Fundamental Solutions for Hermite and Subelliptic Operators

FUNDAMENTAL SOLUTIONS FOR

HERMITE AND SUBELLIPTIC OPERATORS £

OVIDIU CALIN,DER-CHEN CHANG AND JINGZHI TIE

Abstract. In this article, we introduce a geometric method based on multi- pliers to compute heat kernels for operators with potentials. Using the heat kernel, we compute the fundamental solution for the Hermite operator with singularity

at an arbitrary point on Euclidean space and on Heisenberg groups. As a conse- £

quence, we obtain the fundamental solutions for the sub-Laplacian Â in a family of quadratic submanifolds.

1 Introduction

The Hermite operator « arises in many contexts and has been studied by

mathematicians and physicists for quite some time (see e.g., [1], [4], [17] and

[22]). In this paper, we study the fundamental solution of the operator « , where

¾ ¾

Ê ´Ü Ý µ

in , i.e., we seek a distribution « such that

¾ ¿

¾ ¾

· ´Ü Ý µÆ ´Ü Ý µ

(1) «

Since the operator « is not left invariant under the Euclidean group action, we have

to compute the fundamental solution with singularity at any point Ý . One way to achieve this goal is to use the eigenfunction expansion for the operator « (see [4] and [14]). Here we adopt a different approach. First, we use the Hamilton-Jacobi theory to ﬁnd the heat kernel for the heat equation with potentials. The idea is to write down the Hamilton system of equations and study the system qualitatively

£ The research is partially supported by a William Fulbright Research Grant and a Competitive Research Grant at Georgetown University.

223 JOURNAL D’ANALYSE MATHEMATIQUE,´ Vol. 100 (2006) 224 O. CALIN, D.-C. CHANG AND JINGZHI TIE, from the point of view of the conservation law of energy. In general, these systems cannot be solved explicitly. For simple equations, one may characterize

the solutions by ﬁnding the ﬁrst integrals of motions. We demonstrate this method

¾ ¾

´Üµ ¬ Ü by solving the operator with potential Í . This is exactly the case of the Hermite operator. Then we give some examples and comments for general potentials at the end of Section 2. For more details, readers can consult the book [7]. Then we use the heat kernel to construct the fundamental solution for the Hermite operator. This method can also be used to study subelliptic partial differential equations and has been exploited extensively by many mathematicians; see, e.g., [3], [8], [9] and [11].

In Section 4, using results for Hermite operator, we construct the fundamental

£ ¦

solution for the sub-Laplacian Â on a family of quadratic submanifolds

Ò Ñ

¢ ¦

which was ﬁrst studied by Nagel, Ricci and Stein [21]; is a special case of the more general submanifold introduced by Beals, Gaveau and Greiner [2]. We can compute the kernel explicitly in terms of certain geometric invariants.

In particular, our result recovers the formula on the Heisenberg group. We also £ calculate the heat kernel for the operator Â in Section 5. Finally, in Section 6, we

consider the Hermite operator on the Heisenberg group

¾Ò

¾ ¾ ¾

À « · ¬ Ü Ì

¾Ò ¾Ò

¾ ¾

¾ ¾ ¾ ¾

« · ¬ Ü

Ü Ø

½ ½

·¾ Ü Ü

·Ò

Ü Ü Ø Ø

·Ò

¾Ò ·Ò

Here ½ is the basis of the Heisenberg Lie algebra and

½ Ò

for . In order to achieve these goals, we take the Fourier transform Ê

with respect to the Ø-variable in (which is the center of the group) and reduce

¾Ò

Ê « ¬

the operator to a Hermite operator in with parameters , , and the dual Ø variable of . Using results for the Hermite operator (see, e.g., [1] and [14]) and the inverse Fourier transform, we solve the problem. The literature is short of

explicit computations for the heat kernel and fundamental solution of the operators

À À « « and , so it seems worthwhile to give a detailed discussion.

We thank Professor Malcolm F. Adams for discussing the problem with us at the outset and reading an early draft of the paper. He made valuable suggestions which have been incorporated into the paper.

FUNDAMENTAL SOLUTIONS FOR HERMITE OPERATORS 225 ¾

2 Heat Kernel for the Hermite Operator in Ê

We start with the Hermite operator

¾ ¾

À Ü

¾ Ü

¾ Ê where · is a nonnegative real parameter. We associate the Hamiltonian

function as half of the principal symbol

¾ ¾ ¾

´ Ü µ ´ Üµ

(2) À ¾

The Hamiltonian system is

À Ü Ü À Ü

and

Ü Ü ¾ Ê Ü´×µ

As we are interested in ﬁnding the geodesic between the points ¼ , will

satisfy the boundary problem

Ü Ü Ü´¼µ Ü Ü´ µÜ

with ¼

The conservation of energy law is

½ ½

¾ ¾ ¾

Ü ´×µ Ü ´×µ

¾ ¾

where is the energy constant. This can be used to obtain an ODE for the solution

´×µ

Ü :

Ü Ü

¾ ¾

¾ · Ü µ ×

¾ ¾

¾ · Ü

× ¼ × Ø Ü´¼µ Ü Ü´ µÜ

Integrating between and , with ¼ and , yields

Ü Ú

Ù Ú

Ô Ô

´µ

¾ ¾ ¾

¾ · Ù ½·Ú

Ü Ú

¼ ¼

Ô Ô

Ú Ú

with and ¼ . Integration yields

¾ ¾

½ ½ ½ ½

× Ò ´Ú µ × Ò ´Ú µ ´µ × Ò ´Ú µ× Ò ´Ú µ·

¼ ¼

Ú × Ò × Ò ´Ú µ·

It follows that ¼ . This is equivalent to

Ú Ú Ó× ´ µ·Ó× ´× Ò ´Ú µµ × Ò ´ µ

¼ ¼

´µ Ú Ú Ó× ´ µ· ½·Ú × Ò ´ µ

¼ ¼ 226 O. CALIN, D.-C. CHANG AND JINGZHI TIE,

Hence Ö

Ü Ü Ü

Ô Ô

Ó×´ µ· ½· × Ò´ µ

¾ ¾

and Õ

Ü Ü Ó×´ µ· ¾ · Ü × Ò´ µ

¼ ¼

It follows that

´Ü Ü Ó×´ µµ

¾ · Ü

× Ò´ µ

Solving for yields

¾ ¾

´Ü Ü Ó×´ µµ

¾ ¾

¾ Ü

× Ò´ µ

¾ ¾ ¾ ¾ ¾ ¾

Ü ¾ÜÜ Ó×´ µ·Ü Ó×´ µ Ü × Ò´ µ

¼ ¼

× Ò´ µ

¾ ¾ ¾

Ü · Ü ¾ÜÜ Ó×´ µ

× Ò´ µ

Proposition 2.1. The energy along a geodesic derived from the Hamiltonian

Ü Ü

(2) between the points ¼ and is

¾ ¾ ¾

Ü · Ü ¾ÜÜ Ó×´ µ

(3)

