• 沒有找到結果。

Laguerre Calculus and Paneitz Operator on the Heisenberg Group

N/A
N/A
Protected

Academic year: 2021

Share "Laguerre Calculus and Paneitz Operator on the Heisenberg Group"

Copied!
18
0
0

加載中.... (立即查看全文)

全文

(1)

LAGUERRE CALCULUS AND PANEITZ OPERATOR ON THE HEISENBERG

GROUP

DEDICATED TO PROFESSOR TONGDE ZHONG ON HIS 80TH BIRTHDAY

DER-CHEN CHANG, SHU-CHENG CHANG, AND JINGZHI TIE

Abstract. Laguerre calculus is a powerful tool for harmonic analysis on the Heisenberg group. Many sub-elliptic partial differential operators can be inverted by Laguerre calculus. In this article, we use Laguerre calculus to find explicit kernels of the fundamental solution for the Paneitz operator and its heat equation. The Paneitz operator which plays an important role in CR geometry can be written as follows:

Pα= LαLα¯ =1 4 hXn j=1 (Zjj+ ¯ZjZj) i2 + α2T2. Here {Zj}n

j=1is an orthonormal basis for the subbundle T(1,0)of the complex tangent bundle TC(Hn) and T is the “missing direction”. The operator Lα is the Laplacian on the Heisenberg group which is sub-elliptic if α does not belong to an exceptional set Λα. We also construct projection operators and relative fundamental solution for the operator Lαwhile α ∈ Λα.

1. Introduction

The non-isotropic Heisenberg group Hn is the Lie group with underlying manifold Cn× R = {[z, t] : z ∈ Cn, t ∈ R}

and multiplication law

(1.1) [z, t] · [w, s] = [z + w, t + s + 2Im

n X j=1

ajzjw¯j], where a1, a2, · · · , an are positive numbers.

It is easy to check that the multiplication (1.1) does indeed make Cn× R into a group whose identity is the origin e = [0, 0], and where the inverse is given by [z, t]−1= [−z, −t].

The Lie algebra of Hn is a vector space which, together with a Lie bracket operation defined on it, represents the infinitesimal action of Hn. Let hndenote the vector space of left-invariant vector fields on Hn. Note that this linear space is closed with respect to the bracket operation

[V1, V2] = V1V2− V2V1.

The space hn, equipped with this bracket, is referred to as the Lie algebra of Hn.

The Lie algebra structure of hnis most readily understood by describing it in terms of the following basis:

(1.2) Xj = ∂xj + 2ajyj ∂t, Yj = ∂yj − 2ajxj ∂t and T = ∂t;

where j = 1, 2, · · · , n, z = (z1, z2, · · · , zn) ∈ Cn with zj = xj+ iyj; t ∈ R.

Note that we have the commutation relations

(1.3) [Yj, Xk] = 4ajδjkT for j, k = 1, 2, · · · , n.

Next, we define the complex vector fields

(1.4) Z¯j = 1 2(Xj+ iYj) = ∂ ¯zj − iajzj ∂t and Zj= 1 2(Xj− iYj) = ∂zj + iajz¯j ∂t

The first author is partially supported by a research grant from United States Army Research Office and a Hong Kong RGC competitive earmarked research grant #600607.

The second and the third authors were partially supported by a NSC research grant of Republic of China. 1

(2)

for j = 1, 2, · · · , n. Here, as usual, ∂zj = 1 2 µ ∂xj − i ∂yj ¶ and ∂ ¯zj = 1 2 µ ∂xj + i ∂yj.

The commutation relations (1.3) then become

[¯Zj, Zk] = 2iajδjkT with all other commutators among the Zj, ¯Zk and T vanishing.

The Heisenberg sub-Laplacian is the differential operator

(1.5) Lα= −1 2 n X j=1 (ZjZj¯ + ¯ZjZj) + iαT

with Zj and ¯Zj given by (1.4). In the case of aj = 1 for all j’s, the operator Lα was first introduced by Folland and Stein [9] in the study of ¯∂b complex on a non-degenerate CR manifold. They found the fundamental solution of Lα. Using heat kernel technique, Beals and Greiner [4] solved the case when aj’s may be different. We may also obtain the fundamental solution for Lα with different a0

j by using Laguerre calculus, see [5].

In this paper, we consider the Paneitz operator Pαon the Heisenberg group, which is defined as

(1.6) Pα= LαL¯α=1 4 hXn j=1 (ZjZj¯ + ¯ZjZj) i2 + α2T2.

Let us recall some background about the Paneitz operator. Note that for a coframe {θ, ωj, ¯ωj}nj=1 dual to ©T, Zj, ¯Zjªnj=1. Let ϕ ∈ C∞

0 (Hn) be a smooth function with compact support. It is easy to see that

Lnϕ = −1 2 n X j=1 (ZjZj¯ + ¯ZjZj) + inT)(ϕ) = − n X j=1 ¯ ZjZj(ϕ). It follows that Pnϕ = LnL¯ =1 4 h Xn j,k=1 (ZjZj¯ + ¯ZjZj) + 2inT ih (ZkZk¯ + ¯ZkZk) − 2inT i ϕ =n 1 4 £Xn j=1 (ZjZj¯ + ¯ZjZj)¤2+ n2T2oϕ =4 n X k=1 ¯ Zk(Pkϕ). Here (see [19]) Pkϕ = n X j=1 Zk(ZjZj¯ ϕ), k = 1, . . . , n and P ϕ = n X k=1 (Pkϕ)ωk

is an operator that characterizes CR-pluriharmonic functions. Note that a smooth real-valued function u in Hn is said to be CR-pluriharmonic function if for any point p ∈ Hn, there is an open neighborhood U of p

in Hn and a smooth real-valued function v on U such that ¯∂b(u + iv) = 0. Here ¯∂bu = Pn

j=1(¯Zju)¯ωj. Moreover, one has

Z Hn hP ϕ + ¯P ϕ, dbϕi dµ = Z Hn Pnϕ · ϕ dµ, ∀ ϕ ∈ C0(Hn). Here db = π ◦ d : C∞(Hn) → span{ωj, ¯ωj}nj=1 2

(3)

and π is the orthogonal projection onto the subspace span{ωj, ¯ωj}nj=1 in the cotangent space. Moreover, a smooth real-valued function ϕ ∈ L2(Hn) satisfies Pnϕ = 0 on H

n if and only if P ϕ = 0 on Hn.

Note that via the CR Paneitz operator Pn, we are able to get subgradient estimate and establish

Liouville-type theorems of the CR heat equation on Hn(see [7]). For complex geometric aspects about these operators, readers may make references to [12], [16] and [19].

We will apply the Laguerre calculus to find the eigenfunctions, kernels, heat kernel and relative funda-mental solution of the Paneitz operator. Since the Laguerre tensor of Pα is simply diagonal, its inverse is also diagonal. In order to find the fundamental solution, we only need to sum up the corresponding Laguerre expansion of its inverse. We shall write the fundamental solution in terms of the modified complex action

g(s; z, t) and volume element ν(s) of the Heisenberg group, where

(1.7) g(s; z, t) = γ(2s; z) − it = n X j=1 aj|zj|2coth(2ajs) − it and ν(s) = n Y j=1 2aj sinh(2ajs)

were introduced by Beals, Gaveau and Greiner in [2] (see also Calin, Chang and Greiner [6] for detailed discussion).

