The heat kernel for generalized Heisenberg groups

The Journal of Geometric Analysis Volume 6, Number 2, 1996

The Heat Kernel for Generalized Heisenberg Groups

By Jennifer Randall

ABSTRACT. We obtain an explicit expression for the heat kernel h t on a generalized Heisenberg or H-type group N. We also show that ht has an analytic extension hz, z ~ C, Re(z) > 0, and hz is bounded in L P ( N ). We obtain estimates for Ilhzllp, 1 < p < ~ .

1. Introduction

Suppose G is a Lie group with corresponding Lie algebra g which is generated by the vector

fields X1 . . . . , Xa. By a well-known theorem of Hunt [12L the sub-Laplacian s = z j d = l X 2J is the infinitesimal generator of a strongly continuous semigroup { H,: t > 0} of operators on L p (G), the heat semigroup. Further, a theorem of Htirmander [10] tells us that s is hypo-elliptic, from which it follows that (see [6, 13]) Ht is a kernel operator, with kernel hi, known as the heat kernel.

The object of this paper is to study the heat kernel for a class of groups that generalizes the

Heisenberg group, namely H-type groups. This class was introduced by Kaplan [ 14], who suggests

that this is the largest class of groups for which an elementary expression for the fundamental solution of the sub-Laplacian exists. Many interesting groups are H-type groups, including the two-step nilpotent group that appears in the Iwasawa decomposition of a rank-one semisimple Lie group.

In Chapter 1 we find an expression for the Fourier transform in the central variables of the heat kernel for an H-type group. Cygan [4] has obtained a formula for the heat kernel for a general two-step, nilpotent, simply connected Lie group.

In order to obtain the heat kernel for an H-type group from Cygan's general formula, it is

necessary to relate the H- t ype group to a free nilpotent group.

Our method is more direct but applies only to H-type groups. We use the fact that the heat

kernel for the Heisenberg group can be regarded as the Radon transform in the central variables of

the heat kernel for an H-type group together with the known expression for the heat kernel for the

Heisenberg group obtained by Hulanicki [12] and Gaveau [9]. (Ricci [20] introduced this method

@ 1997 The Journal of Geometric Analysis ISSN 1050-6926

288 Jennifer Randall

and used it to find the solution for the sub-Laplacian for an H-type group.) In doing so, we also appeal a great deal to Folland's work [8] on stratified groups and the heat kernel.

Folland [8] has defined generalized Riesz and Bessel potentials and we use our formula for the

heat kernel to obtain expressions for these potentials for H-type groups.

JCrgensen [14] and Kisyriski [17] have shown that the heat semigroup can be extended to a

holomorphic family of operators {Hz: z E C, Re(z) > 0} on L P ( G ) , 1 < p < cx~.

In Section 2 we use our explicit expression for the heat kernel ht to show that, for H-type groups, it may be extended to a function h z which is analytic in the right half of the complex plane and which belongs to L p for p r [1, ~ ] . This is analogous to the Euclidean case where the analytic extension of the heat kernel,

1 h : ( x ) - e -Ix12/4z

(4yrz),/2 , x E R ' ,

is in L p for p E [l, ~ ] and Re(z) > 0.

We obtain L p estimates for hz, the most interesting of which is the estimate for the Ll-norm

II hz 1] 1. An unpublished lemma of M. Christ suggests the following estimate for any stratified nilpotent Lie group of homogeneous dimension Q:

Ilhzlll _< C

(cos(arg z))Q/2

for I arg zl < zr/2 and C a constant.

There is some interest in knowing whether the exponent Q / 2 in this inequality is optimal. Our

result shows it is not, at least for H-type groups. We prove that for an H-type group, homogeneous dimension Q = 2k + 2d, where k is the dimension of the center

C [ k+___23 for k odd Ilhz[]l < (cos(arg(z))a+ e , where e = I k2 + 2 for k even,

I arg(z) l < ; r /2 and C is a constant.

Although the estimate in the paper does not seem to be sharp, it suggests the possibility of

stronger results and should be useful in finding sharp estimates for L p multipliers on H-type groups (cf. [3, 18, 21, 23, 24]).

Since h z is bounded in L p, n z is an operator of type (p, q), p > q > 1, and estimates for the (p , q)-norms of Hz follow immediately from those for the LP-norms of h z.

This paper is based on my Ph.D. thesis and I wish to express my gratitude to my supervisor, Anthony Dooley. Many of the ideas in this paper were suggested by him; my work owes a great deal to his teaching.

The Heat Kernel for Generalized Heisenberg Groups 289

I also wish to thank the referee for helpful suggestions for simplifying proofs and improving

presentation. Lemma 2.4.7 is due to this and gives a sharper estimate for Ilhz 112 than originally obtained.

1. Heat kernel for generalized Heisenberg groups

1,1. Stratified groups and the heat kernel. This and the following section consist mainly

of background material. In this section we state a theorem of Folland's (1.1.1) concerning the heat kernel for stratified groups, and we evaluate an integral (1.1.2) which we use often in subsequent

sections.

Let G be a Lie group with Lie algebra g and identity element 0. We regard g as the space of

left-invariant vector fields on G and denote by d x a fixed right-invariant Haar measure on G. The

heat equation on G is given by

0

with u a function on G x (0, oo). Co is the closure with respect to the sup norm of the space of continuous functions on G that are constant outside a compact set. C 2 is the space of continuous

functions on G that are twice differentiable, and C g is the closure of C 2 fq~00 with respect to the norm

d d

Ilfl12 = I l f l l~ + ~ I IYj f l l~ + Y ~ IIY/Yjflloo, j = l i,j=l

where YL �9 "" Ya is a basis for g.

Hunt's theorem, mentioned in the introduction, tells us that the sub-Laplacian,/2, is the restric-

tion to C02 of the infinitesimal generator of a unique, strongly continuous, contractive, positivity- preserving semigroup of operators on Co. Further, Jcrgensen [14, pp. 121-122] shows that this

semigroup {Ht: t > 0} can be extended to a semigroup of operators on LP(G), 1 0} is called the heat semigroup.

From now on suppose G is a stratified group as in [7, pp. 163-172], the stratification of the Lie

algebra g being given by g = Vl G V2 ~ �9 �9 �9 (~ Vm. Let dilations on G be given by {Yr: r > 0}, where, forYj E Vj, j = 1,2 . . . . . m,

Y,(Yl + Y2 + "'" + Ym) = rY1 + r2Y2 + ' " + r"Ym.

r n Then the homogeneous dimension Q of G is given by Q = Z j = I j (d im Vj). We fix a basis

r X1 .. �9 Xr for V~; the sub-Laplacian is taken to be E = Y~j=l X~.

