THE BEAT EQUATION ON THE SYMMETRIC
SPACE ASSOCIATED WITH SL(n,R)
Patrice Sawyer
Department of Mathematics and Statistici
McGill University, Montreal
September 1989
A Thesis submitted to the Faculty of Graduate Studies and
Research in partial fulfillment of the requirements
for the degree of Doctor of Philosophy.
@Patrice Sawyer, 1989
(
Contents
Abstract
4
Résume
5
Acknowledgements
6
Symbols frequently used
T
Introduction
10
1
The adjoint of the Abel transCorm
21
1.1 Definition of the adjoint of the Abel transforme
21
1.2 The adjoint of the Abel transform for pose n, F)
24
The False Abel Inverse TransCorm
34
2
(
(~
2.1
Defining the "False Abel Inverse Transform" ..
34
2.2
Properties of the "False Abel Inverse Transform" . . . . .
35
2.3
~g(n, li')
=g(n,r(~)/i')
................ .
1
41
3 The heat kernel for Pos(n, R)
52
3.1 Pos(n, a) . . . . . . . . . . .
3.1.1
52
FromPos(n,a)toPo81(n,a) . . . . . . , . . . . . . . . .
3.2 Pos(2, a)
57
3.3 Pos(3, a)
62
4 The asymptotic expansion of the heat kernel of Pos(n, a)
78
4.1 About asymptotic apansioIll
76
4.2 pos(2,a)
80
4.3 Pos(3, a)
84
5 Estimates
90
5.1 Poa(2,a)
·
5.1.1
Lower bound
5.1.2
Upper bound
5.1.3
Final estimates . . . .
5.2.1
Lower bound
5.2.2
Upper bound
· .· · · ·
· · ·. ···
5.2.3 Final estimates . . . .
A Integration on the sphere
·
·
91
·
·
·
·
·
91
·· .·
· .·
94
94
96
96
106
109
111
s6n-l
114
B Technicalities
'
··
· ·.
·· · .·
·· .·
·
·
··
· . · . ···
·
· · · · ·
·· ··
· · ·
5.2 Pos(3, a) . . . . . . . . ...
"
55
2
C The elliptic integral
120
Bibliography
122
3
Abstract
The main topic oC this thesis is the study oC the Cundamental solution oC the heat equation
Cor the symmetric spaces oC positive definite matrice!, Pos( n, R).
Our fi.:st step is to develop a "False Abel Inverse Transform" Ci which transforms functions
of compact support on an euclidean space into integrable functions on the symmetric space.
The transform Ci is shown to satisfy the relation âCi(ji') = Ci(r(â)Ji') (r(â) is the usua!
Laplacian with a constant drift).
Using this transform, we find explicit formulas for the heat kernel in
th~
cases n = 2 and
n = 3. These formulas allow us to give the asymptotic development for the heat kemel as t
tends to infinity. Finally, we give an upper and lower bound of tIte same type for the heat
kerntol. In the case
.....
.....,...
f
~
c#
ft
= 3, the lower bound is completely new .
4
Résumé
Le I~et principal de ceUe thèle est l'étude du noyau de la chaleur pour les elpacel .ymétriques
de matrice. positives définie. Pot( ft, R).
Notre première étape e.t de développer une "pleUdo inverle de la tr&11lformM d'Abel"
g
qui tr&nlforme les fonctiOnt à .upport compact .ur un elpace euclidien en fonctions intégrables
sur l'e.pace .ym;.trique.
n e.t démontré que la transformée g latitrait la relation âg(fi .) =
g(r(â)/i') (r(â) est le laplacien ordinaire avec une pertubation constante).
Noua utililOnt cette tr&nlformée afin de trov,ver de. expressionl explicite. pour le noyau
de la chaleur danl le. cu
ft
= 2 et = 3.
ft
Cu aprelliont noUi permettent de donner le
développement uymptotique du noyau de la chaleur quand t tend verll'infini. Enfin, nous
donnont une bome inférieure et .upérieure du même type pour le noyau de la chaleur. Dans
le cu
ft
= 3, la bome inférieure est complètement inédite.
5
Acknowledgements
1 wi.h to expre•• my gratitude to my &cademic advi.or, Prore.sor Carl S. Herz, ror the
financial .upport he .ecured ror me and, ma.t of aU, ror introducing me to the .ubject or
.ymmetric .p&Ces. Hi. inlight on the .ubject hu been or a great help in thi. the.i•.
1 wo~d like to take thi. opportunity to thank the .taff' or the Mathematic. and Stati.tics
Dc.-partment ror their cooperation during my .tay at McGill Univer.ity. Particular thanJu go
to Allan Youster ror hi. expert ..siltance in many instance•.
The Science 1967 Scholar.hip Crom the National. Science and Engineering Research Council
and the J. W. McConnell Memorial Fe1low.hip, a McGill Univer.ity Scholar.hip, have greatly
contributed to my being able to devote my energie. to re.earch by
pr~...iding
the necel.ary
financial .upport.
Finally, 1 would like to thank David Chu who made it possible ror me to submit this thesis
whlle .taying at the Univer.ity or Prince Edward Island.
6
SYDlbolF frequently used
Symbol
Meaning
a
Lie algebra of the group A.
a+
Positive Weyl chamber of a.
A
Abelian component of the Iwasawa decoml'osition of G.
Positive Weyl chamber of A.
o
A generic root of the symmetric space.
0=
o(H)
= Hl -
H 2 : when it represents a specifie foot of Pos(n, F).