The fundamental solution of Pαis a function Ψα(z, t), smooth away from the origin, so that PαΨα= δ0

in the sense of distribution, where δ0 is the Dirac delta function at the origin. Our main result is following.

Theorem 1.1. The exceptional set of α is given by

Λα=   ± n X j=1 (2kj+ 1)aj: (k1, · · · , kn) ∈ (Z+)n   .

Then for n > 1 and |α| <Pnj=1(2kj+ 1)aj; (k1, · · · , kn) ∈ (Z+)n,

(1.8) Ψα(z, t) = Γ(n − 1) 2nαπn+1 Z −∞ ν(s) sinh(2αs) [g(s; z, t)]n−1 ds,

and for other values of α /∈ Λα, we can obtain Ψα by analytic continuation of (1.8).

We also study the heat kernel of Pα. Combining the semigroup expression of the heat kernel hs(z, t) = exp{−sPα} with the identity:

(1.9) w−β= 1

Γ(β)

Z

0

sβ−1e−swds for w > 0 and Re(β) > 0, one may get the result.

The final version of this paper was in part written while the first two authors visited the National Center for Theoretical Sciences and the Academia Sinica during the year of 2007. They would like to express their profound gratitude to Professors Jin Yu and Jih-Hsin Cheng for their invitations and for the warm hospitality extended to them during their visit in Taiwan.

2. Laguerre Calculus

Laguerre calculus is the symbolic tensor calculus on the Heisenberg group Hn. It was first introduced on H1 by Greiner [13] and extended to Hn and Hn× Rd by Beals, Gaveau, Greiner and Vauthier [3]. The

Laguerre functions have been used in the study of the twisted convolution, or equivalently, the Heisenberg convolution for several decades, see [11] and [21]. The Laguerre functions also played an important role in the Fock-Bargmann and Schr¨odinger representations of the Heisenberg group (see Folland [8] for details). But it was in Greiner [13] that for the first time, Laguerre functions were connected with left-invariant convolution operators on H1, and were used to invert some basic differential operators on H1, namely the Lewy operator

and the Heisenberg sub-Laplacian.

In this work, we shall use the Laguerre Calculus to invert the Paneitz operator. In order to introduce the Laguerre Calculus, we first recall the definitions of the twisted and P.V. convolutions on the Heisenberg group.

Let functions f, g ∈ C∞

0 (Hn), the Heisenberg convolution is given by

(2.1) f ∗ g(x) =

Z

Hn

f (y)g(y−1x)dV (y);

(4)

here dV (y) is the Haar measure on Hn and is exactly the Euclidean measure on R2n+1.

2.1. Twisted Convolution. We focus our attention on the phase space Rn× Rn, which we identify with Cn via ζ ∈ Cn, ζ = u + iv ↔ (u, v) ∈ Rn× Rn. Let A =      a1 a2 . .. an     

be a positive-definite diagonal n × n real matrix. Consider the symplectic form < ·, · > given by the Heisenberg group multiplication law (1.1) and defined by

< z, w >= 2Im(Az · ¯w) = 2Im( n X j=1

ajzjw¯j),

where z, w ∈ Cn. With τ a fixed real constant, we can define the twisted convolution of two functions F and G by

(2.2) (F ∗τG)(z) =

Z

Cn

F (z − w)G(w)e−iτ <z,w>dw;

here dw is the Euclidean measure on Cn. Notice that, in view of the antisymmetry of < ·, · >, we have that

< z − w, w >= − < w, z >; thus

G ∗τF = F ∗−τG, so the twisted product is not commutative.

The twisted convolution arises when we analyze the convolution of functions on the Heisenberg group in terms of the Fourier transform in the t−variable. To see this, let f (z, t) be a test function on Hn. Define

(2.3) feτ(z) = f (z, ·)ˆ(τ ) =

Z

R

f (z, t)e−iτ tdt.

Similarly define e when g is another test function on Hn. Suppose f ∗ g is the convolution of f and g on Hn. Then

(2.4) (f ∗ g)^τ = efτ∗τegτ.

2.2. P.V. Convolution Operators. In order to show the regularity of the solution operator S(f ) = f ∗Ψα, we need to introduce principal value convolution operators on Hn. These operators are the analogues of Calder´on-Zygmund principal value convolution operators on Rn. As we know, the underlying manifold of Hn is R2n+1; but the role of the additive structure in R2n+1 is supplanted by the Heisenberg group

multiplication law (2.2). Moreover, the group law forces us to use non-isotropic dilations on Hn, i.e., x 7→ δ ◦ x = δ ◦ [z, t] = [δz, δ2t]

for all δ > 0. These dilations are automorphisms of the group Hn:

δ ◦ (x · y) = (δ ◦ x) · (δ ◦ y);

but the standard isotropic dilations of R2n+1 are not automorphisms of Hn.

A function f defined on Hn is said to be H-homogeneous of degree m on Hn if

f (δ ◦ [z, t]) = f (δz, δ2t) = δmf (z, t) for all δ > 0. For example, the fundamental solution Ψm of Lm

α is H-homogeneous of degree −2n + 2m − 2. Next we introduce the norm function ρ given by

(2.5) ρ(x) =¡kzk4+ t1/4, where kzk2= n X j=1 aj|zj|2. 4

(5)

Obviously, we have ρ(x−1) = ρ(−x) = ρ(x) and ρ(δ ◦ x) = δρ(x). In addition, the function ρ satisfies the triangle inequality:

ρ(x · y) ≤ C1{ρ(x) + ρ(y)}

for some universal constant C1. The distance function, d(x, y) of points x, y ∈ Hn, is defined to be

d(x, y) = ρ(y−1· x);

It is clear that d(x, y) satisfies the symmetric property: d(x, y) = d(y, x).

Suppose that K ∈ C∞(Hn\ {0}), is H-homogeneous of degree γ. Then K is locally integrable near the origin if γ > −2n − 2. See Folland and Stein [9] for the proof of this statement.

Definition 2.1. Let K ∈ C∞(Hn\ {0}) be H-homogeneous of degree −2n − 2, K is said to have mean value

zero property if

(2.6)

Z ρ(x)=1

K(x)dσ(x) = 0,

where dσ(x) is the induced measure on the Heisenberg unit sphere ρ(x) = 1.

Using Theorem 3 and Corollary 5.24 of Chapter XII in Stein [22], we can get the basic estimate concerning P.V. convolution operators on Hn:

Theorem 2.1. Let K ∈ C∞(Hn\ {0}), H-homogeneous of degree −2n − 2 with mean value zero property.

Then K induces a principal value (P.V.) convolution operator, given by

(2.7) K(f )(x) = (f ∗ K)(x) = lim ε→0 Z d(x,y)>ε f (y)K(y−1· x)dV (y), for f ∈ C∞

0 (Hn). Moreover, the operator K given by (2.7) can be extended to a bounded operator from the

Lp-Sobolev space Lp

k(Hn) into itself, for 1 < p < ∞ and k ∈ Z+.

2.3. Laguerre Functions. The generalized Laguerre polynomials L(α)k (x) are defined by their usual gener-ating function formula:

(2.8) X k=1 L(α)k (x)wk= 1 (1 − w)α+1exp ½ xw 1 − w ¾ , for α = 0, 1, 2, · · · , x ≥ 0, and |w| < 1.