¾ × Ò´ µ

Ü ¼

Setting ¼ , we obtain the following result. Corollary 2.2. The energy along a geodesic derived from the Hamiltonian

(2) joining the origin and Ü is given by

¾ ¾

(4)

¾ × Ò´ µ

We note that if we take the limit in (3), we obtain the Euclidean energy

¾ ¾

¾ ¾ ¾ Ü · Ü ¾ÜÜ Ó×´ µ

´Ü Ü µ

Ð Ñ Ð Ñ

¾ ¾ ¾

¼ ¼

× Ò´ µ ¾ ¾

Ë Ë ´Ü Üµ

2.1 The action function. Let ¼ be the action with initial

Ü Ü point ¼ and ﬁnal point within time . The action satisﬁes the Hamilton-Jacobi

equation

Ë · À ´ÖË µ¼

Note that

½ ½ ½

¾ ¾ ¾ ¾ ¾ ¾

À ´ Üµ ´ Ü µ Ü Ü

¾ ¾ ¾

FUNDAMENTAL SOLUTIONS FOR HERMITE OPERATORS 227

and hence . Using (3), we have

¾ ¾ ¾

Ü · Ü ¾ÜÜ Ó×´ µ

¾× Ò´ µ

¾ ¾

´Ü · Ü µ ÓØ´ µ ÜÜ

¾ × Ò´ µ

ÜÜ

¾ ¾

µÓØ´ µ ´Ü · Ü

¾ × Ò´ µ

Hence

¾ÜÜ

¾ ¾

Ë ´Ü Üµ ´Ü · Ü µ ÓØ´ µ

¾ × Ò´ µ

¾ ¾

´Ü · Ü µ Ó×´ µ ¾ÜÜ

(5) ¼

¾ × Ò´ µ

It follows that

´Ü Ü µ

Ð Ñ Ë ´Ü Üµ

¾ which is the Euclidean action.

Lemma 2.3. We have

¾ ¾ ¾

´ Ë µ Ü ·¾

(i) Ü

ÓØ´ µ

(ii) Ë Ü

Proof. (i) Differentiating in (5) with respect to the Ü-variable yields

Ë Ü Ó×´ µ Ü Ü

(6) ¼

× Ò´ µ

¾ ¾ ¾

Ü Ó× ´ µ·Ü ¾ÜÜ Ó×´ µ

´ Ë µ

× Ò ´ µ

¾ ¾ ¾ ¾

Ü · Ü × Ò ´ µ·Ü ¾ÜÜ Ó×´ µ

× Ò ´ µ

¾ ¾ ¾

´Ü · Ü ¾ÜÜ Ó×´ µµ

¾ ¾

Ü ·

× Ò ´ µ

¾ ¾

Ü ·¾

(ii) Again, differentiating in (6) with respect to the Ü-variable yields

Ë Ó×´ µ ÓØ´ µ

× Ò´ µ 228 O. CALIN, D.-C. CHANG AND JINGZHI TIE,

Now we are in a position to construct the fundamental solution of the heat

equation, which has a representation as follows. From now on, we use Ø as the time

variable. Inspired by the fundamental solution of the usual heat equation Ø , Ü

we seek a heat kernel of the form

Ë ´Ü ÜØµ

Ã ´Ü ÜØµÎ ´Øµ

(7) ¼

Ë ´Ü ÜØµ Î ´Øµ where ¼ is the action function deﬁned in (5). Here will satisfy a

volume function equation and is a real constant. Hence

¼ Ë Ë

Ã Î ´Øµ · Î ´Øµ Ë

Ø Ø

Ë ¼

Î ´Øµ Î ´Øµ

Lemma 2.3 provides

Ë Ë

Ü Ü

¢ £

¾ Ë ¾ Ë ¾ Ë ¾ Ë ¾ ¾

´ Ë µ · Ë ´ µ · Ë

Ü Ü

Ü Ü Ü

¢ £

Ë ¾ ¾

´ Ü ·¾ µ· ÓØ ´ Øµ

We determine the heat kernel using a multiplier method. Let

¾ ¾ ¾

È · « Ü

(8) Ø

« ÈÃ´Ü ÜØµ¼

where is a real multiplier, to be determined such that ¼ for any ¼

Ø .

Ë ¼

ÈÃ´Ü ÜØµ Î ´Øµ Î ´Øµ

Ë ¾ ¾ ¾ ¾ Ë

´ Ü ·¾ µ· ÓØ ´ Øµ Î ´Øµ« Ü Î ´Øµ

Î ´Øµ

Ë ¾ ¾ ¾ ¾ ¾

Î ´Øµ ´ Ü ·¾ µ ÓØ ´ Øµ·« Ü

Î ´Øµ

¾ ¾ ¾ Ë

´¾ ·½µ·´« µ Ü ÓØ ´ Øµ Î ´Øµ

Î ´Øµ

½¾

In order to eliminate the middle two terms in the brackets, we choose

½ ¬ ¾ ¼

and « . Let . Then the operator (8) becomes

¾ ¾ ¾

· ¬ Ü È

(9) Ø Ü

and

ÈÃ´Ü ÜØµÃ ´Ü ÜØµ Î ´ØµÎ ´Øµ·¬ ÓØ ´¾¬Øµ

¼ ¼

FUNDAMENTAL SOLUTIONS FOR HERMITE OPERATORS 229

´Øµ

We choose Î such that

Î ´Øµ

¬ ÓØ´¾¬Øµ Ø¼

Î ´Øµ

Integration yields

ÐÒ Î ´Øµ ÐÒ ×Ò´¾¬Øµ µ Î ´Øµ

×Ò´¾¬Øµ

Using the action (5), the fundamental solution formula (7) becomes

¾¬

¾ ¾

´Ü ·Ü µ Ó×´¾¬Øµ ¾ÜÜ

×Ò´¾¬Øµ

Ã ´Ü ÜØµ

×Ò´¾¬Øµ

¾¬Ø

¾ ¾

¾¬Ø

¡ ´Ü ·Ü µ Ó×´¾¬Øµ ¾ÜÜ

Ø ×Ò´¾¬Øµ

×Ò´¾¬Øµ

¾¬Ø

¬ ¼

We ﬁnd the constant by investigating the limit case , in which case

the operator (9) becomes the usual 1-dimensional heat operator Ø .As

¾¬Ø ½

, the above fundamental solution becomes

× Ò ´¾¬Øµ

´Ü Ü µ

Ã ´Ü Ü Øµ ¬ ¼

¾¬Ø

By comparison with the fundamental solution for the usual heat operator

½ ½

´Ü Ü µ

we ﬁnd . Thus we have proved

Theorem 2.4. Let ¬ . The fundamental solution for the operator

¾ ¾ ¾

È · ¬ Ü

Ø is

¾¬Ø

¾ ¾

½ ¾¬Ø

´Ü ·Ü µ Ó× ´¾¬Øµ ¾ÜÜ

×Ò´¾¬Øµ

Ã ´Ü ÜØµ

× Ò ´¾¬Øµ

¾ ¾

½ ¬

´Ü·Ü µ Ø Ò ´¬Øµ·´Ü Ü µ ÓØ ´¬Øµ

¼ ¼

Ø ¼

× Ò ´¾¬Øµ

230 O. CALIN, D.-C. CHANG AND JINGZHI TIE,

Ó× ´ ØµÓ×´Øµ

The computations are similar in the case ¬ . Using

Øµ ×Ò´¾Øµ

and ×Ò´¾ , we obtain a dual theorem.