Dilations on G • ~ are taken to be {3~: r > 0}, where for (x, t) E G • R, 6,(x , t) =

(VrX, r2t). Thus G • ]1~ has homogeneous dimension Q + 2.


Theorem 1.1.1 [7, pp. 177-178]. The heat semigroup {H,: t > 0} isasemigroupofkernel operators on L P for 1 O, if f E L p or f E Co, there exists a unique function h E C ~ ( G • (0, oo)) such that H t f : ht * f , where h(x, t) = hi(x). Further, the function h has the following properties:

(a) ./a ht(x) dx : 1

(b) h(x. t) > 0

(c) h(x, t) = h(x -1, t).

The function ht is called the heat kernel.

Most of Folland's proof of the above theorem would apply for a general Lie group. Homogeneity of G is used to prove strong continuity of the heat semigroup. However, this also follows from J0rgensen's theorem mentioned above.

Lemma 1.1.2. For fl real and positive and n a positive integer or zero,

f~ [vl2ne_lVl2~d v = (7r) d F(n + d) ~, F(d) fl~+d

Proos

f~2 ]vl2"e-lVl2gdv = fs~._, fo~r2ne-rZ~r2d-l dr drl

F2n+2d - l e - r 2 ~ d r , = O)2d- 1

where O)2e-1 is the surface area of the unit sphere in R 2a, that is, O2d_ 1 = 2 ( ~ ) d / F ( d ) . Thus (see [5, p. 313 (15)]) since the integral is the Mellin transform of e -r2~,

f~ 2(Jr) d I 1 ~. Ivl2"e-el~ -- I '(d) 2 fl~+d l-'(n + d),

which is the required result. [ ]

1.2. H-type groups and the Radon transform. H-type groups can be related to the

Heisenberg group via the Radon transform. We introduce some notation in this section which we shall use for the remainder of this paper.

Definition [ 15]. Suppose n is a 2-step nilpotent Lie algebra with an inner product, denoted (,) . Let ~" be the center of n and _v the orthogonal complement of ~" in n. For v c v, set


f v = (ker ad~) f] _v, and denote by v~ the orthogonal complement of f o in v. Then if ado: v_v ~ is a surjective isometry for every unit vector v �9 v, n is called an H-type algebra or a generalized Heisenberg algebra. The simply connected Lie group N, associated with n__, is called an H-type or

Heisenberg group.

For the H-type algebra, n = _v �9 if, let dim(v) = 2d, d i m ( ( ) = k, and let d N denote a fixed, bi-invariant Haar measure on N. (See [7, p. 164]). N is a stratified group with dilations yr(v, ~) = (rv, r2() , for (v, ~') �9 v_ �9 ~', and homogeneous dimension Q = 2d + 2k.

We realize the Heisenberg group, Hd, of dimension 2d + 1 as the (2d + 1)-dimensional manifold C a x R. The corresponding Heisenberg algebra is denoted h__~. For j = 1 . . . . . d, set Xj = ej and Yj = iej, where ej is the j th standard basis vector in R d. The basis for ~ is taken as {Xl . . . . . Xa, Y1 . . . . . Ya, 1}. The Lie bracket is given by, [Xj, Ye] = 3e.j, [Xj, Xe] = [Yj, Ye] = 0 for g, j -- 1 . . . . . d. The exponential map is taken to be exp(~j__ 1 (otj Xj +/3j Yj) + • = •

The Heisenberg group is an H-type group when endowed with the scalar product

(v + t, v' + t') = - [ i v , v'] + tt', for (v, t), (v', t ') �9 C d x ]R.

A quotient algebra with one-dimensional center [19, p. 3]. Let r/be a unit vector in ~" and let ~(r/) be the hyperplane orthogonal to 1/in ft. Then we denote by n(r/) the quotient algebra n/s The corresponding simply connected Lie group is denoted N 0. For s E ~, (v, s) is the coset of v + s t / in if(r/). The quotient algebra n__(r/) is an H-type algebra, with inner product (,)0 given by ((v, s), (v', s')} o = (v, v') + ss', where v, v' �9 _v, s, s' �9 ll~, and (v, v') is the inner product in n. The Lie bracket in n(r/), [ , ]0, is given by [(v, s), (v', s')]~ = (0, (r/, [v, v'])).

L e m m a 1.2.1 [ 19, p. 4], [20, p. 269]. The quotient algebra n(r/) is isometrically isomorphic to the H-type algebra C d • R.

For f E L I (N) , the partial Radon transform of f with respect to the central variables is the

function on N o given by f , ( v , s) = fe(o) f ( v , s o + v) dv [10, p. 2], [19, p. 5].

Associated with a continuous function ~b on the set {No: r/ c s k-1 } is the function ~ on N

given by

- (2- 2 Js f , , r/, ())dr/.

1.3. The fundamental solution for the heat equation for an H-type group. The method we use to derive the fundamental solution was introduced by Ricci [19] and is based on the

following observations. Since N 0 can be identified with the Heisenberg group of dimension 2d + l,


the solution for the heat equation on N o is known [9, 12]. The heat kernel for N o is the Radon transform of the heat kernel for N.

We start with some preliminary notation and lemmas. The main result is in 1.3.4.

L e m m a 1.3.1. Suppose f ~ L 1 ( N ) is in the domain of • and that f o, the Radon transform o f f , is in the domain of s o, the sub-Laplacian in the enveloping algebra of N o. Suppose, for X E n, f and X f vanish at infinity. Then

Proof. This follows immediately from [19, p. 10, Lemma 8].

Theorem 1.3.2 [12, 9, p. 99]. The heat kernel for the Heisenberg group, h0((v, s), t) (or hto(v, s)), is given by

ho((v, s), t) - exp (2zr)d+22 a s~nh t~. 4tanh t)~

eiZS d)~

for ((v, s), t) C lid x (0, ~ ) .

L e m m a 1.3.3. The heat kernel h o is the partial Radon transform of

h( (v , ( ) , t) = 1 f0~ o-k/2 J~k/2-1)(o-I(I)( ~r )d { - [ v [ z a e x p 2d(27r) k/2+d I(] k/2-1 s-~nh to" 4 tanh ttr

for ((v, () , t) ~ N x (0, o~).

do-

(l)

Proof. 1.3.2. We will show that h is of the form (1).

