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# Jingzhi Tie

Professor
Research Areas:
Research Interests:

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Selected Publications:

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. MathVol. 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, 0qn, 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 ChangApplications 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 ChangEstimates 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 ChangAn 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 Hn . Here  is the sub-Laplacian operator. We show that  for j = 1,2, … 2n where Rj '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 ChangSome differential operators related to the sub-Laplacian on the non-isotropic Heisenberg group
Math. Nachr. Vol. 221 (2001) 19-29.
Abstract: Let ℒα =  + T be the sub–Laplacian on the non–isotropic Heisenberg group Hn where Zj 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{—sα}, the Schrödinger operator exp{—isα} and the operator  + T. 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 ApplicationsVol. 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. GreinerSub-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 ChangHermit operator and Subelliptic Operators,
Acta Math. Sin. (Engl. Ser.) Vol 21, no. 4, 803--818,  (2005).

12.  (Joint with Ovidiu Calin and Der-Chen ChangHermite Operator on the Heisenberg Group,
Harmonic Analysis, Signal Processing and ComplexityFestschrift 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 PDEsVol 31, no.7-9,1047--1069, {\bf 2006}.

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

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

15. (Joint with Der-Chen Chang and Peter C. GreinerLaguerre Calculus on the Heisenberg group and Bessel-Fourier transform on ${\mathbb C}^n$,
Sciences in China, Series AVol 49, no. 11, 1722--1739,  2006.

16. (Joint with M.W. WongThe 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. WongThe Heat Kernel and Green Functions of Sub-Laplacians on the Quaternion Heisenberg Group,
Journal of Geometric AnalysisVol 19, 191-210,  2009.

18. (Joint with Der-Chen Chang and Shu-Cheng ChangLaguerre 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 WuSubgradient Estimate and Liouville-type Theorems for the CR Heat Equation on Heisenberg groups,
Asian Journal of MathematicsVol 14, Number 1, 41-72, 2010.

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

Mathematische NachrichteVolume 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 ApplicationsVol 161, 145-163, 2014

22. (Joint with Duy Nguyen and Qing ZhangStock 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 KuoYau'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 ChangCalabi-Yau Theorem and Hodge-Laplacian Heat Equation in a Closed Strictly Pseudoconvex CR Manifold,
Journal of Differential GeometryVol 97, 395-425, 2014.

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

Education:

Ph.D. in Mathematics, University of Toronto, Canada

M. Sc in Applied Mathematics, University of Victoria, Canada

B. Sc in Mathematics, Lanzhou University, China

Teaching:

Teaching Schedule

Fall 2018
Math 4700  Ordinary Differential Equations

Spring 2018
Math 4720   Ordinary Differential Equations

Math 2700   Introduction to Differential Equations

Fall 2017
Math 4700  Ordinary Differential Equations

Math 3100  Introduction to Analysis - Sequences and Series

Summer 2017
Math 3200 Introduction to Advanced Mathematics

Spring 2017
Math 4720 Introduction to Partial Differential Equations

Math 2260   Integral Calculus

Fall 2016
Math 4700  Ordinary Differential Equations

Summer 2016
Math 3200 Introduction to Advanced Mathematics

Spring 2016
Math 4720 Introduction to Partial Differential Equations

Fall 2015
Math 2260   Integral Calculus
Math 4700  Ordinary Differential Equations

Summer 2015
Math 2700 Introduction to Differential Equations

Spinger 2015
Math 3100  Introduction to Analysis - Sequences and Series

Fall 2014

Math 3200 Introduction to Advanced Mathematics

Math 2500 Multivariable Calculus

Summer 2014
Math 3200 Introduction to Advanced Mathematics

Spring 2014
Math 4720 Introduction to Partial Differential Equations
Math 8150 Complex Analysis

Fall 2013
Math 3100  Introduction to Analysis - Sequences and Series

Summer 2013
Math 3200 Introduction to Advanced Mathematics

Spring 2013
Math 8110  Real Analysis II

Fall 2012
Math 2260   Integral Calculus
Math 8100  Real Analysis

Summer 2012
Math 3200 Introduction to Advanced Mathematics

Spring 2012
Math 8770 Partial Differential Equations

Fall 2011
Math 2250   Differential Calculus

Summer 2011
Math 2500 Multivariable Calculus

Spring 2011
Math 5210/7210 Foundation of Geometry I I

Fall 2010
Math 4100/6100 Real Analysis
Math 5200/7200 Foundation of Geometry I

Summer 2010
Math 3200 Introduction to Advanced Mathematics

Spring 2010
Math 2500 Multivariable Calculus
Math 2700 Elementary Differential Equations

Fall 2009
Math 2260   Integral Calculus

Summer 2009
Math 2700 Introduction to Differential Equations

Spring 2009
Math 2200 Calculus
Math 8150 Complex Analysis

Fall 2008
Math 3100  Sequences and Series
Math 2260  Integral Calculus

Summer 2008
Math 2700  Introduction to Differential Equations

Spring 2008
Math 4720/6720  Introduction to Partial  Differential Equations and Distribution Theory
Math 2250  University Calculus

Fall 2006

Math 5002/7002  Geometry for Elementary Teachers
Math 2250  Calculus

Summer 2006

Math 2700  Introduction to Differential Equations

Spring 2006
Math 4720/6720  Introduction to Partial  Differential Equatoions and Fourier Series

Fall 2005
Math 2200  Introduction to Differential Calculus

Math 8100  Real Analysis

Summer 2005

Math 2700  Introduction to Differential Equations

Spring 2005
Math 5002  Geometry for Elementary Teachers
Fres 1010  The Joy of Discovering Mathematics

Fall 2004
Math 2200  Introduction to Differential Calculus
Math 2210  Introduction to Integral Calculus

Summer 2004
Math 2700  Introduction to Differential Equations

Spring 2004
Math 3100  Introduction to Analysis: Sequences and Seires

Fall 2003
Math 2200  Introduction to Differential Calculus

Math 8100  Real Analysis and Basic Functional Analysis

Summer 2003
Math 2700  Introduction to Differential Equations

Spring 2003
Math 8770  Introduction to  Fourier analysis  and PDE.
Fres 1010  The Joy of Discovering Mathematics

Fall 2002
Math 2200  Introduction to Differential Calculus

Summer 2002
Math 2700  Introduction to Differential Equations

Spring 2002
Math 2200  Introduction to Differential Calculus

Fall 2001
Math 2200  Introduction to Differential Calculus

Summer 2001
Math 2200  Introduction to Differential Calculus

Spring 2001
Math 4720  Introduction to PDE and Boundary Value Problem.

Fall 2000
Math 8100  Real Analysis and Basic Functional Analysis

Spring 2000
Math 2700  Introduction to Differential Equations

Fall 1999
Math 2200  Introduction to Differential Calculus

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