Theorem 2.5. Let . The fundamental solution for the operator

¾ ¾ ¾

È Ü

Ø is

¾Ø

¾ ¾

½ ¾Ø

´Ü ·Ü µÓ×´¾Øµ ¾ÜÜ

×Ò´¾Øµ

Ã ´Ü Ü Øµ

×Ò´¾Øµ

¾ ¾

´Ü·Ü µ Ø Ò´Øµ·´Ü Ü µ ÓØ´Øµ

¼ ¼

Ø ¼

×Ò´¾Øµ

¾ ¾ ¾

¦ Ü

2.2 The harmonic oscillator Ø . Consider the Her-

½ Ò

mite operator in Ê

Ò Ò

½ ½

¾ ¾ ¾ ¾ ¾

¡ Ü Ü

¾ ¾

½ ½

¼ ½ Ò

where for . The associated Hamiltonian is

¾ ¾ ¾

À ´Ü Ü µ Ü

½ Ò ½ Ò

with the Hamiltonian system

À Ü À Ü ½ Ò

and

¼ ¼

Ü´×µ Ü ´Ü Ü µ

The geodesic starting at ¼ and having the ﬁnal point

½ Ò

Ü ´Ü Ü µ Ò

½ satisﬁes the equations

¾ ¼

Ü Ü Ü ´¼µ Ü Ü ´ØµÜ ½ Ò

with

As in the 1-dimensional case, we have the law of conservation of energy

¾ ¾ ¾

´×µ¾ ½ Ò Ü ´×µ Ü

where is the energy constant for the -th component. The total energy, which

is the Hamiltonian, is given by

¾ ¾ ¾

· ¡¡¡ · Ü À Ü Ò

½ (constant)

½ FUNDAMENTAL SOLUTIONS FOR HERMITE OPERATORS 231

Proposition 2.1 yields

¾ ¾ ¼ ¾ ¼

Ü ·´Ü µ ¾Ü Ü Ó× ´ Øµ

¾× Ò ´ Øµ

and hence

¢ £

Ò Ò

¾ ¾ ¼ ¾ ¼

Ü ·´Ü µ ¾Ü Ü Ó× ´ Øµ

¾× Ò ´ Øµ

½ ½

Ü Ü Ø Ë

The action between ¼ and in time satisﬁes the equation or

¢ £

¾ ¾ ¼ ¾ ¼

Ü ·´Ü µ ¾Ü Ü Ó× ´Øµ

¾ × Ò ´ Øµ

Ò Ò

Ü Ü

¾ ¼ ¾

´Ü ·´Ü µ µ ÓØ ´ Øµ

Ø ¾ × Ò ´ Øµ

½ ½

Hence we choose

¾ ¼ ¾ ¼

Ë Ü ·´Ü µ Ó× ´ Øµ ¾Ü Ü

(10)

¾ × Ò ´ Øµ

Let

¾ ¼ ¾ ¼

Ë Ü ·´Ü µ Ó× ´ Øµ ¾Ü Ü

(11)

¾ × Ò ´ Øµ

Ë Ë · ¡¡¡ · Ë Ë Ë

Ò Ü Ü

Then ½ and . Lemma 2.3 yields

Ò Ò Ò Ò

¾ ¾ ¾ ¾ ¾ ¾

´ Ë µ ´ Ë µ ´ Ü ·¾ µ Ü ·¾

Ü Ü

½ ½ ½ ½

Hence

Ò Ò Ò

¾ ¾

Ë Ë ÓØ ´ Øµ

Ü Ü

½ ½ ½

We seek a kernel of the form

Ë ´Ü ÜØµ

Ã ´Ü ÜØµÎ ´Øµ ¾ Ê

(12) ¼

A computation similar to the 1-dimensional case yields

¼ Ë

Î ´Øµ Î ´Øµ Ã Ø

and

¾ Ë Ë ¾ ¾

´ Ë µ · Ë

Ü Ü

232 O. CALIN, D.-C. CHANG AND JINGZHI TIE,

and hence

¢ £

¾ ¾ Ë Ë

Ü ·¾ · ÓØ´ Øµ ¡

In order to ﬁnd the kernel for the heat operator, we once again employ the multiplier