Let h denote the Fourier transform of h in the central variables. That is,

h((v , p), t) --- f h( (v , ( ) , t)e i~r d ( .

Then, (see [10, p. 4])

Suppose h is the function on N x (0, c~) with partial Radon transform h 0 given in

h( (v, arl), t) = ho((v, o- ), t). (2)


Now

F ho((v, o'), t) = h , ( (v , s), t)e is~ ds. OG

(3)

Put

F(X) = exp . sinh tX 4 tanh t)~

Thus

h, ( (v , o'), t) - 1 1 f~fRF(kOeiS{~+O)dXds 2 d (27/') d+l

1

(4yr)~ F ( - a ) "

So, by (2),

h( (v ,o 'O) , t ) -- - - i ( o-

(47t') d sinh to"

_ _ ) d I _Ivl2O- e x p /4~an--n-~ 1"

(4)

Thus

h((v, ( ) , t) - 1 fR h((v , p), t)e-i{P'r (2yr) k ,

(5)

Now put r = I( I and ( = r r/.

Also put p = o'v, v a unit vector, o" = [PI:

1 h((v , o'v), t)o'k-le -i~ do" dr. h((v , ( ) , t) = (2yr) k ~ ,

Thus, by (4),

h((v , r), t) - - - 1 1 ~ (7 d

(2Zr )k+d2df s ,_ l f 0 ( ~ ) exp zlol=o- o-k- l e-iar(n.v) do-dr. 4 tanh to- /

Now (see [21, p. 1541) for X ~ ~ ,

__ ( 2 7 t ' ) k/2 j k / 2 _ l ( ~ , ) " fs~-' ei~(n'V)dv (~,)k/2-1

Thus

h((v , ;:;), t) - (2yr) k/z+d 2 d r k/2-1 ~ exp

! 4 tanh to- [ J(k/2-1)(ar) do',

J

which proves the lemma. [ ]

294

Lemma 1.3.4. distribution

Jenni fer Randal l

The fundamental solution for the heat operator s -- O/Ot on N • R is the

l h ( ( v , f ) , t ) fort > 0 k ( ( v , ( ) , t ) = /o fort < 0

where for ((v, (), t) E N x (0, oo),

h((v, () , t) - (27r)k/2+d 2d i(ik/Z_ 1J(k/2-1)(o'l(I) ~ exp 4tanhtcr

= h,(v, (). (I)

Proof. For fixed r/the partial Radon transform considered in 1.3.3 is a homomorphism of L I(N) into L I(N0). Denote the heat semigroup on N o by/ / to . Then, for f ~ L I(N), Htofo = ht0 * f0 is the Radon transform of Ht f = h t * f . Thus h t (v, ( ) = h ((v, ( ) , t) is the heat kernel on N • (0, oo) and the lemma follows from Folland [7, p. 178].

1.4. Complex powers of operators. Folland [7, pp. 179-185]) defines complex powers of the sub-Laplacian, s and the operator, 2--- s where 2- is the identity operator, on spaces of functions on stratified groups. He shows that under certain circumstances these powers may be expressed as kernel operators. We find explicit expressions for these kernels, the generalized Riesz and Bessel potentials, for an H-type group.

Proposition 1.4.1. The generalized Riesz potential for an H-type group, N, is given by

f0~ ( 4 ) a R~(v, ( ) = I(ll-k/2F(k + d - a/2) try/2_ 1 (4 tanhtr)k/2_~/2+ l

(Ivl4 q- l(1216tanh2 ~/2-k/4+l/Z-a/2 ol-k/2 ( lvl2 ) dcr ' p I k / 2 + d - c t / 2 ~/]Vl4 + 161(12 tanh2o "

for (v, ( ) E N, and 0 < Re(or) < Q where Q = 2d + 2k is the homogeneous dimension of N and Pff is the Legendre function of the first kind (see [5, p. 1201).

Proof. For ot E C, 0 < Re(t~) < Q, the generalized Riesz potential R~ is defined by (see [7, p. 184])

l fo Oc a 1 Ro(v, ~) - r ( ~ ) t~- h((v , ~), t) clt, for(v, () ~ N.


Using the expression for h in (1.3.4) gives

R~(v, ( ) -- f0 ~ 1 t~/2_1 1 1

F(2 ) (2~) k/2+d 2 d

fo e'~ T k/2 ( "g x 1(1~/2_ l J(k/2-1)(rl(I) s-~nh tr

d { _]vl2r

- - exp 4tanh t r d r dt.

Put t r = ~r and then p = 1/t to get

G ( v , ( ) = 1 i~l l-k/2 t

F(2 ) (27r) k/i+d 2 d

(s inha) d J(k/2-1)(crl~lp)exp 4tanh~r / dm

u Then (see [6, p. 185]) for 0 < Re(or) < Q and f in the domain of s

( - - s f = f * R~(x) E L p. []

Proposi t ion 1.4.2. The generalized Bessel potential is

l ~ ( v , ~ ) = 4 1 1 1

(47r) a+k/2 ~ F(~@ ) F(2) ~ ( y k + d - l r r / 2 ( ivl2~r ) �89

f0 f (sin O)k-2 icrlflc~ x (sinh ~r) d a0 4 tanh ~r

x Kk+d-,~/2 2 4 tanh cr

where K~ is the modified Bessel function of the third kind (see [5, p. 51).

Proof. For Re(a) > 0, J~(v, ( ) is defined as follows (see [7, p. 185]):

' f f J,~(v, ( ) -- F(2) t ~ - t e - t h ( ( v , ( ) , t ) d t , provided ( v , ( ) 5~ (0,0).

To calculate J~(v, ( ) we write

fo ~176 T k-l+d h((v , ( ) , t) = M (sinh t r ) d

- Iv l2 r exp 4tanh t r

- - ei~l~lc~176 (sin 0) k/2 dO d r

296

where

Jennifer Randall

M = 2 1

( 47r )d+k/2 ~/-~I~ ( L~- ) "

Now put t r = or. Also let

1~ ~. = crlffl cosO P -- 4 tanh ~

o . k + d - I

and q - (sinh o')d

(1)

Then,

J~ (v, ( ) -- F (~) t~/z-l-k-d e-t qeP/t+ix/t (sin O)k-2 dt dO da .