(3
~
c
Harish-Chandra c-function.
C
The field of complex numberl.
6
dimRF.
6(H)
1
TI sinhma o(H) where mea il the multipiicity
Q.
Q>O
(.
(-
A
Laplace-Beltrami operator for the symmetric space.
F
The base field: C, H or R.
FUi')
The Abel transform of the function / for the aymmetric apace of
7
noncompact type.
Legendre function: ;0(9) =
...,.,
~
he
e-p{B(gla» dlc.
The connected compon:!nt of the group (If isometries of the .ymmetric .pace
containing the identity.
G(Ji')
Adjoint of the Abel transform for the .ymmetric spac:e of noncompact type.
G(n,/i')
Adjoint of the Abel transform for Pos( n, F).
Q(n, li')
"False Abel in,rerse transform" for Pos( n, F).
"f =
IVpl·
r(A)
H
The (nun-commutative) field of quaternionic numbers.
H(g)
H9=
1tn(z)
Hermite polynomial of degree n.
1
The identity matrÎJ':.
K
The maximal compact subgroup of the group G.
K(m)
'l'he elliptic integral.
K(m)
The modified elliptic integral.
Ican
(Iwasawa decomposition) then H(g) :: loga.
Laplacian on the space A.
The maxim~l nilpotent subgroup of the group G.
Pos(n,F)
The set of positive definite matrices over the field F.
The set of positive definite matrkes over the field F of determinant equal
to 1.
---
8
(
1I'(H)
8(11')
1I'(H) = II(Hi - Hi).
i<i
(}
8
8(11') = I!(8H. - 8H).
i<i'
:1
R
The field of real numberl.
r
r = ,(H) = 11H11 = (H,H)1/2.
p
p(H) =
X
Symmetric space of noncompact type.
W
The Weyl group.
•
Signals the end of a proof.
l
La>O
Q.
,
j
L
(~
(
9
Introduction
Our notation will refiect that used by S. Helgason in [6] and in [7] (except in a few instances).
To complement this, especially for the spaces of positive defi.nite matrices, we refer to the
notation uscd by C. S. Herz in [8].
In th.is work, we endeavour to study the heat equation on certain instances of symmetric
spaces of noncompact type. More precisely, if X is the symmetric space, we con si der the
initial value problem on X X (0,00):
âu(z,t) lim u(z,t) =
t-+O+
aBt u(z, t)
(0.1)
I(z)
where â is the Laplace-Beltrami operator on X and
1 is a sufficiently "nice" function.
We are interested in finding the heat kernel (or fundamental solution) of the system.
The heat kernel, Ph will be such that for well behaved functions
1, u(:Il, t) = Pt * 1 (:Il) is
the solution of system 0.1 (* denotes the convolution in the symmE'tric spacej if we write
x = G/ K then 1 * h (gK) =
la l(gT- K) h(TK) dT) .
1
.."..
10
(~
Now, P, is determined by the conditions:
AP,(z)
1im P,*j(z)
t~O+
-
lim P,(z) =
'-0+
p,(z)
In the representation X = G/ K,
As the function
0
~
8
8t P,(z),
f(z),
o if'z # 0 and
o for aU z.
denotes the coset X.
lK P,(k. z)dk also satisfies these conditions, we conclude that Pt i.K-
invariant. To determine the heat kemel, in view of Cartan decomposition G = K A+ X 1 it is
then enough to find p,(a· 0) with a E A+. For simplidty sake, we will write p,(a) (or most
of the time Pe(e H
»with a e A+ (H E a+).
When acting on a X-invariant function, the Laplace-Beltrami operator can be
see~
as a
dift'erential operator on a+. We use the Killing form (or rather a constant multiple of it) to
form a scalar product ( , ) on
8.
For any linear map a E a*, Ha is the element of a which
corresponds to it via the scalar product (that is, for ail H
e
a, (Ha, H) = a(H». Renee,
applied to K -invariant functions on X,
A = LA
+E
ma cothaHa
(0.2)
a>O
where LA is the ordinary Laplacian on A, a a root, ma the multiplicity of a and Ha i. the
dift'erential operator corresponding to that vedor.
In the case where mea = 2 for each a, that is if G is a compla group, thi. equation
11
become.
where r(à)
= LA -12 , 6(H) =
n
ca>o
.inhm • a(H), p = l Eca>oa, 1 2 = (Hp,H,)
= IVpl2 and
(D 0 /}g = D(/' g).
Using that expression, Èskin in [3], shows that the heat kemel can be expressed
&1
(0.3)
where 1 = dima and r = r(H) =
11H11.
For the general case we have the following approach. We consider the Abel transform,
F(hi H) =
eP(H)
lN h(eHn) dn (s. Helgason in [7] writes Fh(e », where h is a K bi-invariant
H
function on G (equivalently h is a K-invariant function on X). We note here that we distinguish the Abel transform !rom the Radon transform. The latter is the integral of the function
on the horocycles of Xi ü h is a K-invariant function on X and ~
(le and H fixed) then the R~don transform of h is Î(~)
= {z : z = 1eeR n· 0, ne N}
= le h( z) dq( z) = lN h( eH n . 0) dn.
For our purposes, the Abel transform will be more practical.
It is DOwn that F(hi .) il Weyl-invariant (see for instance [7]). We have also the following
important property of the Abel transform:
F(âhj H) = r(à) F(hj H).