method. We consider the parabolic operator

¾ ¾ ¾

È « Ü

Ò Ø

where « is a multiplier to be found later. Then

Ë ¼

È Ã Î ´Øµ Î ´Øµ

Ò Ò Ò

Ë ¾ ¾ ¾ ¾ Ë

Ü ·¾ · ÓØ´ Øµ Î ´Øµ· « Ü Î ´Øµ

½ ½ ½

Ò Ò

Î ´Øµ

¾ ¾ ¾ Ë

ÓØ´ Øµ ´« µ Ü Î ´Øµ ´½ · ¾ µ·

Î ´Øµ

½ ½

Î ´Øµ Ò

Î ´Øµ · ÓØ´Øµ

Î ´Øµ ¾

½ ½

« ¡¡¡ « ¬ ¼

where we choose and ½ . Let and choose

¾ ¾

´Øµ

Î satisfying

Î ´Øµ

ÓØ´¾¬ Øµ Ø¼

Î ´Øµ

Integration yields ½ . Hence the fundamental solution for

×Ò ´¾¬ Øµ

¾ ¾ ¾

Ü È Ø

the operator Ò expressed in the form (12) is

¾¬

Ò ¾ ¼ ¾ ¼

´Ü ·´Ü µ µ Ó×´¾¬ Øµ ¾Ü Ü

×Ò´¾¬ Øµ

Ã ´Ü Ü Øµ

½ ¾

×Ò ´¾¬ Øµ

¾¬ Ø

Ò ¾ ¼ ¾ ¼

´Ü ·´Ü µ µ Ó×´¾¬ Øµ ¾Ü Ü

×Ò´¾¬ Øµ

½ ¾

×Ò ´¾¬ Øµ

¬ ¼ ½ Ò Ø

When , , we obtain the kernel of the heat operator ,

which is

½ ½

Ü Ü

Ø¼

Ò ¾

´Øµ FUNDAMENTAL SOLUTIONS FOR HERMITE OPERATORS 233

By comparison, we obtain the value

¬ ¼ ½Ò

Theorem 2.6. Let for . The fundamental solution for the

¾ ¾ ¾

È ¬ Ü Ø

operator Ø is

½¾

½ ¾¬ Ø

Ã ´Ü ÜØµ

Ò¾

× Ò ´¾¬ Øµ

´Øµ

¢ £

½ ¾¬ Ø

¾ ¼ ¾ ¼

¢ ÜÔ ´Ü ·´Ü µ µ Ó×´¾¬ Øµ ¾Ü Ü

Ø ×Ò´¾¬ Øµ

½¾

½ ¬

×Ò´¾¬ Øµ

´¾ µ

¢ £

¼ ¾ ¼ ¾

¢ ÜÔ ´Ü · Ü µ ØÒ´¬ Øµ·´Ü Ü µ ÓØ´¬ Øµ

½ ¼ for Ø .

Theorem 2.6 recovers a result in [11] (see also Chapter 6 in [19]). In a way

similar to the one dimensional case, choosing yields the following result.

¼ ½ Ò

Theorem 2.7. Let for . The fundamental solution for the

¾ ¾ ¾

È · Ü

operator Ø is

½¾

¾ Ø ½

Ã ´Ü Ü Øµ

Ò¾

×Ò´¾ Øµ

´Øµ

¾ Ø ½

¼ ¾ ¼ ¾

Ó×´¾ Øµ ¾Ü Ü Ü ·´Ü µ ¢ ÜÔ

Ø ×Ò´¾ Øµ

¢ £

½¾

Ò ¼ ¾ ¼ ¾

´Ü ·Ü µ Ø Ò´ Øµ·´Ü Ü µ ÓØ´ Øµ

× Ò´¾ Øµ

´¾ µ

½ ¼ for Ø . 234 O. CALIN, D.-C. CHANG AND JINGZHI TIE,

Now using Theorem 2.6 (see also Chapter 4 in [7] and Theorem 2.1 in [14]), one can derive the fundamental solution of the Hermite operator by integration of

the heat kernel.

Ò Ó

´¾ ·½µ ´ µ ¾ ´ µ « ¾

½ Ò ·

Theorem 2.8. For , the

¾ ¾

À « ¡· Ü

Hermite operator « has the fundamental solution

«×

È ´Ü Ý µ × Ã ´Ü Ý µ

× «

½¾

«×

× Ò ´¾ ×µ

´¾ µ

¾ ¾

´Ü · Ý µ

´Ü Ý µ

¢ ÜÔ · ØÒ ´ ×µ ×

¾× Ò ´¾ ×µ ¾

Remarks. (i) The method we used in this section is based on Hamilton-Jacobi theory, which provides deep insight into the geometry induced by the operator. Using this method, one may also construct fundamental solutions for more general operators. For example, we can ﬁnd the fundamental solution for the heat equation with

quartic potential, i.e.,

È ¼

Ø with

Ü Here the behavior is quite different from the quadratic potential case (the Hermite operator), where there is only one energy and one solution between two given points. There are inﬁnitely many energies associated to the quartic case. This

makes the operator È very difﬁcult to invert.

Ü Ü Ø¼

However, given any two points ¼ and and a time , there is a sequence of

´Ü ÜØµ

Ò ¼

energies Ò , which have explicit representations. To each energy, we

Ë Ë ´Ü ÜØµ

Ò ¼

associate an action Ò , which satisﬁes the Hamilton-Jacobi equation

Ë ´Ü ÜØµ

Ø Ò Ò ¼

We can show that

¾ Ò ½

Ë Ò ½

Ò as

¿ ¾ Ø

where

´½µ

Ø ½

and is the principal period of the elliptic functions sn and cn. To each action

Ë Î Ò Ò , we associate a volume function . Finally, we write down the asymptotic FUNDAMENTAL SOLUTIONS FOR HERMITE OPERATORS 235

expansion of the kernel

Ã ´Ü ÜØµ Î ´Øµ

¼ Ò Ò

Ò½

where the constants Ò should be chosen so that

ÐÑ Î ´Øµ ´Üµ Ü ´Ü µ

Ò Ò ¼

Ø²¼

Ò½

for any compact supported function . Using this method, we may handle even more general potentials. Readers may consult Chapter 10 in [7].

(ii) This method can be used to solve a more general operator such as

È · ¾Ü

Ø Ü Ü

The fundamental solution for the operator È is given by

´Ü Ü µ Ü·Ü

½ Ø

¼ ¼

· Ø

¾ ¾Ø ¾ ¾

Ã ´Ü Ü Øµ

(13) ¼

¾ Ø This equation is derived from the famous Euler equation of compressible ﬂuids

(see, e.g., [20] and [23]).

Ø Ü Ü

(iii) If one replaces Ø by and by , the heat operator becomes a Schrodinger¨

½ ½

¾ ¾ ¾

¬ Ü È · Ü

operator. The propagator for the Schrodinger’s¨ operator Ø

¾ ¾

¾ÜÜ

¾ ¾

¬ ´Ü ·Ü µÓØ´¬ ´Ø Ø µµ

× Ò´¬ ´Ø Ø µµ

Ã ´Ü Ü ØØ µ

¼ ¼

×Ò´¬ ´Ø Ø µµ

Ø Ø ¼ ¾ ¢ ½¼ where ¼ and is Planck’s constant. (iv) In [5], Beals, Gaveau, Greiner and Kannai use different methods to study exact fundamental solutions for a class of degenerate elliptic operators, e.g., the

Ò-dimensional Grusin operator

Ò Ò

¾ ¾

· Ü

(14)

Ü Ø

½ ½

On taking the Fourier transform of the above operator with respect to the “missing

Ø Ê directions” , it becomes the Hermite operator in . One may now apply Theorem 2.8 to ﬁnd a solution for this Hermite operator. The fundamental solution for the operator (14) can then be obtained by taking inverse Fourier transform. We omit the details.

236 O. CALIN, D.-C. CHANG AND JINGZHI TIE,

3 -complex on quadratic submanifolds

In this section, we study the sub-Laplacian on a class of quadratic submanifolds

Ò Ñ

¦ ¢ ¦

. Each carries the structure of a nilpotent Lie group such that the

£ ´¼Õµ sub-Laplacian acting on forms is a left invariant second order differential

operators on the group. Such a group is isomorphic to the quotient of the product of

one-dimensional Heisenberg groups ½ by a central subgroup. The complex

£ ´¼Õµ ´¼Õµ and the corresponding sub-Laplacian operators which map forms to forms was studied by Nagel, Ricci and Stein [21]. Beals, Gaveau and Greiner have introduced a more general type submanifold in [2]. In this section, we follow the

notation introduced in [21].