1 Now let -; = x. Then (see [6, p. 146 (29)]),

J~(v,r - M/0 f0 f0 F(~) (sin O) k/2 xk+d-a/2-1e -1/x q e -px+ixx d x dO dry

f0 fo F(~) q (sin 0)k-22

k+d-ot/2 2

Kk+d_,~/2(2v/-p -- i)~ ) dO dtr.

In view of (1), this is the required expression for J~(v, ( ) .

F o r f E L p, 1 < p < ~ , ( 5 - - f f . p ) -~ f ~ f �9 Ju (see [7, p. 185]).

.

The layout of this section is as follows. In Section 2.1 we obtain an expression for the analytic

continuation, h z, of the heat kernel ht for an H-type group. Sections 2.2 to 2.5 are devoted to finding

estimates for Ilhz lip, 1 0} which reduces to the heat semigroup on the positive real axis. Since, for f E L P ( N ) , g E N , and z E C, Re(z) > 0, the functions

Z --+ H z f ( g ) and z --+ (hz * f ) ( g ) are both holomorphic and coincide on the positive real axis, H z is a kernel operator on L P ( N ) , with kernel hz.


2.1. Analytic extension of the heat kernel.

297

Notation 2.1.1. For n, s positive integers, n > s, we write

fo ~ X n A(n, s) = (sinh x) ,~ dx.

We obtain an explicit expression for A(n, s) in the appendix.

2.1.2. We shall frequently use the following results.

Let w = rye i4a and suppose tr is real and positive and Iq~l y = tr sin ~b:

7r < g. P u t x = crcos~b and

(i) sinh x < I sinh 09[ < cosh x.

(ii) [sinh o9[ < sinh x - - COS q~ "

(iii) sinh x x. Thus

[sinh oo] < sinh x , /1 + sin2 ~b

- - V COS2 ~ "

[ ]

2.1.3. We often use the following upper bound for the Bessel function (see [1, p. 362]):

1 . l u l l lm(z)] 1

[J~(z)l < J~'* ~ for v ___ - - l-'(v -'k l) 2"

L e m m a 2.1.4. For (v, () ~ N, z ~ C, with Re(z) > 0, write

he(v, ( ) - - - 1 1 fo ~ - - (4:l") a -(2.Tt')k/2

exp - - d r . pk/2-1J(k/2-1)(rp) s-~nh zr 4tanh z r


Then Z --+ hz(v , ( ) is the analytic continuation of the function t --+ h t (v , ( ) , where ht is the heat

kernel on N.

Proof . The integral converges on compact subsets in the right half of the complex plane. We

see this as follows:

/ exp < - - p k--~-~-lJ~k/2-1)(rp) s inhz r 14 tanh z r - V(k/2)

.~d+k-I X

I sinh zrl a"

Thus for k > 1

fo CX~ .g k /2 .~ d [ pkj2, J, exp \s lnn z r / 4 tanh z r

21-k/2 < A ( k + d - 1, d). - F(k /2 ) (Re(z ) )~+d

d r

Thus the integral converges on compact subsets in the right half of the complex plane.

Hence the function z --+ hz(v , ( ) is analytic for Re(z) > 0. Since h z reduces to ht on the positive real axis, this proves the lemma. [ ]

L e m m a 2.1.5. For p < c < d the analytic continuation o f ht may be written

hz(v ,~) = (2Yrz)d+k/22 a Z pk/2-1J~k/2-1) ~ exp 4Z tanh ,k

Proof. The above expression is analytic for Re(z) > 0 and reduces to ht (v, ( ) on the positive real axis. This can be seen by putting t~r = ~ in the expression 1.3.4 (1) for ht (v , ~). The lemma follows by uniqueness of the analytic continuation.

L e m m a 2.1.6. For )~ E R, 09 E C, and Re(m) > O, m, n positive integers,

/ I (sinh x w ) n exp - - t a n h x w d x - (w)m+n+l (sinh x)" exp og--tanh x d x .

Proof . The proof is similar to that of 2.1.5; the result may also be obtained by a contour integral, [ ]


2 . 2 . L ~ n o r m o f hz.

299

L e m m a 2 . 2 . 1 .

2A(k + d - 1, d) < I l hz l l~ < (4Zr )k/2+dF (k /2)lzld+k --

where 0 = arg(z).

2A(k + d - 1, d)

(4:rr) ~/2+d F (~) [Z I a+k (cos 0)d+k

Proof . We have shown, in the proof of 2.1.4, that

2A(k + d - 1,d) Ihz(x, ()1 <

(4zr )a+~/2F ( ~ )(Iz I cos0) k+a'

from which the right-hand inequality follows immediately.

To prove the left-hand inequality we find Ih z (0, 0)1.

We write the expression for hz(v, ( ) in 2.1.4 as follows:

hz(v, () - - (27r)k/2+d 2d (rp)l-k/2J(k/2_l)(rp)rk-1 r --10121 - exp d r . s-~nh z r 4 tanh z r

Now (see [1, p. 360]),

2_~ lim co - v J~ (co) - -

~o--,0 F(v q- 1)'

for co ~ C, provided v is not a negative integer.

Thus

1 1 1 f0 ec "t "d+t-I hz(O, 0) = (2rr)t/2+d 2d+k/e_l F(k /2 ) (sinh zr) d dr.

Or, by 2.1.6,

2 1 1 fo ~ rd+k-I hz(O, 0) = (47r)k/2+d F(k /2) (Z) d+~ (sinh r ) --------~ d r .

Now II h z II o0 ~ I hz (0, 0) I. Hence, we have the result. [ ]

For s a real positive number, Strichartz [23] has considered the kernel h-is of the operator e -isc on the Heisenberg group, Hd. He shows that, as opposed to the Euclidean case, there are no global estimates for e -isc. However, for (v, r) c lid, the kernel h_i,(v, r) is bounded and C a for

I r l < ds.

300

2.3. The L 2 n o r m o f hz.

Jennifer Randall

L e m m a 2.3.1. The L 2 norm Ilhzll2 is given by

A (k + d - l , d) IIhr I1~ = (7r)d_k/eF(~)23d+k_ 1 (Izl cos O) d+k' where arg(z) = 0.