Now, ifPe(A)
-
1
=
F(Pei H ) exp (i(H.\, H)dH, then
:le(l) = 1. F(:t PeiH ) exp(i(H>.,H)dH
12
(0.4)
--------
= f. F(âPei H) exp(i(H,\, H)
dB
=
f.
=
f. F(PeiH)r(â)exp(i(H",H»dH
r(â)F(Pei H ) exp (i(H", H» dH
= -«B",H,,) + "Y2);,(~).
On the other hand,
10 Po(.\) = 1 (the';" are the Ipherical functionli
,;,\(~O)
= 1). The above equation il standard
and is a consequence of computations contained in Chapter 1.
This tells us that ;t(~) = exp ( -( (~,~) +12) t)
50,
that in view of Plancherel formula,
(0.5)
where c is Harish-Chandra c-function (see [7]) «~,~) = (H", H,,).
R. Gangolli, in [5], shows that P, defined by the equation 0.5 is the heat kemel for the
symmetric space G/ K and obtain from it, when G is a complex group,
(0.6)
where X il the Iymmetric Ipace 8Slociated to G, ';0 is the Legendre function and C ÏI a
positive constant. Thil il equivalent to the formula 0.3 given by Èskin (once we eliminate
the dift'eren tiaI operator).
(
( ."
'
13
In general, however, to study the behaviour or PI using equation 0.5 is diflicult, mainly
due to the presence of the spherical miction
'A (and not because of Plancherel meuure
Another approach (see for instance [11]), especially valid when the dimension or a (the
rank orthe symmetric space) is 1, is the Collowing: From ',he relation
L
F(Pli H) exp ( (HA, H) dB = exp( -( (A, A) + 7 2 ) t), we conclude that
F(Pli H) =
e-'Y
21
(4d)-r/2 exp( -,.2/(4t» where 1 = dima.
Now, if the rank of X is 1, explicit computations are possible. For a function h which
is K-invariant, F(h;.) can be expressed as a combination of simple Abel transformations.
These transformations have inverses, so the inverse of the Abel transform can be found.
However, if the rank of the symmetric space is greater than 1, finding the inverse of the
Abel transform is not always an easy task (when possible).
Notwithstanding the previous remark, our approach is, to sorne extent, based on the Abel
transCorm and on the idea of finding an inverse.
Again, we concem ourselves with K-invariant functions on the symmetric space X. -ll.
can be extended to a dense subset of the space L2 (a+ , 6( H) dH) (the square integrable Kinvariant functions on X) as a self-adjoint positive operator while the operator -r(ll.) can
be extenden i;o a dense subset of the space L 2 (a/W, dH) (the space of square integrable
Weyl-invariant functions on a) as a self adjoint positive operator.
----.....
14
1t il then natural, UlÎllg the Uluallcalar product on L2 Ipaces, to define for Weyl-invariant
functioDJ, the adjoint of the Abel tranlform:
(F(hj .), J)L2(a/W,cUl)
=
(h, G(Jj ·»L2(a+.6(B)cUl)·
(0.7)
Thil il in order to exploit equation 0.4i for aU h E Cgo(K\X) (the K-invariant smooth
functionl on X of compact lupport)
(AGUi .),h)
=
(G(Ji ·),Ah)
=
(f, F(Ahi .»
=
(f, r(A) F{hi
=
(r{A) J, F(hj.»
=
(G(r(A) Jj .), h).
=
G(r(A)Ji·)·
.»
ThuI,
AG(fi·)
(0.8)
The property Ihown in equation 0.8 is one "Yould expect trom the inverse of the Abel
transforme However, for
J E Cc:{a/W), GU;·) is generally not integrable. A good candidate
for the inverse of the Abel tranaform Ihould satisfy equation 0.8 and transform functions of
compact lupport into integrable functioUl.
We will not try to find the inverse of the Abel transform, but rather a transform having
these propertiel calling the resuit a "False Abel Inverse Transform" .
15
......
In Chapter 1, we fuit elaborate on the definition of the adjoint of the Abel tn.nlfonn,
G(J; .), and itl propertiel. In [4], I. M. Gelfand and M. A. Neumark ule a fonn of induction
on the integration over the group K = SU(n), to compute the Ipherical functionl for the
symmetric spaces corresponding to SL(n, Cl. We extend thil idea to expresl G(R,I;') (the
adjoint (Jf the Abel transform for the symmetric space pose n, F» in terms of G( J'l, - l,llEli .)
where Iiti is the function 1 restricted to a Ilubset of Ai
G(n,/iH)
=
1
JDom(H)
G(n-l,/IElie)1/1(H,e)tIe.
(0.9)
The object ofthls is, by mod.ifying the domain ofintegration Dom(H) in 0.9, to define the
"False Abel Inverse Transform". We want to change the domain in such a way that property
0.8 is preserved, but that the transform of functions of compact support be integrable. We
stress here that although the fonn of the "False Abel Inverse Transform" is arrived at through
considerations of the Abel transfonn and its adjoint, the use we will make of it together with
the justifications will not depend on the existing theory. This is the program for the following
r.hapters.
In the second chapter, we define formally the "False Abel Inverse Transform" fi(n,!;')
for the space Pos(n, R) and prove some of its properties, in particular, that
(0.10)
Our proof of 0.10, is quite difl'erent from the standard proof of 0.4 (and hence of 0.8).
In the third chapter, we use the "False Abel Inverse Transform"
16
....