Ñ ¢ Ò

Deﬁnition 3.1. Let be an real matrix. Let

½ ¾ Ñ

´ µ ¼

½ Ò Ê Ò

¾ Ò

for , so that ½ is a set of non-zero vectors. Write

Ò Ò

½ ¾ Ñ ¾ Ñ

´Þ µ Þ Þ ¾ Ê

½ ½

Ò Ò

½ Ñ Ñ

´Þ Ûµ Þ Û Þ Û ¾

½ ½

The quadratic manifold associated with is the set

Ò Ñ

¦ ´Þ Ûµ ¾ ¢ ´Û µ´Þ µ

¦ ¦ ¢ ½

The simplest example of is the domain in deﬁned by

Ó Ò

¦ ´Þ Ûµ ´Û µÞ

½ Im

¼ Ê Ò ¾ Ñ ½ In this case, , where is a vector in . When and ,we

have a similar example

Ó Ò

¼ Ò ¾

Í ´Þ Þ µ ¾ ¢ ´Þ µ Þ

·½ Ò·½ Ò·½

Ò Im

which is the boundary of the non-isotropic Siegel upper half space

Ò Ó

¼ Ò ¾

Í ´Þ Þ µ ¾ ¢ ´Þ µ Þ

Ò·½ Ò·½ Ò·½

½ Ò In this case, for .

FUNDAMENTAL SOLUTIONS FOR HERMITE OPERATORS 237

Ò ¦

Another example is the Cartesian product of copies of ½ given by

Ò Ò Ò ¾

¦ ¦ ¢¡¡¡¢¦ ´Þ Ûµ ¾ ¢ ´Û µÞ

½ Im

Ò Ò

Ò Ê Ê Ò

In this case, the vectors ½ in are the standard basis elements of .

´½ ¼µ ´¼ ½µ ¦

The space of vector ﬁelds of type and which are tangent to are

linear combinations of the operators

·¾ Þ

Þ Û

¾ Þ Ä

Þ Û

for ½ .

¦ ´Þ Ø ·

As usual, we may introduce coordinates on by identifying the point

Ò Ñ

´Þ µµ ¾ ¦ ´Þ Øµ ¾ ¢ Ê Ø ´Û µ

with the point , so that Re . When using these

coordinates, we have the following identiﬁcations:

½ ½

° and

Û ¾ Ø Û ¾ Ø

Ò Ñ

¦ Ê ¢ ¾Ò · Ñ

Then is a generic submanifold of with real dimension ,

Ò Ñ

Ê dimension , and “missing directions”. With this notation, the space of

´½ ¼µ ´¼ ½µ ¦ ´Þ Øµ vector ﬁelds of type and on in the coordinates are spanned,

respectively, by

· Þ ¡Ö Þ ¡Ö ½ Ò

Ø Ø

(15) and for

Þ Þ

Ö ´ ½ Ò µ

Here . Obviously, the vector ﬁelds and ,

Ø Ø

½ Ñ

annihilate the deﬁning functions

Ò Ò

Û Û

¾ ¾

´Þ Ûµ Û Þ Þ

½ ½

½ Ñ ¦ for . Hence they are tangential to the boundary of . Now deﬁne real

vector ﬁelds by

·¾Ü ¡Ö ½ Ò

·Ò Ø

¾Ü ¡Ö

·Ò Ø

(16)

·Ò

Ì ½ Ñ

Ø 238 O. CALIN, D.-C. CHANG AND JINGZHI TIE,

Then

½ ½

´ µ ´ · µ

·Ò ·Ò

and

¾ ¾

The commutator structure of the vector ﬁelds , and can be easily calculated.

For ½ , , one has

¼ ¼ ¼ ¼

Ð Ð

and

¾Æ ¡Ö

Ð Ð Ø ¦

It is known from [21] that the manifold is non-degenerate if the vectors

Ê ¦ Ò

½ span . The necessary and sufﬁcient condition for being non-

¾Ò

degenerate is that the vector ﬁelds ½ satisfying the generating property ¦

(see Chow [15]). Moreover, each manifold carries the structure of a nilpotent

Lie group with the group law

´ µ ¡ ´ µ · · ·¾

Ò Ñ

´ µ ´ µ ¾ ¢ Ê

¾Ò

for and . It follows that the vector ﬁelds ½

¾ Ñ ½ given by (16) form a ﬁnite dimensional step nilpotent Lie algebra.

As usual, the sub-Laplacian is deﬁned as

£ £

£ ·

½ ¾

½ Õ

Let Õ denote the set of increasing -tuples .A

´¼ form , can be written uniquely as

¡¡¡

Â Â Â

½ Õ

Â ¾ Â ¾

Õ Õ

½ ¾

where is the dual 1-form of the vector for . For each , set

£ ·

(17) Â

¾Â ¾Â

¦ ´¼µ

Then a standard calculation shows that the sub-Laplacian on acting on

forms is given by

£ ¡¡¡ £ ¡¡¡

Â Â Â

½ Õ ½ Õ

Â ¾ Â ¾

Õ Õ

£ ¦ ¾

Thus the study of the on reduces to the study of second order operators

Ò Ñ

£ ¢ Ê ¾ ¼ Õ Â on for , . Using this notation, one may deﬁne the sub-Laplacian as follows (see [21]).

FUNDAMENTAL SOLUTIONS FOR HERMITE OPERATORS 239

Ä ¦ Deﬁnition 3.2. The partial-Laplacian on is given by the non-negative

operator

½ ½

¾ ¾

· · Ä

·Ò

¦ Ä Ä £

The sub-Laplacian on is given by . The operator is given by

Ò Ò

Â Â

£ Ä · ´¯ µ ¡Ö Ä · ¡Ö

Â Ø Ø

(18)

½ ½ £

4 Sub-Laplacian Operators Â

« ´« « µ Ê Ñ

Let ½ be a vector in with

« ¯

½ ¾Â ¾Â

« Â £

Thus depends on the multi-index . Then the operator Â can be written as

¾ ¾ Â

· ·« ¡Ö £ Ä · ¡Ö

Ø « Ø

·Ò

½ £

In this section, we study the invertibility of the operator Â . This operator is a sum Ò

of squares of ¾ “horizontal” vector ﬁelds and is therefore not elliptic. However,

Ù £ Ù

it is hypoelliptic (see Hormander¨ [18]), i.e., the solution of « is smooth

¾ ´À µ «

whenever Ò , provided does not belong to some exceptional set.