Proof . As in 1.3.3, hz(v, v) denotes the Fourier transform of hz(v, ( ) with respect to the

central variables. By the same reasoning used in 1.3.4, it can be seen that the analytic extension of

the heat kernel of the Heisenberg group is the partial Radon transform of h z (v, ( ) , which we denote

hz, o(v, t), where q is a unit vector in ( .

Now

hz(V, r 0 ) = hz,o(v, r )

(4zr) a sinh z r 4 tanh z r

The last step follows by the same argument as that used in 1.3.3 to obtain ht(v, crrl).

By the Plancherel theorem for R k,

,hz, = ,hz,

= s 1 6 3 thz(V,P)12dpdv

= f ~ = ~ f s , , f o ~ l h z ( v , r ~ ) 1 2 r k - t d r d g d v ,

where S k-I is the unit sphere in ]R k and we have put p = r~' , with IP[ = r .

W e substitute for Ihz(v, rff)l from (1) and then integrate over IR ea and S k-t to get

2 fo ~ r d+k- I Ilhzll~ -- F(~)(aJr) a (sinh 2x) d d r

where x = r Izl cos 0. Recalling the notation 2.1.1, the result follows immediately. [ ]

(1)

(2)

R e m a r k . Compare the above expression for tlhz ll2 with the value of the L2-norm of the analytic continuation of the heat kernel on ]K 2d, viz. Ilhz tl2 = 1/(8rr tzl cos O) d. Note that A(d - 1, d) does not converge. [ ]

The Heat Kernel for Generalized Heisenberg Groups 30 ]

2.4. T h e L ! n o r m of h z. The main result of this paper is the following estimate for II hz II1

Proposition 2.4.1.

Ilhzlll

The L l-norm for h~, Ilhz t11, satisfies the following estimate:

/ k+3 for k odd C where ~ -~- I k 2

I + 2 for k even (cos(arg z)) d+e

and C is a constant depending on d and k.

2.4.2. To derive the estimate for Ilhz lit we split I[hz II1 into the sum of two integrals, I and J . We then obtain the estimates

(see 2.4.7) and

(see 2.4.20) where

C I <

- (cos(arg z)) d

K J <

- (cos(a~z))d+e

/ k+2 for k odd ~ = [ k 2

+ 2 for k even.

Combining these gives the estimate for Ilhz II1. In carrying out the above calculations we need the following lemmas.

L e m m a 2.4.3. Forot realandpositive fo Jo(x) dx < 1.5.

Proof. Suppose Yn is the nth positive zero of J0, and let

foYl fYn+l So = Jo(x ) dx and S, = Jo(x ) dx. JYn

Then (see [5, p. 60]) the series Y'~,~o Sn is an alternating series with monotone decreasing terms.

Also Jo(0) = 1. Thus, for ot > 0,

Yo f? Jo(x) dx < Jo(x) dx.

Now (see [1, p. 409, p. 492]) Yi "~ 2.4 and thus f~' Jo(x) dx < 1.5. []

302

L e m m a 2.4.4. k , k <2d.

Jennifer Randall

For the H-type algebra n_n_ = v_ + ( with dim(v) = 2d, and d i m ( ( ) =

Proof . For the linear map adv" _v ~ ( ,

dim(ker ad~) q-d im(g) = dim(v) .

Since ker ad~ contains v at least

dim(g) < dim(y_).

S o m e expressions for IIh~ I) 1.

L e m m a 2 . 4 . 5 . Ilhz)ll : Ilhei~

Proof . By homogeneity, Ilhrzlll = Ilhzlll. [ ]

7r From now on we consider only Z = e i~ 101 < 7"

2.4.6. We can write Ilhzlll = I + J where

2(Tr)k/2 �89

' - LYo Ihz(v, ()[pk-ldp dv

and

2(n)k /2 ]hz(v, ( )lpk-l dp dr,

with p = I gl. We see this as follows:

Since hz(v, g) depends only on p,

Ilhzlll = Wk-l 2~ Ih~(v, ()lpk-ldp + Ihz(v , ()[pk-ldP dv ,

where Wk-I - - r(k) �9


Estimate for I.

c Lemma 2.4.7. I < ~ where C is a constant depending on k and d.

Proof.

2 ,k k-1 ~ d [hz(v, ()[ < (4zr)d+k/2F(k/2) ~ exp

-< Aexp{ -Ivlzc~ 1 4 '

where A is a constant.

Thus, in view of 2.4.6,

zs0 / cos0/ I < A' ~ exp 4 Pk-idpdv =

where A' and C are constants depending on d and k.

Thus the lemma is proved. [ ]

1 It follows from the expression for hz (v, ( ) in 2.1.5 that, with I z I = I and p < ~ < d,

[v[~. cos 0 d~.

4 tanh

(cos O) d

Derivation of the estimate for J . We have divided the derivation of this estimate into

three parts. In the first part, preliminaries, we introduce some notation and prove lemmas that we use

later. The second part consists of the derivation of an estimate for I MI defined in 2.4.10. In the final

section we use the preceding results to obtain the estimate

A [ k+3 for k odd J < withe = /,- -5-

-- (c~ d+e [ ~ ' 7 + 2 for k even.

Preliminaries.

and

2.4.8. Put

/7 17

P(/7) ---- sinh Z/7' q(/7) -- tanh z/7

[-Ivl2 ) q$~(z) -- (p(r)) d exp ~-- -~q(z) .

304 Jennifer Randall 2.4.9. Denote by ePj(r) the coefficient of (_L~)J exp(_ I_~q(r)) in ~b~e)(r), where ~b~ e)

is the gth derivative of ~b~.

2.4.10.

Then (see 2.4.7 and 2.1.4)

Let M : f o ( p Z ) k / 2 J < t / 2 _ l ) ( p r ) g p ~ ( r ) d r .

j _ - - 21-k/2+2d fR f~176 F(k/2) ~ [Mt d p d r .

2.4.11. For v, ~, C I~ and v + X a positive integer, we put

The recursion relation for J~ (see [5, p. 11]), becomes

J~(u) = (-1)~+~u~f~,~(u).

Lemma 2.4.12. For n an integer greater than one and v E IR and v -- 1 - - m + )~ > O,

f " f u"L,~(u)au = ~__oeJ-'-2~L_,_~,~(u) + ~ U"->2mL_I_m,~(U) clu, j=O

where 2 m < n - 2 and the otj ' s and fl are constants depending on n and m.