'0 define a "candidate"
for the heat kemel for POI(n, R). We also show that the problem of finding the heat kemel
for POI(n, R) is equivalent to that of finding the heat kemel for POil (n, a) (the positive
definite matrices with entries in R, with determinant equal to 1)1.
The complexity of our formulas maltes it difficult to malte explicit computations in the
general case. We then study the case n = 2 and the case
ft
= 3.
Although the case n
=2
i. already well mown, the proofs involved in both cases require similar steps, the technical
difliculties being however much greater with
ft
= 3.
We will write our "candidate" as
(0.11)
C e-.,2t t-(climX)/2 exp(r 2/(4t», with the appropriate constant, is the fundamental solution for the ordinary heat equation (with a drift _,2) on Rdimx. In order to prove that Pt
i. indeed the heat kemel, we will have to investigate E(t, H) when t is close to O.
In Chapter 4, we show that our formulas allow us to give the asymptotic expansion for
the heat kernel of Pos(2, R) and of Pos(3, R) (or rather the asymptotic expansion of the
function E(t,H» in negative powers of..fi.
00
E(t,H) ~
E (-1)mbm(H)(v'i)-2m-q as t ....
00,
m=O
The coefficients 6m (H) are positive and explicitly given. We expect that this expansion
1.
1
lThi. il an important conlideration: molt relulb in thil thelil are about the Iymmetric Ipace POI(n,R)
and not about POII(n,R), the Iymmetric Ipace ulociated to SL(n,R).
17
should provide a lot of information about the behaviour of the heat keme1. We find out that
the term 60 is a constant multiple of the Legendre function and that E(e, H) S 60 (H) (Vtt'.
In the paper [1], Jean-Philippe Anker gives the foUowing upper bound on heat
the Iymmetric Ipacel U(p,q)/U(p) X U(q): for t
> 0 and H
Pe(e ll ) ~ C e--,2e t- I/ 2 exp ( _,2/( 4e» e-P(B)
)u~IDel
for
E a+,
II
W(m ... +m, ... )/2(t, a)
aEE:
where Et il the let of indivisible positive roots of H, 1 = dim 8,
root a,
m2a
> 0, z
is the multiplicity of the
the multiplicity of 20 and
w,(t,z) =
for t
ma
~
iftS1+z
{
ift~l+z
0 and. E R.
Ifwe replace the function w,(t,z) by the function t-' (1 + z) (1 + z + t)'-1
pendent of t and z), the upper bound is equivalent to
C e--y2t e-(climX)/2 exp ( _,2 I( 4t» ~(H)
II (1 + 0 + t)(m ... +m:a ... )/2-1.
(0.12)
aeE;
We made use of the estimates given by Jean-Philippe Anker in [2]; there exilta Cl
and C2
> 0 such that
C1 e-P(B)
II
(1+a)~t/>o(eB)SC2e-p(Il)
aeE:
II (1+0)
aeE:
for aU HE a+.
-
>0
18
(0.13)
(
He alIo couJec:turel that the upper bound in the equation 0.12 holela for aUsymmetric
.paca OfDoDcompact type. Thi. i. clearly true for the comples cale (.ee equatioD 0.6). Using
equation 0.5, it CaD be .hown that thi. upper bound i. alway. valid if t il .ufficiently large. We
mow (He [7]), that if ~
e .*, Ic(~)I-l S Cl + C21~12p with p = L (mca + mi&/I~'
A simcaEEt
ilar aDalYlil Ihowi that if I~I S 1, Ic(~)I-l < C 1~IIEtl. FinaUy, if .\ e •• , ItPA(eH)1 S tPo(e B ).
Bence, if t
~
1,
S ..,(eH ) [C 2 1
JI A1::::; 1
S ..,(eH ) [C2
= ..,(eB )
1..e
exp ( -(~,~) e) 1.\1'11::1d.\ + C" (
JIAI~l
exp ( -(.\,~) 1) 1.\1 2IE:1 d.\ + C" e- I
[C't-1/2-IE11
+Cil e- t 1
JIAI~l
exp(-«(~,~)
1
op( -(~,~) e) 1.\/4p d~]
JIAI~l
op( -( (~,.\) - 1) t)
1.\1 4P d.\]
-1» 1~I4pd.\]
which confirma the upper bound given in the equatioD 0.12 when t i. sufticiently large. This
derivatioD wu outliDed to UI by Noël Lahoué in a perlonal communication.
In Chapter 5, we prove for Poa(2, R) and POI(3, R), that pte eB) il between two constant
multiplel of
e--,2t t-(dbDX)/2 exp ( _r2 /(4t» ;0(8)
11
(1 +a
+ e)(ma +m
2 C1 )/2-1.
GEE:
While IODle upper bounds mit for POI( n, R) for aU R, the lower bound given here in the
cue R = 3 (Theorem 5.4) i. completely new.
In the lame chapter, for each of the two cuel under consideration (Pos(2, R) and
19
--
Po1{3, a)), we have a .ub.ectioD. titled "Final eatimate.".
We write
with 9 and C choaen in .uch a way that limc...oo Ve(H)
= 1.
Thi. i. po••ib!e due to tbe
uymptotic expUllion oC eTc up(r J /(4t»p,(e B ) (u t ~ (0) whieh hu Ct-,,~ u fuit
term.
We then proceed to atate our e.timate. and uymptotic e%pUlliom on PcC eB) in tenns
oC Ve(H). We alao give the Collowing e.timate. on Ve(N) whieh can .erve imtead or the
uymptotic expanaioDi.