Ò Ñ

´Þ Øµ ¢ Ê 4.1 Fourier Analysis on . For a nice function on ,

the partial Fourier transform with respect to the variables Ø is deﬁned by

Ø¡

´Þ µ ´Þ ØµØ

Ñ Ê

and the inverse transform given by

Ø¡

´Þ Øµ ´Þ µ

´¾ µ

Ê Ã

We seek a distributional kernel Â such that

£ Ã Ã £ Á

Â Â Â Â £

if Â is invertible, and

£ Ã Ã £ Á Ë

Â Â Â Â Â

240 O. CALIN, D.-C. CHANG AND JINGZHI TIE,

£ Ë Ä ´ µ

if Â is not invertible and is the orthogonal projection of onto the null

£ Ã Â

space of Â . Because of the group structure, we only need to ﬁnd at the origin Ã and can then translate Â to any point by the group multiplication. Integration by

parts yields

Ø¡

Ø Ø

£ Ø

Now take the partial Fourier transform of Â with respect to and obtain

¾Ò

£ « ¡

with

·¾ ´ ¡ µÜ ¾ ´ ¡ µÜ

·Ò ·Ò

and

Ü Ü

·Ò

¾ ¾

We compute the term ﬁrst:

·Ò

¾ ¾

·Ò

¾ ¾

¾ ¾ ¾

· · ´ ¡ µ Ü Ü ´ ¡ µ ´Ü · Ü µ

·Ò

¾ ¾

Ü Ü Ü Ü

·Ò

Â ·Ò

Since , we may rewrite as

¾Ò Ò

¾ ¾

£ ´ ¡ µ Ü Ü « ¡ · ´ ¡ µ Ü

Â ·Ò

Ü Ü Ü

·Ò

½ ½

£ ´Ü Ü µ

Ò·

Since the operator Â is rotationally invariant on the plane , we can Ã

assume that Â is polyradius, i.e.,

Ü Ü Ã ¼

·Ò Â

Ü Ü

·Ò

Thus we can reduce Â to the operator

¾Ò

¾ ¾

À ´ ¡ µ Ü « ¡ Â

(19)

¾Ò

which can be realized as the resolvent of the Hermite operator on Ê . À

4.2 Fundamental solution of Â . Applying Theorem 2.8, we ﬁnd that À

the fundamental solution of Â at the origin is

µ ´

Ò ¾Ò

¾ ¡

¾ ´«¡ µ×

× ÜÔ ¡ ÓØ ´ ¡ ×µÜ

´¾ µ × Ò ´ ¡ ×µ

½ ½

FUNDAMENTAL SOLUTIONS FOR HERMITE OPERATORS 241

¡ ¾ £ £

for , where is set

´ µ

¾Ò

£ ¡ ´¾ ·½µ ´ µ ¾

½ ¾

(20)

£ ¡ ¾ £

If , then Â is not invertible. We study this case in a forthcoming paper.

Replacing by in (20), we can write Â in the form

´ µ

Ò ¾Ò

¡ ½

´«¡ µ× ¾

ÜÔ ¡ ÓØ ´ ¡ µ

× Ò ´ ¡ µ

½ ½

Next we take the inverse Fourier transform with respect to and obtain

½ ¡

´«¡ µ×·Ø¡

Ñ Ò·Ñ

¾ × Ò ´ ¡ µ

Ê ¼

¾Ò

¡ ÓØ ´ ¡ µ ¢ ÜÔ

Replacing by and integrating the above integral with respect to ﬁrst, we

have

½ ½

´«¡ µ·´Ø¡ µ ×

Ñ Ò·Ñ Ò·Ñ

¼ Ê

´ µ

Ò ¾Ò

¡ ½

¢ ÜÔ ¡ ÓØ ´ ¡ µ

× Ò ´ ¡ µ

½ ½

½ ¡ ½

´«¡ µ

Ñ Ò·Ñ Ò·Ñ

¾ × Ò ´ ¡ µ

¼ Ê

¢ ÜÔ ´ µ ¡

¾Ò

´ µ ¡ ÓØ ´ ¡ µ

where .

Changing the order of the integrations and replacing by ,wehave

¡ ´ · ¾µ

«¡

Ñ Ò·Ñ Ò·Ñ ½

¾ × Ò ´ ¡ µ ´ µ ¡

£ Â

Then Â is the fundamental solution of at the origin when it is invertible, i.e.,

£ £ Æ

Â Â Â Â ¼ 242 O. CALIN, D.-C. CHANG AND JINGZHI TIE,

The fundamental solution at arbitrary point can be obtained by translation through

the group law over . Here

¾Ò

´Ü Ø µ ´Ü µ Ø ¡ ¡ ÓØ ´ ¡ µÜ Ø ¡

½ is the modiﬁed complex action function, which is the solution of the Jacobi-

Hamilton equation. The function

Ú ´ µ

× Ò´ ¡ µ

is the volume element associated to the action function which is the solution of

the transport equation.

Ñ ½ Ñ ½ ¦ À

4.3 Formula when . In the case , ,wehave ; Ã

and the fundamental solution Â becomes

«¡

´Ò ½µ Ú ´ µ

«¡

Ã ´Ü Øµ

Â Ò

Ò·½ Ò Ò

¾ × Ò ´ µ ´Ü µ Ø ¡ ´Ü Ø µ

Ê Ê

This recovers the formula obtained by Beals and Greiner [4] (see also [6]). In

½ ½Ò particular, when for , we may obtain an exact form for the

kernel. In this case, the kernel is

«×

´Ò ½µ ½

Ã ´Ü Øµ ×

Ò·½ Ò

¾Ò

¾ × Ò ´×µ

Ü ÓØ ´×µ Ø

«×

´Ò ½µ

Ò·½

´ Ü Ó× ´×µ Ø × Ò ´×µµ

¾Ò

¾ ¾

Ü Ü

where . Set

¾ ¾ ¾

Ö ´Ü Ø µ Ö ´Ü Øµ

· and

with . Using the identity

¾ ¾

Ó× ´× · µ Ó× ´×µÓ× · × Ò ´×µ× Ò

we have

«×

´Ò ½µ

Ã ´Ü Øµ ×

(21) Â

Ò·½ ¾ Ò

¾ Ö Ó× ´× · µ ½

Upon changing the contour, the formula (21) becomes

«×

½ ´Òµ

Ã ´Ü Øµ ×

Ò·½ ¾Ò Ò

¾ Ö Ó× ´×µ ½ FUNDAMENTAL SOLUTIONS FOR HERMITE OPERATORS 243

The above integral can be evaluated as

½ Ò · « Ò «

¾Ò «

Ã ´Ü Øµ Ö

Ò·½

¾ ¾ ¾

Ò « Ò·«

¾ ¾

´Ü · Øµ ´ µ

Ò«

´µ

From the above formula, we see that the kernel can be extended from Re

Ò £

to the region , where

£ ¦´ ·¾ µ ¾ ·

This coincides with the result obtained by Folland and Stein [16], except for a

constant factor arising because our vector ﬁelds have a different constant in front

of (see also [6] and [12]). In particular, when , one has

¾ Ò¾ ¾ Ò¾ ¾ Ò¾

´ µ ´ · µ ´ µ ´ · µ

¼ Ò Ò

4.4 Product of Heisenberg groups ½ . In the case and non-

¡ ´ Ø ´µµ degenerate, we can introduce the new variables and .