Definition 2.2. Let z = |z|eiθ and k, p = 0, 1, 2, · · · . Then we define f Wk(p)(z, τ ) = 2|τ | π · Γ(k + 1) Γ(k + p + 1) ¸1/2 (2|τ ||z|2)p/2eipθe−|τ ||z|2L(p)k (2|τ ||z|2) (2.9) f Wk(−p)(z, τ ) = 2|τ | π (−1) p · Γ(k + 1) Γ(k + p + 1) ¸1/2 (2|τ ||z|2)p/2e−ipθe−|τ ||z|2L(p)k (2|τ ||z|2) (2.10)

2.4. Laguerre Calculus on H1. The most important property of the fWk(p)(z, τ ) is the following theorem by Greiner [13]:

Theorem 2.2. Let p, k, q, m = 1, 2, · · · . Then (2.11) Wf(p∧k)−1(p−k) ∗|τ |Wf(q∧m)−1(q−m) = δ (q) k · fW (p−m) (p∧m)−1, Wf (p−k) (p∧k)−1∗−|τ |Wf(q∧m)−1(q−m) = δm(p)· fW (q−k) (q∧k)−1,

where a ∧ b = min(a, b) and δ(q)k denotes the Kronecker delta function, i.e, δk(q) = 1 if k = q and vanishes

otherwise.

Thus the twisted convolution of two functions of fWk(p)is another function of the same type. This surprising result justifies the use of Laguerre function expansion on the Heisenberg group in analogy with Mikhlin’s use of the spherical harmonics on Rn.

Let Wk(p)(z, t), ± p, k = 0, 1, 2, · · · , be the inverse Fourier transform of fWk(p)(z, τ ) with respect to τ , i.e.,

Wk(p)(z, t) = 1 Z −∞ eitτWfk(p)(z, τ )dτ. 5

(6)

These are the kernels of the generalized Cauchy-Szeg¨o operators on H1. In particular,

W0(0)(z, t) = S++ S−, where S±= 1

π2 ·

1 (|z|2∓ it)2

denotes the Cauchy-Szeg¨o kernels in H1.

The following result implies that the generalized Cauchy-Szeg¨o kernels indeed induce principal value convolution operators.

Theorem 2.3. The generalized Cauchy-Szeg¨o kernels Wk(p)(z, t) are in C∞(H

1\ {0}) and have zero mean

value property.

Formula (2.11) is reminiscent of matrix multiplication. To exploit this similarity, Greiner [13] introduced the Laguerre matrix:

Definition 2.3. We define the positive Laguerre matrix

M+

³ f

W(p∧k)−1(p−k)

´

of fW(p∧k)−1(p−k) to be the infinite matrix which has one at the intersection of the p-th row and k-th column and zeros everywhere else. The negative Laguerre matrix M−

³ f

W(p∧k)−1(p−k) ´can be defined as the transpose of the positive Laguerre matrix.

Following this definition, Theorem 2.2 takes the following form

(2.12) M+ ³ f W(p∧k)−1(p−k) ∗|τ |Wf(q∧m)−1(q−m) ´ = M+ ³ f W(p∧k)−1(p−k) ´· M+ ³ f W(q∧m)−1(q−m) ´

The Laguerre matrix for any left-invariant convolution operator can be defined as in [13] and [3]. In particular, we can define the Laguerre matrix for any left-invariant differential operator on H1, since we can

write it into the form of convolution operator. We omit the detail here and the interested readers can consult [3] and [5]. We only list the following results:

Theorem 2.4. If F and G are two P.V. convolution operators on H1, and M( ˜F) and M( ˜G) denote the

Laguerre matrices of F and G respectively; then

(2.13) M(eF ∗τG) = M(ee F) · M( eG) = M+(eF) · M+( eG) ⊕ M−(eF) · M−( eG).

A simple consequence of this theorem is

Corollary 2.1. Let I denote the identity operator on C∞

0 (H1), then I is induced by the identity Laguerre

matrix W±(eI) = (δ(p)k ).

2.5. Laguerre Calculus on Hn. We define the n-dimensional version of the exponential Laguerre functions on Hn by the n-fold product:

(2.14) Wfk(p)(z, τ ) = n Y j=1 ajWfk(pjj)( ajzj, τ ) where fW(pj) kj ( a jzj, τ )’s are given by (2.9).

We will compose two functions of the type (2.14) via twisted convolution: f Wk(p)∗τWfm(q) =   n Y j=1 ajWfk(pjj)( ajzj, τ ) ∗τ   n Y j=1 ajWfm(qjj)( ajzj, τ )   = n Y j=1 a2j Z R2 e2iajτ Im(zjw¯j)Wf(pj) kj ( aj(zj− wj), τ ) fWm(qjj)( ajwj, τ )dwj = n Y j=1 ajWfk(pjj)∗τWfm(qjj)( ajzj, τ ) Consequently, Theorem 2.2 implies that

(7)

Theorem 2.5. Let kj, pj , mj and qj= 1, 2, 3, · · · for j = 1, 2, · · · , n. Then (2.15) Wf(k∧p)−1(p−k) ∗|τ |Wf(m∧q)−1(q−m) = δ (q) k · fW (p−m) (p∧m)−1, Wf (p−k) (k∧p)−1∗−|τ |Wf(m∧q)−1(q−m) = δm(p)· fW (q−k) (q∧k)−1,

where k = (k1, k2, · · · , kn), p = (p1, p2, · · · , pn), m = (m1, m2, · · · , mn) and q = (q1, q2, · · · , qn). Here

(k ∧ p) − 1 = (min(p1, k1) − 1, . . . , min(pn, kn) − 1)

and δk(p)=Qnj=1δ(pj)

kj is the n-fold Kronecker delta function.

Instead of the Laguerre matrix, Beals et. al. introduced the definition of Laguerre tensor for the convolution operators on Hn. We omit the details here.

Then we have the n-fold version of the Theorem 2.4:

Theorem 2.6. (The Laguerre calculus on Hn) Let F and G induce the convolution operators on Hn. M(eF)

and M( eG) denote the Laguerre tensors of F and G respectively. Then M(eF ∗τG) = M(ee F) · M( eG). Corollary 2.2. The identity operator I on C∞

0 (Hn) is induced by the identity Laguerre tensor:

M±(eI) = (δ(p1) k1 · · · δ

(pn) kn ).

2.6. Left-invariant differential operators. A left-invariant differential operator P on Hnis a polynomial

P(X, Y, T) with constant coefficients, or in complex coordinates, a polynomial in vector fields T and Zj, ¯Zj. We can have the following representation for P as a convolution operator on Hn:

(2.16) P = PI = X |k|=0 PWk(0,··· ,0)1,··· ,kn∗, where I = X |k|=0 Wk(0,··· ,0)1,··· ,kn

is the identity operator on C∞

0 (Hn). In particular, T and Zj, ¯Zj, j = 1, 2, · · · , n can be represented as

convolution operators and written in the Laguerre tensor forms. This is our next proposition. Proposition 2.1. (1) M( eT) is the iτ multiple of the identity Laguerre tensor:

M±( eT) = iτ (δ(p1) k1 , · · · δ

(pn) kn ).