There m.ta po.itive con.tanta A, B and C IUch that
o
BQt(H)
where Qc(H)
~
1- Ve(N):5 A(l
~
V,(H)
~
+r)/HCt ~ 1 + r,
CQc(N) ift
~ 1
+,
=II«l +,,)/t)-1/2, the produd being takenover the rootl" .uch that 1+" ~
t.
Thi. Cormulation, and the proof for t?le cue n = 2, wu communicated to u. by Carl S.
Herl. It i. intere.ting to note that the proof oC the fuit part i. exact!y the lame Cor the case
ft
= 3 if one use. the corre.ponding eltimate•.
20
.....
Chapter 1
The adjoir1t of the Abel transform
1.1
Definition of the adjoint of the Abel transforme
As explained in the Introduction, the goal of this chapter is to study the adjoint of the Abel
transform. The Abel transform corresponds to the integration on horocycles of K -invariant
functions on the symmetric space X (with a factor; as pointed out in the Introduction the
Abel transform is not quite the Radon transform). Therefore, the adjoint will correspond to
the integration over the boundary of K -invariant functions defined on the space of horocycles.
We define in this section thp. adjoint of the Abel transform for any symmetric space of
noncompact type.
Let h be a function in Cc(G) bi-invarilmt under K and
under the action of the Weyl group W.
21
.
l
f be a function in Cc(A) invariant
t
Th,! Abel tran.form of h il
Now,
L
F(hj a) I(a) da
=
LIN
eP(1oc m) h(an) I(a) dn da
L
- IK IN e 2P{lot m) h(1:an) e-P{B(lamn» l(eB{laGn»
- f
dndad1:
e-P(H(sr» h(g) l(eB(g» dg
JG
1 1 h(kak) e-P(H(~Ia» l(eB(!m"» 6(a) da d1c diM
_ l
JKIM
=
1 1
lK lA+
= L+
=
6(e H ) = 6(H) =
JK JA+
1
lA+
h(a)e-p(H(ala» !(e H(mla»6(a)dad1c
h(a)
[IK e-P{H(ak» l(eB(ala»
d1c] 6(a) da
h(a) G(fj a)cS(a) da.
II sinhma a(H) where ma is the multiplicity orthe root a. These intea>O
gration formulas appear in [7].
We then have
L
F(hja) !(a) da =
=
1
lM
fa
Rence)
-
22
h(a) G(!; a) 6(a) da
h(g) G(Jjg) dg.
(
Definition 1.1 The adjoint 01 the Abel ~rtJru/orm U defined by the equation
Note: We have, with proper normalization,
1 I(lc)dlc = 1
lx
JI l(lcm)dmdlcM.
lKIM M
If 1
is aM-invariant function on K, it can be considered as a f'unction on KIM (the bound-
1 e-P{H(a1l» l(e H (a1l» dlcM (we
lKIM
dm = 1). However, for our computations, the form. given
ary). So the Definition 1.1 could be given as G(fi a) =
assume
1
lK
dlc =
1
lKIM
dlcM =
1
lM
in Definition '.1 is going to be more practical.
In particular, the spherical functiollB tan be expressed in terms of the adjoint as
For aIl f, h satisfying the ab ove conditions (with, in addition, sorne obvious smoothness
conditions), we have:
la h(g)G(r(tl)fjg)dg = LF(hj a) r(â)f(a)da
=
=
L
r(â)F(hi a) f(a) da
L
F(âhj a) f(a) da
=
L
=
la
23
âh(g) G(fj g) dg
h(g) âG(fi g) dg.
We conclude that
âG(fi·) = G(r(â)/i .).
(1.1)
For the sake ofpracticality, if HE a, we will write F(fi H) for F(hi eH) and G(fi H) for
1.2
The adjoint of the Abel transform for Pos(n, F)
We set to compute G(n, li H)
=
L
e-P(H(eHIa» l(eH(eHIa» die, the adjoint of the Abel trans-
form for the symmetric space corresponding to GL+(n, F) (F = R, Cor H). We will write
5
= dimRF.
We aim to express G(n, li H) in a manner that will suggest a definition for a "False Abel
Inverse Transform" which conserves the property given in equation 1.1. The main result of
this chapter is Theorem 1.1 ttiving such an expression.
In what follows, K is the unitary subgroup of SL(n, F) and KI that of SL(n - 1, F).
Our goal, as mentioned in the Introduction, is to adapt an idea of 1. M. Gelfand and M. A.
Neumark, in using induction to compute the integration in Definition 1.1.
10 The induction process in the integration:
~
Lemma 1.1
"
t
r
lx
1.
~
1(Ie)dle = Cn - l
1 [1
J5 1.-1 JKI
~
1
~
"ï
"
~
~
>
fIl
t
i
--
24
1([J(z )leli z]) dle l ] dl/(z)
(~
where J(z) E Fn)(n-l) U cho.en
a deue ,ub,e'
QI
0/ s6n-l.
tI.,(z)
'0 Ual" [J{z)jz1 E X
and J(z) tlepentU .moolAly on Z over
u U:.e X-invarian' meGlure on s'n-l
a .ubgroup o/SO(6n,R»); Cn -l U cho,en
'0 U&ae Cn - 1 J15
1.-1
(K can be con.idered
d.,(z) = 1.