Proof. In view of 2.4.11,

d f u"L~(u)du = f u"-lyuL-l,~(u)du

= f ( . 1)un-2fv_l,X(u)du.

Repeated use of the above equality proves the lemma. []

Lemma 2.4.13. For s a positive integer, s > 2,

d P, sP0 = (p~)r = ~ ( ~ - t 0),

.~P~ = ( q ' ) ( , - l P ~ - J ) = (q , )S (pa )


and, for s > 1,

sPj = ~ r ( ~ - l P i ) + ( q ' ) ( , - , P j - 1 ) , j = 1 , 2 , . . - s - 1.

The proof is done easily by induction.

Lemma 2.4.14. For s > O, 0 <_ j <_ s e P j ( r ) is a linear combination o f the terms a ( s m , n ) ~- rm(sinh zr)n-lc~ zr

tsinh zT~+J+~ ,where m + n < d + j + s 0 < m < d + j a n d l < n < j + s and b(s s, t) - - r S ( s i n h z r ) ' ~sinhz~)d+J +e'where s q- t < d + j + s O < s < d q- j , and O < t < j q- s

P r o o f , Clearly (see 2.4.8) the lemma is true for ~ = 0, 1 and 2. Now assume the lemma is

true for s = r > 0. Then, by 2.4.13, the lemma is true for s = r + 1. Hence the lemma is proved. []

Corollary 2.4.15. For n a positive integer, g~ > O, "rn~b~e)(r) tends to zero as r tends to

infinity. We appeal to this corollary frequently in obtaining an expression for M.

Lemma 2.4.16.

where C is a constant.

For 0 < z < 1 and ~ a positive integer or zero

l e e j ( r ) l _< C (1)

Proof. Let a and b be functions of a complex variable, o9, given by

O9 O9 a ( w ) - - and b ( w ) - -

sinh o9 tanh o9

Then a (09) and b(og) and all their derivatives are analytic for Io91 < 7r/2. Now p ( r ) = l a ( r z )

and q ( r ) = ! b ( r z ) .

Thus

d n P ( r ) - p(n)( r ) = z~-Ja(n)(rz) , d r n

and

q<n)(r) = zn - lb (m(r z ) .


Now since a(n)( rz) and b(n)(rz) are analytic for Irzl < 7r/2, they will be bounded for 0 < [rz[ < 1. Thus for Iz[ = 1 and 0 < r < 1 p(n)(r ) and q ( m ( r ) are bounded. The required

result follows in view of 2.4.13.

E s t i m a t e s f o r I MI , We show that

IMI < ~5 f ( r , v ) d r

where C is a constant and f o F ( r , v)dr is an integrable function of v.

Notation 2.4.17.

Put

It is convenient to introduce the following notation:

cos u for t even 0t(u) = is in u f o r t o d d ,

and, for constant real vectors ot = (or0 . . . . . Olk/2_s_l[2) and fl = (rio . . . . . fl.~-l), depending on k and integer s, set

k - l - 2 j - s sRk , - 2(u) --ps [ j=o O t j U fk/2-l-j-s'l/2(U) + ~-'flJu'*-a-lrlJ+l(u)j=o f o r s > 0

and

oRk, �89 (U) = U k-1 A/2-1 , �89 (U).

Where possible, we will suppress the superscripts or,/3 in .~ R~, ~2, for typographical reasons.

Lemma 2.4.18. For k an odd positive integer,

k-1

fo IMI < j=0 ~Olj r k/2-1/2-j (*+~' dr. (1)

For k an even positive integer,

k/2- I ~ k/2 oo

Iml < j~o.= p j+2 yj fo f0 _

where the Ogj 'S, y j 'S and (Tj 'S are constants depending on k.

(2)


Proof . We consider the case k odd in detail. We first derive the following result: for k odd, s an integer such that 0 < s < k/2 1

2 '

IMI = f ~ sRk,�89 (u)~b~S)(r)dr (3)

where u = pz.

We prove (3) by induction on s. For s = 0 (3) follows from 2.4.10 and 2.4.11 with L = ! 2 '

Assume

Iml = n / / n) .g "t" , k 1

for 0 < n < - - - . (4) 2 2

Now

f i f 1 .Rk,�89 = - .Rk,�89 = pU~[ll(U) + I2(u)] (5) P

where, for n > 0

k /2 -n - �89 f U k - l - n - 2 j .; , . . . . l l ( u ) = ~ otj Jk/z_r_i-n,�89

and

n - I

Iz(u) = E ~ j f j = 0

and, for n = 0,

11(u)= f ,~ ' fk/2_l,�89 12(u) = O.

Note that, for r and ~ positive integers or zero,

f z ur~s du = yjur-Jrle+l+j(u) j = 0

for constants yj.


An application of 2.4.12 gives

Ii(u) + 12(u) =

k/2-(n+l)-�89 E ,~, uk-l-2j-(n+l) re gj dk/Z-l-j-(n+l),�89 (lg) j=O n

+ ~ crj'-J rb+l (u), j=O

where the yj's and crj's are constants depending on k and n.

Thus (5) implies

f a,/~ y,a . RT, 7 (pz )dv = .+1 R~ 7(pz) .

Integration by parts in (4) gives

IMI = n+lRk, l(pT)~l):n)('C) : - fo~~ l(pT)~t):n+l)(.~)dT~ .

For n odd, r = 0 since r is an even function. For n even ,+iR~.�89 (0) = 0. Thus (3) is true

for all integers s such that 0 < s < k/2 - I /2 . Now put s = k/2 - 1/2 in (3) to get

1 I M I - pkl2-il2 f0 ~ d r

- - flj Uk/2-1/2-joj(U)r ) , j=O

where the flj's are constants, depending on k.

Two further integrations by parts yield the required estimate, (1), for M. The derivation of the bound, (2), for M for k even is similar. The chief difference is that, for k odd the method which we used enabled us to reduce the Bessel functions in our integrals to J_ �89 an elementary function. For k

an even integer we cannot do this but we express the integrand in terms of J0 and then use the bound 2.4.3. We omit the details. [ ]

E s t i m a t e f o r J . We shall carry out the integration with respect to p and v in the expression

for J in 2.4.10, using the estimates for [M 1 in 2.4.18. The estimate is given in Lemma 2.4.20. It is convenient to prove the following result first.