(2) Zj, j = 1, 2, · · · , n, has the following Laguerre tensor representation:

M(eZj) = M+(eZj) ⊕ M−(eZj) where

M−(eZj)(p1,··· ,pn) k1,··· ,kn = q 2aj|τ |p1/2j δ(pj) k1 · · · δ (pj+1) kj · · · δ (pn) kn and M+(eZj) = M−(eZj) t.

(3)j, j = 1, 2, · · · , n, has the following Laguerre tensor representation: M(ej) = −M(eZj)t.

Theorem 2.7. Let P = P(Z, ¯Z, T) = P(Z1, · · · , Zn, ¯Z1, · · · , ¯Zn, T) denote a left-invariant differential

operator on Hn,i.e., P is a polynomial in the vector fields T and Zj, ¯Zj j = 1, 2, · · · , n. Then

(2.17) M( eP) = P(M(eZ), M(eZ), iτ ),¯

where we set M(eZ) = (M(eZ1), · · · , M(eZn)) and M(eZ) = (M(e¯ Z¯1), · · · , M(eZn)).¯

2.7. The Heisenberg sub-Laplacian. As an application of the Laguerre calculus, Beals et al. [3] obtained the fundamental solution of the sub-Laplacian:

(2.18) Lα= −1 2 n X j=1 (ZjZj¯ + ¯ZjZj) + iαT.

via its Laguerre tensor. Taking the partial Fourier transform with respect to the t-variable, one has

(2.19) Leα= eLαeI = X |k|=0−1 2 n X j=1 (eZjZje¯ + eZj¯ Zj) − ατe  Yn j=1 ajWfk(0)j ( ajzj, τ ) ∗τ. 7

(8)

A simple calculation yields (2.20) 1 2(eZjZje¯ + eZj¯ Zj) fe W (0) k ( ajzj, τ ) = (2k + 1)|τ |ajWfk(0)( ajzj, τ ). Thus, (2.19) and (2.20) imply

(2.21) Leα= X |k|=0   n X j=1 (2kj+ 1)|τ |aj− ατ   n Y j=1 ajWfk(0)j (√ajzj, τ ) ∗τ. Consequently the Laguerre tensor of the convolution operator induced by Lα is

(2.22) M( eLα) = |τ |     n X j=1 (2kj+ 1)aj− αsgn(τ ) δ(p1) k1 · · · δ (pn) kn ,

which is invertible as long as ±α 6= Pnj=1(2kj+ 1)aj, where k = (k1, k2, · · · , kn) ∈ (Z+)n. According to

Theorem 2.6, the inverse Laguerre tensor of (2.22) is

(2.23) M( eL−1α ) = |τ |−1     Xn j=1 (2kj+ 1)aj− αsgn(τ )   −1 δ(p1) k1 · · · δ (pn) kn    , If we write it in the Laguerre series expansion:

(2.24) Le−1α (z, τ ) = |τ |−1 X |k|=0  Xn j=1 (2kj+ 1)aj− αsgn(τ )   −1 n Y j=1 ajWfk(0)j ( ajzj, τ ).

Then we can sum this series in the sense of Abel, take the inverse Fourier transform with respect to τ , and find the fundamental solution of Lα. We shall not repeat here, and refer to [3] and [5] for details. In fact, the above calculation makes sense for any polynomial of Lα. We will carry out this extension, and find the fundamental solution of the Paneitz operator LαL¯α in the next section.

3. The fundamental solution of the Paneitz operator

3.1. The Laguerre tensor of the Paneitz operator. First we will find the Laguerre tensor of the operator

from the Laguerre tensor of Lα. Similar to (2.19), we can take the Fourier transform with respect to t, and write e as a twisted convolution form:

(3.1) e = eLαLαeI = X |k|=0−1 2 n X j=1 (eZjZje¯ + eZj¯ Zj) − ατe    −1 2 n X j=1 (eZjZje¯ + eZj¯ Zj) + ατe   n Y j=1 ajWfk(0)j (√ajzj, τ )∗τ Then, (2.20) yields (3.2) Peα= X |k|=0        n X j=1 (2kj+ 1)|τ |aj   2 − α2τ2      n Y j=1 ajWfk(0)j ( ajzj, τ ) ∗τ. Consequently the Laguerre tensor of the convolution operator induced by Pα is

(3.3) M( ePα) = τ2        Xn j=1 (2kj+ 1)aj   2 − α2    δ(p1) k1 · · · δ (pn) kn    , which is invertible as long as α does not belong to the exceptional set Λα, where

Λα=   ± n X j=1 (2kj+ 1)aj: k = (k1, k2, · · · , kn) ∈ Zn+   . 8

(9)

According to Theorem 2.6, the inverse Laguerre tensor of (3.3) is (3.4) M( eP−1 α ) = τ−2        Xn j=1 (2kj+ 1)aj   2 − α2    −1 δ(p1) k1 · · · δ (pn) kn    , and we write its kernel eΨα(z, τ ) in the Laguerre series expansion:

(3.5) Ψα(z, τ ) = τe −2 X |k|=0      n X j=1 (2kj+ 1)aj   2 − α2    −1 n Y j=1 ajWfk(0)j (√ajzj, τ ).

In order to find the fundamental solution of Pα, we may sum this series and take the inverse partial Fourier transform with respect to the τ -variable. First we introduce the following integral representation of

A−1: (3.6) 1 A = 1 Γ(m) Z 0

e−Asds for Re(A) > 0. Let A(k) =Pnj=1(2kj+ 1)aj. Note that

1 A2(k) − α2 = 1 µ 1 A(k) − α− 1 A(k) + α.

Assume that |α| < a1+ a2+ · · · + an. Then we apply (3.6) and write (3.5) in the following form:

(3.7) Ψα(z, τ ) =e 1 2ατ2 X |k|=0 Z 0 h e−(Pnj=1(2kj+1)aj−α)s− e( Pn j=1(2kj+1)aj+α)s i ds n Y j=1 ajWfk(0)j ( ajzj, τ ). Next we interchange the summation and integration, and use the definitions of fWk(0)j ,

e Ψα(z, τ ) = 1 2α|τ |2 Z 0 X |k|=0 h e−(Pnj=1(2kj+1)aj−α)s− e( Pn j=1(2kj+1)aj+α)s iYn j=1 ajWfk(0)j ( ajzj, τ ) =|τ | n−2 2απn Z 0 X |k|=0 h e−(Pnj=1(2kj+1)aj−α)s− e(Pnj=1(2kj+1)aj+α)s i ds× × n Y j=1 2aje−aj|τ ||zj|2L(0) kj (2aj|τ ||zj| 2) =|τ | n−2 2απn Z 0 [Φα(z, τ ; s) − Φ−α(z, τ ; s)] ds where Φα(z, τ ; s) = eαs n Y j=1

2aje−ajs−aj|τ ||zj|2 X kj=0 (e−2ajs)kjL(0) kj (2aj|τ ||zj| 2).

Applying the generating formula for the Laguerre polynomials X k=0 L(p)k (x)zk = 1 (1 − z)p+1 exp ½ xz 1 − z ¾ to Φ(z, τ ; s), we obtain Φα(z, τ ; s) = eαs n Y j=1 2aje−ajs 1 − e−2ajsexp ½ −aj|τ ||zj|2 · 1 + 2e −2ajs 1 − e−2ajs ¸¾ = eαs  Yn j=1 aj sinh(ajs)   exp   −|τ | n X j=1 aj|zj|2coth(ajs)    9

(10)

This yields that (3.8) Ψα(z, τ ) =e |τ |n−2 απn Z 0 sinh(αs)  Yn j=1 aj sinh(ajs)   exp   −|τ | n X j=1 aj|zj|2coth(ajs)   ds.