Prool: We write "U) for the meuure in the right hand side of the equation. All we need
to prove ia that
"U 0 Li) = "U) (Li
stands for left multiplication by 1:). To choose J(z)
smoothly posel no problem: one pick. linearly independent vedors {!l, ... , In-l} and, for
every z E s6n-l which doe. not belong to the .pan ofthele vector., apply the Gram-Schmidt
process to the set {ft, ... , ln-l' z} starting with z (it can be done in a "smooth" manner) to
form a matra j(z); we set J(z) = j(z)/(det[j(z)j Z])l/Cn - 1) (In the case 6
do something else: The case n
= 1 we have to
= 2 is no problem. For n > 2, det[j(z); z] will be constant on
conneded components of sn-l - span{/h ... ' In-l}. On the connected components where
the determinant is -1 we just have to e ..change the mst two columns of j(z».
The choice of J(z) (as long as it issmooth) does not matter. Indeed, if j(z) has the same
property, then
1 [1 l([i(z)1:1;z]) d1:1]4.,(z) = 11 1 [1 I([J(Z)1:1 ••1:1jZ]) 41:11d.,(z)
J51. -1 JKI
J5 .- JKI
= 1 [1 I([J{z)1:1 j z])d1:l ] d.,(z).
JSI.-l JXl
since, for each z, we can find 1:1•• E Kl with j(z) = J(z)1:1...
25
",
.
FinaUy, Ü
~
e K,
=
=
1 [1
1SI.-l
1Ka f([~J(Z)~li"Z])d't1d,,(z)
1 [1 f([J(iz)~l;iz])d~l]d,,(z)
1SI.-l
1Ka
1 [1 1([J(z)ili~])dil]d,,(z).
1SI.-a
JKa
2° Introducing an auxiliary transrorm:
Deflnition 1.2 The Gram detemainant of a mahu Q for the fir.t p row. and column.t il
given. by
We will wnte âo(Q)
=1.
Deflnition 1.3 Ij Q it a po.itive definite mcdN, we write
It is easy to see that Q(n, fi Q) depends only on the eigenvalues of Qin any order.
The relevance of the Gram determinant in our situation come !rom the next relation: if
, e GL(n., F),
(1.2)
This is easily checked if one writ~s ,
-
= 'an (Iwasawa decomposition)i ,-, = n-a2n and
by direct verification one proves the equation 1.2.
26
Our computations are in order to exploit the relation
(1.3)
with h(c)
= h(a1/ 2 ) (tbis is a Itraightforward consequence of Definition 1.1, Deflnition 1.3
and equation 1.2).
T. S. Bhanu Murti ule the Gram determinlmt explicitly in [10} to describe the Plancherel
measure for the Iymmetric space corresponding to SL(n, R).
In what followl, the eigenvalues of Q are li
> lz > ... > ln > 0 (in particular, the eigen-
values are distinct).
From Definition 1.3 and Lemma 1.1,
Q.(n, fi Q) = Cn - 1
(
[(
1SI.-1 1Kl
f(~,,([J(z)kli z]*Q[J(Z)k1i z])h~p~n dkt) dll(z).
(1.4)
We will show that the inner integral corresponds to a Q-transform of order n -1 and that
the outer integral can be parametrized by the eigenvalues of J(z)*QJ(z).
We exp and further the tenn [J(Z)k1iZ}*Q[J(z)k1i z}:
kiJ(z)*QJ(z)kt kiJ(z)*Qz ).
[J(Z)k1i z]*Q[J(z)k 1 ; z] =
(
Lemma 1.2 The notlltion 6eing
10 11 E z.1. {:} 11 = J(z)z for
41
z*QJ(z)k1
c60fJej
60me
z E Fn,
27
l
z*Qz
(1.5)
30 J(z)"'QJ(z) ÎI po,iCive dejinite.
Proo!:
10 Since [J(z); z]
Renee, (z,J(z)%)
e K,
= (J(z)·z,%) = O.
20 J{z)w = 0 ~
10
=0 and J(z)J(z)'" + zz· = 1.
(z,y) = 0 ~ y = J{z)J{z)*, + zZ·JI = J{z)(J{z)·,).
we have J(z)· J(z) = 1, J(z)*z
= J(z)· J(z)w
= O.
30 If w ::; 0 then (J(z)·QJ(z)w, w)
=(QJ(z)w, J(z)w) > 0 linee Q il pOlitive definite
and J{z)w ::; O.
H {pï} 1 SiSn-1 are the eigenvalues ofthe matrix J(z)·QJ(z) then
n
n-1
i=1
= n -1,
ifp = n.
ifp
iii
detQ
This is clear from equation 1.5.
30 We will now investigate the relation between z and the eigenvalues /Ji. Our goal il, for
appropriate functions, to express the measure tlII(Z) on 5 6n- 1 in terms of the Pi.
We piek V1,
V2, ••• , Vn
an orthonormal buis over F for which Qv,. = .\,v. (the canonical
n
basis if Q is a diagonal matrix). We will write
li:
= E u.v•.
We will assume that u. ::; 0
.=1
for allie (we are neglecting a set of measure 0). This ensures that no eigenvector of Q is
perpendicular to
lI:.
Let v be an eigenvector of J·(z)QJ(z) with eigenvalue pj J*(z)QJ(z)v = l'v hence
-~
r 1
28
Ql(z)v = pl(z)v + 'z (suppole,
= l(z)z e zJ.; (Ql(z)v- pJ(z)v,J(z)z)
=(J*(z)Ql(z)v- pJ-(z)l(z)v,z) = (J*(z)QJ(z)v- pv,z) = 0).