Then

¾ ¾

´ µ ÓØ´ µ´ · µ

·Ò

The fundamental solution becomes

´¾ ¾µ

Ò ¾Ò

¾ Ø´µ

«¡´ ×µ

¢ £

¾Ò ½

¾ ¾

×Ò´ µ

ÓØ´ µ´ · µ ¡ ´ µ

½ ·Ò

If we assume that , where is the identity matrix, then we obtain

«¡×

´¾ ¾µ

¾Ò ½

Ò ¾Ò

¾ ×Ò´ µ

¾ ¾

ÓØ´ µ´ · µ ¡

½ ·Ò £

5 Heat Kernel of Â

´ µ

The heat kernel is the distributional kernel × satisfying

´ µÆ ´ µ · £ ´ µ¼ ÐÑ

× ×

Â and

×¼

244 O. CALIN, D.-C. CHANG AND JINGZHI TIE, £

Just as the fundamental solution of Â can be derived from that of the Hermite operator, we can also derive the heat kernel from that of the Hermite operator.

Following the same line of the argument as in last section, we have

µ ´

Ò ¾Ò

´«¡ µ×

¾ ¡

¡ ÓØ´ ¡ ×µÜ È ´Ü µ ÜÔ

´¾ µ ×Ò´ ¡ ×µ

½ ½

Next take the inverse Fourier transform with respect to to obtain

¡ ½

´«¡ µ×·Ø¡

È ´Ü Øµ

Ñ Ò·Ñ

¾ ×Ò´ ¡ ×µ

¾Ò

Ò Ó

¢ ÜÔ ¡ ÓØ´ ¡ ×µÜ

It seems unlikely that the above formula can simpliﬁed further in general. We

consider the special cases.

À Ñ ½

5.1 Heisenberg group Ò . Take , . The above formula

becomes

´ µ

Ò ¾Ò

È ´Ü Øµ ÜÔ « × · Ø ÓØ´ ×µÜ

Ò·½

¾ ×Ò´ ×µ

½ ½

Replacing by ,wehave

È ´Ü Øµ

´ µ

Ò ¾Ò

ÜÔ « ÓØ´ µÜ Ø

Ò·½

¾´×µ ×Ò´ µ ×

½ ½

This coincides with the formula found by Beals and Greiner [4].

À Ñ Ò ´¼ ¼ ½ ¼ ¼µ

5.2 Product of ½ . Take and , i.e.,

1isonthe -th position. Then we have

È ´Ü Øµ

Ò ¾Ò

¾ ×Ò´ ×µ

´ µ

¾ ¾

´« ¡ µ× · Ø ¡ ¢ ÜÔ ÓØ´ µ´Ü · Ü µ

·Ò

FUNDAMENTAL SOLUTIONS FOR HERMITE OPERATORS 245 À

The integration can be separated into products of the heat kernel on ½ naturally:

È ´Ü Øµ

Ò ¾Ò

¾ ×Ò ´ ×µ

¨ ©

¾ ¾

¢ ÜÔ « × · Ø ÓØ ´ µ´Ü · Ü µ

·Ò

Ò ¾Ò

¾ ´×µ ×Ò ´ µ

Ò Ó

¾ ¾

¢ ÜÔ « ÓØ ´ µ´Ü · Ü µ Ø

·Ò ×

6 Hermite Operator on the Heisenberg Group Ã

In this section, we calculate the fundamental solution « for the Hermite

À Ñ ½

operator on the Heisenberg group Ò . This corresponds to the case of and

½ Ò

½ ¾Ò

, in Section 4. The left-invariant vector ﬁelds are

·¾ Ü

·Ò

Ü Ø

¾ Ü ½ Ò

·Ò

Ü Ø

·Ò

¬ ¬

·Ò

Assume that . We deﬁne the operator

¾Ò

¾ ¾ ¾

¬ Ü Ì À « ·

½ to be the Hermite operator on the Heisenberg group.

First we note that

¾ ¾ ¾

· Ü ¾ Ü

·Ò

Ü Ø Ü Ø

¾ ¾ ¾

Ü · ·¾ Ü

·Ò

Ü Ø Ü Ø

·Ò

It follows that

¾Ò ¾Ò

¾ ¾

¾ ¾ ¾ ¾

Ü ¬ À « · Ü

Ü Ø

½ ½

Ü ·¾ Ü

·Ò

Ü Ü Ø Ø

·Ò

246 O. CALIN, D.-C. CHANG AND JINGZHI TIE,

½Ò

·Ò

Here for .