L e m m a 2.4.19. For 0 < j < s and 2 < f~ < k /2 + 2, set

fo' ItPj('c)[ fl ~176 [ePj(T) i .ct_2 d.~ tAj = dr and eBj = (~(r))d+j (/~ (-~))d+j


where f l ( r ) = R e ( ~ ) . Thus

r cosh x sinh x f l ( z ) : 4[ sinh 2 Zr[ where x : r cos 0.

Then

(i) e Aj < cj where is constant, and - - (cos0) a+e' Cj a

(ii) e Bj < dj where dj is a constant. -- (cosO)d+e,

Proof of (i). Using 2.4.16 and 2.1.2 (ii) and putting x ----- r cos0 we see that

c :coso(s x). eAJ <- (cos 0 ) d+j+l dO - - d x .

Now, fo r0 < x < 1, 1 < sinhx < 1.2. Thus

cj cj

eAj < (COS0)d+ j < (COs0)d+ e,

and (i) is true.

and

Then

Proof of (ii). We use Lemma 2.4.14 and the notation introduced there. Put

f ( s r, m) = f ~ re-21a(s r, m)l d r (f(r))"+J J1

g(s s, t) = f / ~ re-21b(e ' s, t)l d r .

J l (t~(r))~+J

~nj ~ E Olr'mf(~" r, m) -b fls:g(s s, t) (1) r,m,s,t

w h e r e Olr, m and fls,t are constants and the sum is taken over permissible values of r, m, s and t. We

obtain estimates for f ( s r, m) and g(s s, t).

Estimates for f ig, r, m) and g(s s, t). We prove that

A f(s r, m) <

(cos 0) ~+e'


where A is a constant�9 We first consider the case when m > s which ends in (6)�9 The case m < /~

is given in (7). Then (8) gives the corresponding result for g(s s, t).

By 2.4.14

fOG f (g, r, m) = r ~-d-j+e-2 [ sinh Z.~[d+j+m-e-l[ cosh

(sinh x)d+J(cosh x) a+j

Suppose m >/~. Then, using 2.1.2, we get

f ( s r, m) < 1 fo ~ X r-d-j+e-2

(COS O) r-d-j+m-1 (sinh X) d+j+g-m

d r . (2)

dx. (3)

Now, by 2.4.14, d + j + s -- m > d > 1, thus (3) implies

1 l ~ xr+m-2d-2j-I dx.

f ( s r, m) < (cosO)r_d_j+m_ 1 sO sinh x (4)

We have, by Lemma 2.4.4, that d >

r + m < d + j + L Thus

k 5" Recall also that 2 < s k < 5 + 2 and (see 2.4.14) that

2 d + 2 j - ( r + m ) + l > d + j - g . + l > j - 1 . (5)

X 3 Suppose j = 0 or 1. It follows from (4) that, since x + - - < sinh x,

3 ! -

l 6 f ( s r, m) < (COSO)r+m_(d+j)_ 1 sO X 2d+2j-(r+m)+3 dx.

CI Cl That is, f ( s r, m) < (cos gY~+~+, < ~ where Cl is a constant. Thus, since s > 2, f ( s r, m) < CI C2 (co~)~+e Also for j > 1, (4) gives f ( & r, m) < �9 -- (cos'-ff?+J "

Thus for m > s

f ( s r, m) < (6) (cos0)d+ e

for c a constant.

For m </~, a similar argument gives

f ( s r, m) < b

(cos 0) d+e' (7)

where b is a constant.


An analagous proof shows that

g(e, s, t) < (cos O) d+e '

where e is a constant.

The required estimate follows from (1), (6), (7), and (8).

We now obtain the estimate for J

(8)

Lemma 2.4.20.

K [ ~+__23 for k odd J < (cos O) d+e where ~ = | k 2 - ~ + 2 for k even

and K is a constant depending on d and k.

Proof.

Set

We first establish the following result.

H&n = - - r n p2+m d v d p d v ,

where ~, n, and m are positive integers or zero. Then

~=o fo ~ ]ePJ(r)]vn d r (1) He,. < c (f l(r))d+j

where c is a constant depending on d, g, and m, where f l ( r ) is given in Lemma 2.4.19.

To prove (1), carry out the integration with respect to p in the expression for He,. to get

He,. <_ (2 + m) ~ r n d r do. (2)

Now,

I~b~e)(z)] < ~ l e p j ( r ) ] - T e x p - 13(v) .

Again changing the order of integration in the right-hand side of (2) and using Lemma 1.1.2, we get

e f0 ~ c2 He,n ~ j~=o T n I, pj(v)l ~-a77dr,


where the cj 's are constants, depending on d and m. If we let c be the maximum of the cj's, the result (1) follows.

Now using the estimates for IMt in 2.4.18 in the expression for J in 2.4.10, we get, in view of

(1), for k odd

~+~ {f0 l J_< ~ - ~ c t=O

and for k even

~ e , ( r ) 1 (r d+'

fl ~176 T k/2-1 } d r + ~ Pt(r) (~(7~))d+t d r ,

J _< c, ~ l~/2+2P,(r) I (~(r))d+ ' d r + Ik/2+2P,(r) I t=0

k/2 j + 2 o0

,=0 ((fl(r))d+t d r ,

T'k/2 ] (/3 (r))d+, d r

where c, Cl, and c2 are constants, depending on d and k.

An application of 2.4.19 gives the desired estimate. Proposition 2.4. l follows from 2.4.6, 2.4.7, and 2.4.20. [ ]

Remark. The L 1 norm for the analytic continuation of the heat kernel o n ]]~2d is given by

1 Ilhzlll - (cos 0) d"

This is consistent with our results.

2.5. Estimate for Ilhzllp, 1 2. As usual hz is the analytic continuation of the heat kernel on an H-type group N of homogeneous dimension Q = 2d + 2k,

where k is the dimension of the center of N; p' is the conjugate exponent of p, - 7 r / 2 < 0 = arg < 7r/2 and C and D are constants.

Forl 2,


and

C Ilhzllp S (cos O)(2/p-l)(cos O)Q/2P[z[ Q/2p'

i fk < 2 .

For 2 < p < c~,

D ]]hzllp <

- ( I z l cosO)O/=p'"

Remark. If hz is the heat kemel on R n, for 1 < p < oo,

Ilhzllp = \ ~ / \ 4 z r l z l / '

where Q -- n is the homogeneous dimension of ]I{ n.