The condition "II :/: 0 for aU
~
impliel, in particular, that J(z)v il not an eigenvector of
Q (Iince by Lemma 1.2,2°, l(z)vl. z). AI a consequence,' #: o.
ft
E 'IIVJ.. The relation QJ(z)v = pl(z)v +'z gives us Et=1 A",,,v,, =
11=1
u.
E:=lp'leV" + E:=1 'U"VII 10 A",,, = l"" + ,,,,, and hence, '" = .. A. _ l' (1' = A.. would
imply "Ji = 0 or .. = 0). The eigenvalue l' determines v up to a constant factor sinee
=l(z)-J(z)lI = .. E \ UII J(z)*v". Inotherwords, l(ztQJ(z)has n-l distinct eigenWe write J(z)v =
ft
11
11=1
"le - l'
values.
1
'j
(1.6)
n
~tJi
•=1
,,
"
=
1 with t ..
> o.
,
These equations imply that l' il Iqueezed between two eigenvalues of Q (otherwise
t
A t.
•=1 .. -
would be strictly positive or strictly negative) .
l'
Let Pl
> ... > Pn-l be the eigenvalues of J*(z)Ql(z). We have for i < j
29
1
L
,
1
hence we cannot have i
<i
and ~p
> l'i > pj > .\pH
since the ab ove expression would be
either strictly positive or strictly negative.
We tben have >'1
Let
> 1'1 > .\2 > 1'2 > ... > .\n-l > l'n-l > .\n.
1
JI. = -,
lin =O.
Pa
if 1 ~ i Sn -1,
ifi =
R.
We apply the resuIt in [15, page 202];
We set A =
[1 _~i~AI]
i we have
A[tl' . .. ,t,S" = [0, ... ,0, 1]*
50, by another application
of the result in [15, page 202],
tp
= (A- 1 [O, ... ,O,1]")p
= de! A (_l)ptn det [Aii]i~n,#p
= n~.;l (Pa - ~p)
ni~p(~' - ~p) .
At last, using straightforward calculus,
=
lli<p(Pi - pp)
lli<p(.\i - .\p).
n
4° z =
.,..
~
~
f1II!'IIo
,~
1,
..0.
E UiVAI, so once the {VAl} are fixed, the {UAI} provide a parametrization for s6n-l.
AI=l
30
For functionl whicll dependl only on t. =
=
4v(z)
IUJtl 2 (as in the case of the Pi):
B" (tl ... tn)"/2-1 dt1 .• • 4tn-l (Lemma A.1 in Appendix A)
",<pCPi -pp)
= B" p=1
n. j','" _- "PlA..1]6/2-1
j
n..•<p'
(A' _ A ) dpI •• . dPn-1
TI" [n~.':'ll#li
p
,~p
= B"
n-1
n
i=1
;=1
TI(A, - .A; )1-1 II II j", -
,<;
.A;jl/2-1
II (#li - l'p) dl'1 .• -4""-1,
i<p
Here B" il a positive constant wbich depends only on n.
We then have
r(6n/2) (>'.-1
(r(6/2»" 1>..
Q(n,fiQ) -
'~(~,I')dl'I
(>'1
. "1>.2
.
Q(n-1,fdetQi diag[1l1J .. ·,Pn-l])
..• dPn-1
where fdetQ(P) = l(diag[P,detQ/detP]) and
=
"(A,p)
[II<~,
i<i
,,-1 "
- >.;)]1-6 II II IPi - ~jI6/2-1 Il<Pi - pp).
i=l ;=1
i<p
The constant factor in the above equation is computed in Appendix A, Lemma A.2 (it is
cleu !rom the definition that Q(n, 1j Q) = 1).
For FE a, if plc(F)
(p(H) =
21 E
= -26 E
(Iii - Fi) then p,,(F)
=p"-I(F) + 6(tr F -
nFn)/2
i<i~"
a(H)
= Pn(H».
a>O
1
At last, making use of relation 1.3, we return to G(n, fi H); with the change of variable
Il = e2(, we obtain
(~
(~
31
l'
=
(e- Pa f)(exp(diag[X/2, tr H - tr X/2]))
=
exp ( -6[tr H - n(tr H - tr X/2)J/2) [e-;;:l!ldeteH (e X)
= exp(6«n - 1)/2) tr H) exp( -6(n/2) tre) [e-;'::I!lddeH (eX).
We need also to compute
'i)(e
2H
, e 2t )
n-l n
_
-
II(e2Hi - e2Hi )1-6 II II le2Ei _ e2HiI6/2-1 II(e2Ei _ e2Ep )
i<i
i=1 ;=1
i<p
exp(~)l- 6)(H, + Hj» [II 2 sinh(H, - H;)]1-6
i<i
i<i
n-l n
.exp(l: ~)6/2 - l)(e, + H;)}
i=1 ;=1
n-1 n
·II II 12 sinh(ei -
H;)16/2-1 exp(~)ei +ep » II[2sinh(ti - ep )]
_=1;=1
-
i<p
i<p
21- n exp((l- 6)(n - 1) tr H) [6(H)](1-6)/6
· exp«S/2 - l)(n - 1) tr H) exp«6/2 - l)n tre)
n-l n
·i=1
II ;=1
II Isinh(ti -
21 -
n
H i )1 6 / 2 - 1 exp«n - 2) trt) [6(e)p/6
exp( -Sen - 1)/2 tr H) [6(H)](1-6)/6 exp((6n/2 - 2) tre)
n-1 n
· II II Isinh(ei -
H;)16/2-1 [6(t)P/6.
i=1 i=1
32
(
All or tms put together gives the
Theorem 1.1 Il 1 ia a continuou. /-ndion. on. A and BEa Ihen
G(I,/i H)
=
l(e B ) aM, iln ~ 2,
G(n,/i H )
=
A(n - 1,6) [6(8)]1/1-1
l
Ba-1
.. .