¾ ¾

· ´Ü Ü µ

·Ò

The operator is invariant under rotations in the plane,

·Ò

Ã ´Ü Ü µ

·Ò

so the fundamental solution « is also invariant under rotations in the

Ã ´ÜØ Ý ×µ plane (see [10], [13] and [19]). Therefore, we may assume that « is

independent of the rotational variables. It follows that

Ã Ã

« «

Ü Ü ¼ ½Ò

·Ò

Ü Ü

·Ò

Let

¾Ò ¾Ò

¾ ¾

¾ ¾ ¾ ¾

Ê « · ¬ Ü

Ü ·

Ü Ø Ø

½ ½

À Ø

Taking the partial Fourier transform of « with respect to the -variable, one has

¾Ò ¾Ò

¾ ¾ ¾ ¾ ¾

Ê ¬ Ü Ü

· « ·

½ ½

¾Ò

¾ ¾ ¾ ¾

« · · ´ · ¬ µÜ

¾Ò

¾ ¾

« · Ü

¾Ò

¾ ¾

Ê · ¬

This is exactly the Hermite operator on with parameter and

« « · ½ Ò

·Ò

. Recall that for . Hence the exceptional set of Ê

« is

¾Ò

¾ ¾

£ ´« µ « · · ¬ ´¾ ·½µ ¾ ´ µ

´« µ ¾ £ Ê

Using Theorem 2.8, we know that for , the fundamental solution for «

¾Ò

½¾

´«· µ×

Ã ´Ü Ý µ

´¾ µ × Ò´¾ ×µ

¾Ò

£ ¢

´Ü Ý µ

¾ ¾

¢ ÜÔ µ ØÒ´ ×µ × · Ý ·´Ü

¾ × Ò´¾ ×µ

FUNDAMENTAL SOLUTIONS FOR HERMITE OPERATORS 247 À

Next, we can ﬁnd the fundamental solution for the operator « with singular

Ý ¼µ

point ´ by taking the inverse Fourier transform with respect to . We note that

¾ ¾

· ¬

and that it depends on .Now

Ã ´Üµ Ã ´ÜØ Ý ¼µ

« «

¾Ò

½ ¾

½ ½

´«· µ× Ø

Ò·½

´¾ µ ×Ò´¾ ×µ

¼ ½

¾Ò

¢ £

´Ü Ý µ

¾ ¾

¢ ÜÔ ·´Ü · Ý µØ Ò´ ×µ ×

¾ ×Ò´¾ ×µ

¾Ò

½ ¾

¾ ¾ ¾ ½ ¾

½ ½

´ · ¬ µ

Ø ´«· µ×

¾ ¾

Ò·½

¾ ½ ¾

´¾ µ

×Ò´¾×´ · ¬ µ µ

½ ¼

¾Ò

¾ ¾ ¾ ½ ¾

´ · ¬ µ

´Ü Ý µ

¢ ÜÔ

¾ ¾

¾ ½ ¾

· ¬ µ µ ×Ò´¾×´

¾ ¾ ¾ ¾ ¾ ½ ¾

·´Ü · Ý µ Ø Ò´×´ · ¬ µ µ ×

Ý ×µ

The fundamental solution at any point ´ is

Ã ´Ü Ø Ý ×µÃ ´Ü Ø × Ý ¼µ

« «

The above formula is rather complicated and cannot be simpliﬁed in the general case.

REFERENCES

[1] R. Beals, A note on fundamental solutions, Comm. Partial Differential Equations, 24 (1999), 369–376. [2] R. Beals, B. Gaveau and P. C. Greiner, The Green function of model step two hypoelliptic operators and the analysis of certain tangential Cauchy-Riemann complex, Adv. Math. 121 (1996), 288–345. [3] R. Beals, B. Gaveau and P. C. Greiner, Complex Hamiltonian mechanics and parametrices for subelliptic Laplacians, I,II,III, Bull. Sci. Math. 121 (1997), 1–36, 97–149, 195–259. [4] R. Beals and P. C. Greiner, Calculus on Heisenberg manifolds, Ann. Math. Studies 119, Princeton University Press, Princeton, New Jersey, 1988. [5] R. Beals, B. Gaveau, P. C. Greiner and Y. Kannai, Exact fundamental solutions for a class of degenerate elliptic operators, Comm. Partial Differential Equations 24 (1999), 719–742. [6] C. Berenstein, D. C. Chang and J. Tie, Laguerre Calculus and Its Applications on the Heisenberg Group, International Press, Cambridge, MA, 2001. [7] O. Calin and D. D. Chang, Geometric Mechanics on Riemannian Manifolds, Applications to

Partial Differential Equations, Birkhauser¨ Boston, Boston, MA, 2005.

´ ·½µ [8] O. Calin, D. C. Chang and P. Greiner, On a step ¾ Sub-Riemannian manifold, J. Geom. Anal. 14 (2004), 1–18. 248 O. CALIN, D.-C. CHANG AND JINGZHI TIE,

[9] O. Calin, D. C. Chang, P.Greiner and Y.Kannai, On the geometry induced by a Grusin operator, in Complex Analysis and Dynamical Systems II, Comtemp. Math., 382 (2005), 89–111. [10] O. Calin, D. C. Chang and J. Tie, Hermite operator on the Heisenberg group,inHarmonic Analysis, Signal Proceedings and Complexity, (I. Sabadini, D. Struppa and D. Walnut, eds.), Birkhauser¨ Boston, Bdoston, MA, 2005, pp. 37–54. [11] D. C. Chang and P. Greiner, Analysis and geometry on Heisenberg groups,inProceed- ings of the Second International Congress of Chinese Mathematicians (C. S. Lin and S. T. Yau, eds.), International Press, Cambridge, MA, 2004, pp. 379–405. [12] D. C. Chang and J. Tie, Estimates for spectral projection operators of the sub-Laplacian on the Heisenberg group, J. Anal. Math. 71 (1997), 315–347. [13] D. C. Chang and J. Tie, Estimates for powers of sub-Laplacian on the non-isotropic Heisenberg group, J. Geom. Anal. 10 (2000), 653–678. [14] D. C. Chang and J. Tie, A note on Hermite and subelliptic operators, Acta Math. Sin., (Eng. Ser.), 21 (2005), 803–818. [15] W. L. Chow, Uber¨ Systeme von linearen partiellen Differentialgleichungen erster Ordnung,

Math. Ann. 117 (1939), 98–105.

[16] G. B. Folland and E. M. Stein, Estimates for the complex and analysis on the Heisenberg group, Comm. Pure Appl. Math. 27 (1974), 429–522. [17] J. Glimm and A. Jaffe, Quantum Physics. A Functional Integral Point of View, 2nd ed., Springer-Verlag, Berlin, New York, Heidelberg, 1987. [18] L. Hormander,¨ Hypoelliptic second-order differential equations, Acta Math. 119 (1967), 147–171. [19] L. Hormander,¨ Riemannian Geometry, Department of Mathematics, Lund, Sweden, 1990. [20] T. P. Liu, Hyperbolic and Viscous Conservation Laws, CBMS-NSF Regional Conference Series in Applied Mathematics, 72, SIAM, Philadelphia, PA, 2000. [21] A. Nagel, F. Ricci and E. M. Stein, Singular integrals with ﬂag kernels and analysis on quadratic CR manifolds, J. Funct. Anal. 181 (2001), 29–118. [22] S. Thangavelu, Lectures on Hermite and Laguerre Expansions, Princeton University Press, Princeton, NJ, 1993. [23] Y. Zeng and T. P. Liu, Large Time Behavior of Solutions for General Quasilinear Hyperbolic- Parabolic Systems of Conservation Laws, Mem. Amer. Math. Soc. 125, 1997.

Ovidiu Calin DEPARTMENT OF MATHEMATICS EASTERN MICHIGAN UNIVERSITY

YPSILANTI, MI 48197, USA

ÑÐ Ó ÐÒ ÑÙÒÜºÑºÙ

Der-Chen Chang DEPARTMENT OF MATHEMATICS GEORGETOWN UNIVERSITY

WASHINGTON, DC 20057-1233, USA

ÑÐ Ò ÑØºÓÖØÓ ÛÒºÙ

Jingzhi Tie DEPARTMENT OF MATHEMATICS UNIVERSITY OF GEORGIA

ATHENS, GA 30602-7403, USA

ÑÐ Ø ÑØºÙºÙ (Received November 17, 2005)