Lemma 2.5.2: p - q Norms for H z. Suppose {Hz: z ~ C, z # 0, [argz[ < 2} is the

analytic extension of the heat diffusion semigroup. Then, for each Z E C with Re(z) > 0, and for

p >_ q > 1, H z i s a n o p e r a t o r o f t y p e ( p , q ) with ( p , q ) normsatisfying IIHzll~pq) <_ IIhzllr, where r - qP qp+p_q "

This follows from Young's inequality.

Mel l in transform of cosechPx .

Appendix

Lemma A.1. Reduction formula for

fo ~176 X n In p : - - d x . ' sinh p x

Suppose n, p are positive integers, n > p > 2 and let m be an integer such that 1 < m < ~ - 1

for p even, and 1 < m < ~_! for p odd.

Write

ln_2s,p-2m = I s ( m ) (1)

ae = (p -- s (2)

314

Then

Jennifer Randall

1 s Im_s(m)b,(m) I,,p -- l__i~m ~ x=o

(3)

where b~ (m) is defined as follows.

Let Qm = {1,2 . . . . . m}, and.for s < m, denote a subset of Q,, containing s elements by Qm,s. Then

2 m - l

bo(m) = 1-I (n - j) j=o

= (2m~-I - - j ) ) (O.,.,g~, ( 1-I. a2g)) b.(m) ( - 1 ) ' (n \esQ.,.

forl <s <m--1 m

bin(m) = ( - 1 ) m l - Ia2g e= l

(4)

P roo f . The proof is by induction on m.

f 1 dx - cosh x sinh p x (p -- 1) sinh p-I x

(p - 2) [ dx p - 1 J s i n h p - 2 x "

Thus

In, p fo ~ X n d 7xx [ (p - cosh x -i Gff-, x ] dx (p -- 2) [ ~ x"

( p - - 1) J0 sinh p-2 x dx

- -x n cosh x ~ f ~ n x n - 1 cosh x

o

+ Jo (p • i'-) s-~nh p---I x (p -- 1) sinh p-I x

(p -- 2) d x I In,p_2

(p -- 1)

-nx "-l l o f o ~ n ( n - l ) x n-2 (p - 1)(p - 2) sinht'-z x + (p - l ) ( p - 2) sinhp-2 x

dx ( p - 2 ) . (p -- 1) ln'p-2"

It follows that

/ n , p ( p - - 1 ) ( p - - 2 )

{n(n - 1 ) I n _ 2 , p _ 2 - - (p - 2 ) 2 I n , p _ 2 } .

This is applied recursively to give the desired result. [ ]

Corol lary A.2.

(i) For p odd,


Explicit expression for In,p.

315

. , p -

p I

1 ~?-]bs(P~2 1 ) ~ ( 4 - 2 P - n - Z s ) ( n + 2 s - p + l ) ! ~ ( n + 2 s - p + 2 ). ( p - 1)! ,=0

(ii) For p even,

s

1 ~obs - 1 2p-n-Z~-l(n+2s - p + 2 ) ! ~ ( n + 2 s - p + 2 ) l . , p - - ( P - 11! :

where ~ is Riemann's zeta function (see [5], Vol. 1, p. 32).

References

[1] Abramowitz, M., and Stegun, L. A. (eds.), Handbook of Mathematical Functions. National Bureau of Standards, Washington, D.C., 1964.

[2] Balakrishnan, A. V. Applied Functional Analysis. Springer-Verlag, Berlin, 1966. [3] Cowling, M. Harmonic analysis on semigroups. Ann. Math. 117, 267-283 (1983). [4] Cygan, J. Heat kernels for class 2 nilpotent groups. Studia Math. 64, 227-238 (1979). [5] Erd61yi, et al. (eds.) Higher Transcendental Functions, vols. 1 and 2. McGraw-Hill, New York, 1953. [6] Erd61yi, et al. (eds.) Tables of Integral Transforms, vols. 1 and 2. McGraw-Hill, New York, 1954. [7] Folland, G. B. Lectures on Partial Differential Equations. Springer-Verlag, Berlin, 1983. [8] Folland, G. B. Subelliptic estimates and function spaces on nilpotent Lie groups. Ark. Math. 13, 161-207 (1975). [9] Gaveau, B. Principe de moindre action, propagation de la chaleur et estim6es sous elliptiques sur certains groupes

nilpotents. Acta Math. 139, 95-153 (1977). [10] Helgason, S. The Radon Transform. Birkh~user, Boston, 1980. [11] H6rmander, L. Hypoelliptic second order differential equations. Acta Math. 119, 147-171 (1967). [12] Hulanicki, A. 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). [13] Hunt, G. A. Semigroups of measures on a Lie group. Trans. Amer. Math. Soc. 81,264-293 (1956). [14] JCrgenson, P. Representations of differential operators on a Lie group. Z Funct. Anal 20, 105-135 (1975). [ 15] Kaplan, A. Fundamental solutions for a class of hypoelliptic PDE generated by composition of quadratic forms. Trans.

Amer. Math. Soc. 258, 147-153 (1980). [16] Kaplan, A., and Ricci, E Harmonic analysis on groups of Heisenberg type. In Leeture Notes in Mathematics, 992,

Springer, Berlin, 416~35 (1983). [ 17] Kisyfiski, J. Holomorphicity of semigroups of operators generated by sub-Laplacians on Lie groups. In Lecture Notes

in Mathematics, 561,283-297 (1976). [18] Mclntosh, A. Operators which have an H~ functional calculus. Proc. Centre Math. Anal 14, 210-231 (1986). [19] Ricci, E Harmonic analysis on generalized Heisenberg groups. Preprint. [20] Ricci, E Commutative algebras of invariant functions on groups of Heisenberg type. J. London Math. Soc. 32(2),

265-271 (1985). [21] Stein, E. M. Topics in Harmonic analysis. Ann. of Math. Studies (63), Princeton University Press, Princeton, NJ,

1970.

316

[22]

[23] [24]

Jennifer Randall

Stein, E. M., and Weiss, G. Introduction to FourierAnalysis on Euclidean Spaces. Princeton University Press, Princeton, NJ, 1975. Strichartz, R. S. L J' Harmonic analysis and Radon Transforms on the Heisenberg Group. Preprint. Varopoulos, N. Th. Hardy-Littlewood theory for semigroups. J. Funct. Anal. 63, 240-260 (1985).

Received June 8, 1992

University of New South Wales, Kensington, N.S.W. 2033, Australia

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The heat kernel for generalized Heisenberg groups