Induced by the Engel Groups; Hamiltonian Equations,

Mathematische Nachrichte, Volume 286, Issue 14-15, 1381–1406, 2013.

21 (Joint with Duy Nguyen and Qing Zhang), An Optimal Trading Rule Under A Switchable Mean-Reversion Model,

Journal of Optimization Theory and Applications, Vol 161, 145-163, 2014

22. (Joint with Duy Nguyen and Qing Zhang) Stock Trading Rules under a Switchable Market,

Mathematical Control and Related Fields,  Vol 3, Number 2,  209-231, 2013.

23. (Joint with Shu-Cheng Chang and  Ting-Jung Kuo) Yau's Gradient Estimate and Liouville Theorem for Positive Pseudoharmonic Functions in a Complete Pseudohermitian $(2n+1)$-manifold,

Submitted to Communications in PDEs, October 2013.

24. (Joint with Der-Chen Chang and Shu-Cheng Chang) Calabi-Yau Theorem and Hodge-Laplacian Heat Equation in a Closed Strictly Pseudoconvex CR Manifold,

Journal of Differential Geometry, Vol 97, 395-425, 2014.



25. (Joint with Shu-Cheng Chang, Yen-Wen Fan and  Chin-Tung Wu), Matrix Li-Yau-Hamilton Inequality for the CR Heat Equation in Pseudo-Hermitian (2n+1)-Manifolds, Mathematische Annalen, Online First, May 2014.