3.2. The fundamental solution: the case n ≥ 2. To simplify the notation, we introduce the new function

γ(z, s) =

n X j=1

aj|zj|2coth(ajs). We next take the inverse Fourier transform with respect to τ and find

Ψα(z, t) = 1 2απn+1 Z −∞ |τ |n−2 Z 0 sinh(αs)  Yn j=1 aj sinh(ajs) eitτ −|τ |γ(z,s)dsdτ. Change the order of integrations, we obtain

Ψα(z, t) = Γ(n − 1) 2απn+1 Z 0 sinh(αs) hYn j=1 aj sinh(ajs) i ½ 1 [γ(z, s) − it]n−1 + 1 [γ(z, s) + it]n−1 ¾ ds.

For the second part of the integral, we substitute s by −s and note that sinh(−αs) = − sinh(αs) and

γ(z, −s) = −γ(z, s) since sinh and coth are odd functions; hence

Z 0 hYn j=1 aj sinh(ajs) i · sinh(αs) [γ(z, t) + it]n−1ds = Z −∞ 0 hYn j=1 aj sinh(−ajs) i · sinh(−αs) [−γ(z, t) + it]n−1(−1)ds = Z 0 −∞ (−1) (−1)n+n−1 hYn j=1 aj sinh(ajs) i · sinh(αs) [γ(z, t) − it]n−1ds = Z 0 −∞ hYn j=1 aj sinh(ajs) i · sinh(αs) [γ(z, t) − it]n−1ds. We can write Ψα(z, t) in a compact form

(3.9) Ψα(z, t) = Γ(n − 1) 2απn+1 Z −∞ hYn j=1 aj sinh(ajs) i sinh(αs) [γ(z, s) − it]n−1ds

As we mentioned before, we may rewrite the fundamental solution in terms of the modified complex action

g(s; z, t) and volume element ν(s) which were defined in (1.7): g(s; z, t) = γ(2s; z) − it = n X j=1 aj|zj|2coth(2ajs) − it and ν(s) = n Y j=1 2aj sinh(2ajs).

Substituting s by 2s, one obtains

Ψα(z, t) = Γ(n − 1) 2nαπn+1 Z −∞ hYn j=1 2aj sinh(2ajs) i sinh(2αs) [g(s; z, t)]n−1ds = Γ(n − 1) 2nαπn+1 Z −∞ ν(s) sinh(2αs) [g(s; z, t)]n−1 ds.

It seems impossible to find the exact formula for the above integral in general. We will consider the special case of aj= a for all j. In this case,

Ψα(z, t) = Γ(n − 1) 2απn+1 Z −∞ h a sinh(as) in sinh(αs)

[a|z|2coth(as) − it]n−1ds.

(11)

Differentiate with respect to t and we obtain ∂Ψα ∂t (z, t) = Γ(n)ian 2απn+1 Z −∞ sinh(αs)

[a|z|2cosh(as) − it sinh(as)]nds. Denote

ρ = (a2|z|4+ t2)1

4 and e−iφ= ρ−2(a|z|2− it) with φ ∈¡−π

22

¢

. Using the identity

cosh(s + iφ) = cosh(s) cos φ + i sinh(s) sin φ, we can write ∂Ψα ∂t (z, t) = Γ(n)ian 2απn+1ρ2n Z −∞ sinh(αs) [cosh(as − iφ)]nds. Next we apply the integral formula:

Z −∞ eαs coshν(as + b)dx = 2ν−1 aΓ(ν)e αb aΓ³ 1 2ν − α 2a ´ Γ³ 1 2ν + α 2a ´ Hence, Z −∞ sinh(αs) [cosh(as − iφ)]nds = 1 2 Z −∞ · eαs [cosh(as − iφ)]n e−αs [cosh(as − iφ)]n ¸ ds = 2 n−1Γ³n 2 −2aα ´ Γ³n 2 +2aα ´ 2aΓ(n) ³ e−iαaφ− eiαaφ ´ = 2 n−1Γ³n 2 −2aα ´ Γ ³ n 2 +2aα ´

aΓ(n) [−i sin( α aφ)].

This yields that

∂Ψα ∂t (z, t) = Γ(n)an 2απn+1ρ2n · 2n−1Γ³n 2−2aα ´ Γ ³ n 2 +2aα ´ aΓ(n) sin( α aφ) =2 n−2an−1Γ³n 2 −2aα ´ Γ ³ n 2 +2aα ´ απn+1ρ2n sin( α aφ).

Recall that ρ4= (a2|z|4+ t2)1/4 and ρ2e = a|z|2+ it. We can write Ψα(z, t) explicitly in term of z and t

by integration with respect to t by requiring that limt→−∞Ψα(z, t) = 0. In the case of a = α, we can find more explicit formula for the fundamental solution by noting that when a = α,

Ψα(z, t) = Γ(n − 1) 2πn+1 Z −∞ h a sinh(as) in−1 1

[a|z|2coth(as) − it]n−1ds =Γ(n − 1)a n−1 2πn+1 Z −∞ 1

[a|z|2cosh(as) − it sinh(as)]n−1ds =Γ(n − 1)a n−1 2πn+1ρ2n−2 Z −∞ 1 [cosh(as − iφ)]n−1ds =Γ(n − 1)a n−1 2πn+1ρ2n−2 · 2n−2hΓ(n−1 2 ) i2 aΓ(n − 1) = a n−22n−3hΓ(n−1 2 ) i2 πn+1 ³ a2|z|4+ t2 ´n−1 2 11

(12)

3.3. The fundamental solution: the case n = 1. In the case, (3.10) Ψα(z, τ ) =e 1 απ|τ | Z 0 sinh(αs) · a sinh(as) ¸

exp©−|τ |a|z|2coth(as)ªds.

We note if we take the inverse Fourier transform directly, the integral will diverge so the fundamental solution is not a regular function. We will give the fundamental solution in the form of generalized function. Recall the Fourier transform with respect to t and its inverse:

e f (z, τ ) = Z R e−itτf (z, τ ) and f (z, t) = 1 Z R eitτf (z, τ )dτ.e

We need to compute the inverse Fourier transform of e

f (z, τ ) = 1 |τ |e

−γ(z,s)|τ |.

The following computation will yeild the formula for the inverse Fourier transform. We apply the following formula for the inverse Fourier transform (see (33c) page 153 of [17]).

F−1( 1 |τ |) = − ε π− ln |t| π and F −1(e−γ|τ |) = γ π(γ2+ t2),

where ε is the Euler’s constant:

ε = Z 1 0 1 − cos y y dy − Z 1 cos y y dy.

Then the inverse Fourier transform of ef (z, τ ) is the convolution

h ε π− ln |t| π i h γ π(γ2+ t2) i = − γ π2 Z −∞ ε + ln |u| γ2+ (t − u)2du.

We will compute the integral first.

Z −∞ ε γ2+ (t − u)2du = Z −∞ ε γ2+ u2du = επ γ .

We apply the residue theorem to compute the integral

Z −∞ ln |u| γ2+ (t − u)2du = Z 0 ln u (u − t)2+ γ2du + Z 0 ln u (u + t)2+ γ2du.

We apply the residue theorem and find that (see formula 7.2.11 of [1]).

Z 0 ln u (u − t)2+ γ2du = − 1 2Re n³ Res |{z} z=t+iγ + Res|{z} z=t−iγ ´£ (ln z)2 (z − t)2+ γ2 ¤o = −1 2Re n ln2(t + iγ) 2iγ + ln2(t − iγ) −2iγ o = 1 4γRe{i[ln 2(t + iγ) − ln2(t − iγ)]}

Since 0 ≤ arg(z) < 2π, then for t > 0,

ln(t + iγ) = lnpt2+ γ2+ iθ and ln(t − iγ) = lnpt2+ γ2+ i(2π − θ),

where θ = arctan(γ/t) ∈ [0, π/2]. Hence

Z 0 ln u (u − t)2+ γ2du = π − θ ln(t 2+ γ2). Similarly for t < 0,

ln(t + iγ) = lnpt2+ γ2+ i(π − θ) and ln(t − iγ) = lnpt2+ γ2+ i(π + θ),

where θ = arctan(γ/|t|) ∈ [0, π/2]. Then

Z 0 ln u (u − t)2+ γ2du = θ ln(t 2+ γ2). 12

(13)

Similarly, we have Z 0 ln u (u + t)2+ γ2du = ( π−θ ln(t2+ γ2), t < 0 θ 4γln(t2+ γ2), t > 0 . Hence Z 0 ln u (u − t)2+ γ2du + Z 0 ln u (u + t)2+ γ2du = 2π − θ ln(t 2+ γ2),

where θ = arctan(γ/|t|). Summarize the calcluation, we have h ε π− ln |t| π i h γ π(γ2+ t2) i = − 1 π2 h επ + (π 2 θ 4) ln(t 2+ γ2)i.

Hence the fundamental solution Ψα(z, t) = − a απ3 Z 0 sinh(αs) sinh(as)γ(z, s) Z −∞ ε + ln |u| γ2(z, s) + (t − u)2duds = −a 2|z|2 απ3 Z 0 Z −∞ sinh(αs) cosh(as) sinh2(as) ε + ln |u|

a2|z|4coth2(as) + (t − u)2duds

= −a 2|z|2 απ3 Z 0 Z −∞

sinh(αs) cosh(as)(ε + ln |u|)

a2|z|4cosh2(as) + (t − u)2sinh2(as)duds

= − a απ3 Z 0 sinh(αs) sinh(as) h επ +³ π 2 θ 4 ´

ln[a2coth2(as)|z|4+ t2] i

ds.

4. The Heat Kernel of Paneitz operator In this section, we will compute the heat kernel hs(z, t) = exp{−sPα}δ0.

We first take the Fourier transform with respect to the t-variable and write the heat kernel ehs(z, t) as a twisted convolution operator.

e hs(z, τ ) = exp{−s ePα}eI = X |k|=0 exp{−s ePα}   n Y j=1 ajWfk(0)j ( ajzj, τ )   = X |k|=0 e−sτ2[Pnj=1aj(2kj+1)]2+sα2τ2 n Y j=1 ajWfk(0)j ( ajzj, τ ). Apply the Fourier integral formula:

e−sξ2 = 1

4πs

Z

−∞

e−x24s−ixξdx for ξ =Pnj=1(2kj+ 1)aj and sτ2 for s:

e−sτ2[Pnj=1(2kj+1)aj]2 = 1 4πsτ2 Z −∞ e−4sτ 2x2 −ix Pn j=1(2kj+1)ajdx. We can write ehs(z, τ ) = X |k|=0 esα2τ2 1 4πsτ2 h Z −∞ e−4sτ 2x2 −ix Pn j=1(2kj+1)ajdx iYn j=1 ajWfk(0)j (√ajzj, τ ) = e 2τ2 4πsτ2 Z −∞ e−4sτ 2x2 h X |k|=0 n Y j=1 aje−i(2kj+1)ajxWfk(0)j ( ajzj, τ ) i dx = esα 2τ2 4πsτ2 Z −∞ e−4sτ 2x2 hYn j=1 aj|τ | π e −aj|τ ||zj|2−iajx X kj=0 e−2ikjajxL(0) kj (2aj|τ ||zj| 2)idx 13

(14)

We can not sum up the series of Laguerre polynomials since |e−2iajx| = 1. We can insert a convergence factor e−²kj and let

gs,²(z, τ, x) =h n Y j=1 aj|τ | π e −aj|τ ||zj|2−iajx X kj=0 e−(²+2iajx)kjL(0) kj(2aj|τ ||zj| 2)i.

Then apply the generating formula, we obtain

gs,²(z, τ, x) = n Y j=1 aj |τ | π e −aj|τ ||zj|2−iajx 1 1 − e−(²+2iajx)exp n e−(²+2iajx) 1 − e−(²+2iajx)2aj|τ ||zj| 2o = |τ |n πn n Y j=1 aje−iajx 1 − e−(²+2iajx)exp n −1 + e−(²+2iajx) 1 − e−(²+2iajx)aj|τ ||zj| 2o = |τ | nen²/2 2nπn hYn j=1 aj sinh(² 2+ iajx) i expn− |τ | n X j=1 aj|zj|2coth(² 2+ iajx) o

Take the limit ² → 0+, we have

lim ²→0+gs,²(z, τ, x) = |τ |n 2nπn hYn j=1 aj sinh(iajx) i exp n − |τ | n X j=1 aj|zj|2coth(iajx) o .

Since sinh(ix) = i sin x and coth(ix) = −i cot x, we have e hs(z, τ ) = esα2τ2 4πs |τ |n−1 2nπn Z −∞ e− x2 4sτ 2 hYn j=1 aj i sin(ajx) i expni|τ | n X j=1 aj|zj|2cot(ajx) o dx.

Next we take the inverse Fourier transform with respect to τ and obtain the heat kernel associated with the Paneitz operator: hs(z, t) = (−i)n (2π)n+1√πs Z 0 τn−1eα22 cos(tτ ) Z −∞ hYn j=1 aj sin(ajx) i expn x2 4sτ2+iτ n X j=1 aj|zj|2cot(ajx) o dxdτ.

It seems impossible either find either of the integrals explicitly. In the special case of aj = a for all j = 1, 2, · · · , n, hs(z, t) = (−i) n (2π)n+1√πs Z 0 τn−1eα22 cos(tτ ) Z −∞ h a sin(ax) in exp n x 2 4sτ2 + iτ a|z| 2cot(ax)odxdτ.

The integration with respect to τ is the following

Z 0 τn−1eα22 x2 4sτ 2+iτ a|z| 2cot(ax) cos(tτ )dτ. It may be possible to write the above integral in terms of some special functions.

5. Projection and Relative fundamental solution Greiner and Stein [14] have shown that

(5.1) 1 2(eZjZje¯ + eZj¯ Zj) fe W (±p) k ( ajzj, τ ) = £ (2k + 1) + p(1 ± sgnτ )¤|τ |ajWfk(±p)( ajzj, τ ) for k, p = 0, 1, 2, · · · . This implies 1 2 n X j=1 (eZjZje¯ + eZj¯ Zj)e n Y j=1 f W(±pj) kj ( ajzj, τ ) = nXn j=1 £ (2kj+ 1) + pj(1 ± sgnτ )¤|τ |aj oYn j=1 f W(±pj) kj ( ajzj, τ ) and the Panietz operator Pα= LαL¯α satisfies

e ³Yn j=1 f W(±pj) kj ( ajzj, τ ) ´ = τ2 n³Xn j=1 £ (2kj+ 1) + pj(1 ± sgnτ )¤aj ´2 − α2 oYn j=1 f W(±pj) kj ( ajzj, τ ) 14

(15)

Hence the spectrum of the operator e is the set σ( ePα) =nλk,p= τ2 h³Xn j=1 £ (2kj+ 1) + (|pj| + pj· sgnτ ) ¤ aj ´2 − α2i, k ∈ Zn + and p ∈ Zn o .

The operator Pαis not invertable if α ∈ Λ with Λ = n ± n X j=1 £ (2kj+ 1) + (|pj| + pj· sgnτ ) ¤ aj, k ∈ Zn+ and p ∈ Zn o .

This implies that if αk,p= ±

n X j=1 £ (2kj+ 1) + (|pj| + pj· sgnτ ) ¤

aj for some k ∈ Zn+ and p ∈ Zn, then

e Pαk,p ³Yn j=1 f W(pj) kj ( ajzj, τ ) ´ = 0. Let e(+)= e for τ > 0 and ePα(−)= ePαfor τ < 0. Then for τ > 0,

αk,p= ± n X j=1 £ (2kj+ 1) + (|pj| + pj)¤aj = ( ±Pnj=1£2(kj+ pj) + 1¤aj pj ≥ 0 ±Pnj=1(2kj+ 1)aj pj ≤ 0 . And for τ < 0, αk,p= ± n X j=1 £ (2kj+ 1) + (|pj| − pj) ¤ aj= ( ±Pnj=1(2kj+ 1)aj pj ≥ 0 ±Pnj=1[2(kj− pj) + 1]aj pj ≤ 0 .

We summarize this result in the following proposition.

Proposition 5.1. (i) When τ > 0, ePαu = 0 has the following linearly independent set of L2-solutions nYn j=1 f W(pj) kj ( ajzj, τ ), k ∈ Zn+ and p ∈ Zn satisfy ± n X j=1 £ (2kj+ 1) + (|pj| + pj)¤aj= α o . (ii) Similarly, when τ < 0, ePαu = 0 has the following linearly independent set of L2-solutions

nYn j=1 f W(pj) kj ( ajzj, τ ), k ∈ Zn+ and p ∈ Zn satisfy ± n X j=1 £ (2kj+ 1) + (|pj| − pj)¤aj= α o .

Given α, the set that index L2-solutions of eP

αu = 0 is quite complicated and is given by Σα= n ± n X j=1 £ (2kj+ 1) + (|pj| + pjsgn(τ )) ¤ aj = α, k ∈ Zn+ and p ∈ Zn o .

In case of aj= 1 for all j’s, the set Σαis empty if n is odd and α is not an odd integer or n is even and α is not an even integer. In general this set can not be classfied by some simple rules.

We now consider the case of n = 1 and a1= 1. Then Σα is empty if α is not an odd integer. So we let

α = 2m + 1. Then for τ > 0, 2m + 1 = ±[(2k + 1) + (|p| + p)] = ( ±[2(k + p) + 1] p ≥ 0 ±(2k + 1) p < 0. Similarly for τ < 0, 2m + 1 = ±[(2k + 1) + (|p| − p)] = ( ±[2(k − p) + 1] p < 0 ±(2k + 1) p ≥ 0.

Let m ∈ N. Then we have

(i) τ > 0 and α = ±(2m + 1), the set of linearly independent L2-solutions of eP(+)

±(2m+1)u = 0 is n f W(−p) m (z, τ ), p = 0, 1, 2, · · · o n f Wk(m−k)(z, τ ), k = 0, 1, 2, · · · , m o . 15

(16)

(ii) τ < 0 and α = ±(2m + 1), the set of linearly independent L2-solutions of eP(−) ±(2m+1)u = 0 is n f W(p) m (z, τ ), p = 0, 1, 2, · · · o nWfk(k−m)(z, τ ), k = 0, 1, 2, · · · , mo.

The Laguerre matrix of ePα: M( ePα) = τ2¡£(2j − 1)2− αδk j ¢ is diagonal. If α = ±(2m + 1), we let f M2m+1= 1 τ2 X k6=m 1 (2j + 1)2− (2m + 1)2Wf (0) k (z, τ ). Then Lagueree calculus will yield

e

P2m+1Mf2m+1+ eSm= I, where eSm= fWm(z, τ ). We now sum up fM2m+1:

f M2m+1= 1 τ2 X k6=m 1 (2k + 1)2− (2m + 1)2Wf (0) k (z, τ ) = 1 τ2 · 2|τ | π e −|τ ||z|2 X k6=m L(0)k (2|τ ||z|2) 4(k − m)(k + m + 1) = e −|τ ||z|2 2|τ |π · 1 2m + 1 X k6=m ³ 1 k − m− 1 k + m + 1 ´ L(0)k (2|τ ||z|2) We first consider the case of m = 0. We need to sum up

F (ω) =X k=1 ³ 1 k− 1 k + 1 ´ L(0)k (2|τ ||z|2). Let ω = 2|τ ||z|2. Then F (ω) = X k=1 ³ 1 k− 1 k + 1 ´ L(0)k (ω) = X k=1 L(0)k (ω) k L(0)k (ω) k + 1 = X k=0 L(0)k+1(ω) k + 1 X k=1 L(0)k (ω) k + 1 = L(0)1 (ω) + X k=1 1 k + 1[L (0) k (ω) − L (0) k+1(ω)] We apply the formula

ωL(p+1)k (ω) = (k + p + 1)L(p)k (ω) − (k + 1)L(p)k+1(ω) with p = 0 and obtain

F (ω) = L(0)1 (ω) − ω X k=1 L(0)k (ω) (k + 1)2.

Apply the formula

1 (k + 1)2 =

Z

o

e−(k+1)ssds and apply the generating formula of the Laguerre polynomials:

F (ω) = L(0)1 (ω) − ω X k=1 L(0)k Z 0 e−s(k+1)sds = L(0)1 (ω) − ω Z 0 se−sh X k=1 L(0)k (ω)e−skids = L(0)1 (ω) − ω Z 0 se−s h 1 (1 − e−s)2exp n e −s 1 − e−sω o − L(1)0 (ω) i ds = L(0)1 (ω) + ωL(1)0 (ω) Z 0 se−sds − ω Z 0 se−s (1 − e−s)2exp n e −s 1 − e−sω o ds 16

(17)

Note that Z 0 se−sds = 1 and L(0)1 (ω) + ωL(1)0 (ω) = 1. We obtain F (ω) = 1 − ω Z 0 se−s (1 − e−s)2exp n e −s 1 − e−sω o ds. This yields f M1(z, τ ) = e −ω/2 2|τ |πF (ω) = e−ω/2 2|τ |π ω 2|τ |π Z 0 se−s (1 − e−s)2exp n −1 + e −s 1 − e−s ω 2 o ds = e −|τ ||z|2 2|τ |π |z|2 π Z 0 s (sinh s)2e −|τ ||z|2coth s ds

We take the inverse Fourier transform with respect to τ .

M1(z, t) = 1 Z −∞ eitτMf 1(z, τ )dτ = 1 (2π)2 Z −∞ |τ |−1eitτ −|z|2|τdτ −|z| 2 2 Z 0 s (sinh s)2 Z −∞ eitτ −|τ ||z|2coth sdτ ds.

The second part can be integrated with respect to τ :

Z −∞ eitτ −|τ ||z|2coth s dτ = 1 |z|2coth s − it+ 1 |z|2coth s + it= 2|z|2coth s |z|4coth2s + t2. Hence we have M1(z, t) = 1 (2π)2 Z −∞ |τ |−1eitτ −|z|2 dτ −|z| 4 π2 Z 0 s coth s |z|4cosh2s + t2sinh2sds.

We can compute the first integral by the convoluation. It is similar to the integral we computed when we derived the fundamental solution.

1 (2π)2 Z −∞ |τ |−1eitτ −|z|2|τdτ = 1 h ε π− ln |t| π i h |z|2 π(|z|4+ t2) i = − 1 3 h επ + (π 2 θ 4) ln(t 2+ |z|4)i,

where θ = arctan(|z|2/|t|). Summarize the computation, we have the relative fundamental solution:

(5.2) M1(z, t) = − 1 3 h επ + (π 2 arctan(|z|2/|t|) 4 ) ln(t 2+ |z|4)i|z|4 π2 Z 0 s coth s |z|4cosh2s + t2sinh2sds

The integral converges for z 6= 0.

References

[1] M. Ya. Antimirov, A. A. Kolyshkin and R. Vaillancourt, Complex Variables, Academis Press, San Diego • London • Boston • New York • Sydney • Tokyo • Toronto, 1997.

[2] R. Beals, B. Gaveau and P. C. Greiner, Complex Hamiltonian mechanics and parametrics for subelliptic Laplacians, I,II,III, Bull. Sci. Math., 121: 1-36, 97-149, 195-259(1997).

[3] R. Beals, B. Gaveau, P. Greiner and J. Vauthier, The Laguerre calculus on the Heisenberg group, II, Bull. Sci. Math., 110: 255-288(1986).

[4] R. Beals and P. C. Greiner, Calculus on Heisenberg Manifolds, Annals of Math. Studies #119, Princeton Univ. Press, Princeton, NJ, 1988.

[5] C. Berenstein, D.-C. Chang and J. Tie, Laguerre Calculus and its Application on the Heisenberg Group, AMS/IP series in advanced mathematics #22, International Press, Cambridge, Massachusetts, ISBN 0-8218-2761-8, 2001.

[6] O. Calin, D. C. Chang and P. C. Greiner, Geometric Analysis on the Heisenberg Group and Its Generalizations, AMS/IP series in Advanced Mathematics, #40, International Press, Cambridge, Massachusetts, ISBN-10: 0-8218-4319-2, 2007. [7] S.-C. Chang, J. Tie, and C.-T. Wu, Subgradient Estimate and Liouville-type Theorems for the CR Heat Equation on

Heisenberg groups, preprint, 2008.

[8] G. B. Folland, Harmonic Analysis in Phase Space, Annals of Math. Studies #122, Princeton Univ. Press, Princeton, NJ, 1989.

[9] G. B. Folland and E. M. Stein, Estimates for the ¯∂b complex and analysis on the Heisenberg group, Comm. Pure Appl. Math., 27: 429-522(1974).

(18)

[10] B. Gaveau, Principe de moindre action, propagation de la chaleur et estimees sous elliptiques sur certains groupes nilpotents, Acta Math., 139: 95-153(1977).

[11] D. Geller, Fourier analysis on the Heisenberg group, Proc. Natl. Acad. Sci. USA, 74: 1328-1331(1977).

[12] C. R. Graham and J. M. Lee, Smooth Solutions of Degenerate Laplacians on Strictly Pseudoconvex Domains, Duke Math. J., 57: 697-720(1988).

[13] P. C. Greiner, On the Laguerre calculus of left-invariant convolution operators on the Heisenberg group, Seminaire Goulaouic-Meyer-Schwartz, exp. XI: 1-39(1980-81).

[14] P. C. Greiner and E. M. Stein, On the solvability of some differential operators of type ¤b, Proc. International Conf., Cortona 1976–77, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 4: 106–165(1978).

[15] A. Hulanicki, The distribution of energy in the Brownian motion in the Gaussian field and analytic hypoellipticity of certain subelliptic operators on the Heisenberg group, Studia Math., 56: 165-173(1976).

[16] K. Hirachi, Scalar Pseudo-hermitian Invariants and the Szeg¨o Kernel on 3-dimensional CR Manifolds, Lecture Notes in Pure and Appl. Math., Marcel-DEkker, 143: 67-76, 1992.

[17] R. P. Kanwal, Generalized Functions: Theory and Applications, Third Edition, Birkh¨auser, Boston • Basel • Berlin, 2004. [18] H. Koch and F. Ricci, Spectral projections for the twisted Laplacian, Preprint, arXiv:math. AP/0412236, (2004). [19] J. M. Lee, Pseudo-Einstein Structure on CR Manifolds, Amer. J. Math., 110: 157-178(1988).

[20] W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, Springer-Verlag, Berlin • New York • Heidelberg, 1964.

[21] J. Peetre, The Weyl transform and Laguerre polynomials, Le Matematiche, 27: 301-323(1972).

[22] E.M. Stein, Harmonic Analysis - Real Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, New Jersey, 1993.

[23] S. Thangavelu, Lectures on Hermite and Laguerre Expansions, Mathematical Notes, #42, Princeton University Press, Princeton, New Jersey, 1993.

[24] S. Thangavelu, Harmonic Analysis on the Heisenberg Group, Progress in Mathematics, #159, Birkh¨auser, Boston-Basel-Berlin, 1998.

E-mail address: chang@georgetown.edu E-mail address: scchang@math.nthu.edu.tw E-mail address: jtie@math.uga.edu

Department of Mathematics, Georgetown University, Washington D.C. 20057, USA Department of Mathematics, National Tsing Hua University, Hsinchu, Taiwan 30013, ROC Department of Mathematics, University of Georgia, Athens, GA 30602-7403

參考文獻

相關文件

The students were either copying the notes on the board //or correcting the mistakes on the test(quiz), //but Peter and Jack weren't. The cat which/ that almost drowned

By correcting for the speed of individual test takers, it is possible to reveal systematic differences between the items in a test, which were modeled by item discrimination and

Monopolies in synchronous distributed systems (Peleg 1998; Peleg

Corollary 13.3. For, if C is simple and lies in D, the function f is analytic at each point interior to and on C; so we apply the Cauchy-Goursat theorem directly. On the other hand,

Corollary 13.3. For, if C is simple and lies in D, the function f is analytic at each point interior to and on C; so we apply the Cauchy-Goursat theorem directly. On the other hand,

This study will base on the perspective of the philological education to discuss 788 characters that were commonly used in the daily life of the early Tang era, for highlighting

In this chapter, the results for each research question based on the data analysis were presented and discussed, including (a) the selection criteria on evaluating

For Experimental Group 1 and Control Group 1, the learning environment was adaptive based on each student’s learning ability, and difficulty level of a new subject unit was