B.
lBI
G(n - l, l'rHi e)
82
n-l "
.II II Isinh(e, -
H,;)I'/2-1 [6(e)]1/1 d(.
i=1 ;=1
We use the convention "rB(et )
Ide\B(e t
= I(exp(diag[e, tr H - tre])
(denoted previously by
».
In particular:
G(2,/jdiag[r,-rn = A(6) [sinh(2r)J1-1
f:p I(exp(diag[,,-..]))
[cosh(2r) - cosh(2.. )}'/2-1 d,.
A straightforward consequence is
-rn = B(6) 1 I(exp(diag[.. , - ..1) [sinh(2,)]1-1 [cosh(2r) - cosh(2")1' / 2- 1 dot.
00
F(fi diag[r,
33
Chapter '2
The False Abel Inverse Transforrn
2.1
Defining the "False Abel Inverse Transform"
In the Theorem 1.1, the adjoint of the Abel transCorm is expressed in a Conn involving
induction. This suggests that we define the "False Abel Inverse Transfonn" Cor H E a+ the
following way:
Definition 2.1 The "Fal&e Abel Inver,e Tra.MfofTn": if HE a+ "'en
"(l,/jH) = l(e H )
and, for n ~ 2,
g(n,/jH) = (6(H)] l -1
1B--2
B_-l
,,-1
. •.
Loo O(n - 1, It.rHi e)
BI
ft
.II II \sinh(ei Î=l ;=1
We recal1 that ItrB(e() = J(exp(diag[e, tr H - tre]).
--• h,
34
H;)\I-l [6(t)1 t
de·
(
For n
= 1 and n = 2,
g(l,J; H) _
J(e B ),
g(2,J;H) _
[linh(Hl - H2)]i- l (00 J(exp(diag[e,Hl + H2 - eD)
JBI
.[ainh(e - Hl) linh(e - H2)]f- l de
_
[sinh(Hl
-
H2)]i- 1 (00 J(e F )
JBI
'[linh(Fl - Hl) linh(F1 - H 2)]f- l dF1.
(2.1)
In particular, if H = diag[r, -r] then
Q(2, li H) -
[sinh(2r)]t- l
100 f(exp(diag[e, -em [sinh(€ - r) sinh(€ + r)]f- il(
l
= 21 -f [sinh(2r)]t-1
2.2
LOO I(exp(diag[€,-e])
[cosh(2€) -
cosh(2r)]f-l~.
Properties of the "False Abel Inverse Transform"
We will need sorne information about g(n,/i H) (from now on we assume that H E a+).
The results of this section concem mainly the support of the function g (n, 1; .) and some
upper bound estimates.
Definition 2.2 Let D(n)(H) be the support of J --. g(n, Jj H) seen cu
D(l)(H) =
{H} and, Jor n ~ 2,
D(n)(H) =
{F=
(w
o
0
0
dittnbution:
) :wED(n-l)(e),HiSeiSHi_l,lSiSn-l}.
trH - trw
To ,impliJy notation, we adopt the convention thot Ho =
35
00.
Some properties or DCn)(B):
Lemma 2.1
1° For t > 0, t· DCn)(H) = D(n)(t. H).
4° If c(n)(H) = {F e a :
P
f'
.=1
.=1
1: Fil ~ L HII (1 $. r $. n -
1) and tr F =tr H} "'en
Proof:
1° and 2° are straightforward using induction.
3° is a straightforward consequence of 1° and 2° .
4° Proofby induction. The result is clearly true for n = 1. Assume true for n -1 (n ~ 2):
f'
H Fe D(n)(H) then tr F
(1 $. r $. n - 2,
eas in Definition 2.2).
n-l
FinallY'L F,. =
11=1
not equal (for n
~
f'
•
=tr H and, by the induction hypothesis, L F,. ~ 2: e,. ~ L
11=1
n-l
n-I
11=1
11=1
L ell ~ L
"=1
HII
11=1
BII' The two sets are
2) •
The next lemma shows that, to some extent, the "False Abel Inverse Transrorm" preserves
compact support.
Lemma 2.2 1/ /
e Cc(A)
"'en {H
e a+ : O(n, fi H):F O}
Ï8
bounded.
Proor: Let E be the support of f. It is enough to show that {JI e a+ : D(n)(B) nE
=F 0}
is bounded. Let T = min tr F and S = max FI' Now Fe D(n)(B) nE::::> Hl $. FI $. S and
FeE
....
FeE
36
(
H,. =
n
n-l
.=1
.=1
E F. - :E HIt ~ T -
(fi -1)5. Hence D(")(H) nE =F" => T - (n -1)5 ~ H,. < Hn-l
< ... < Hl ~ S.
We will now inveatigate hoW' rut Q(n, /;.) decreue. at infinity,
The following lemma about the .inh function will be Uled repeatedly in finding upper and
lower bounds:
Proor, Euy calculUi exercise.
Another pre1iminary lemma:
Lemma 2.4 If a
< b then Uaere eZÎltl G cOnltGnt 0 < ON < 00 depending oRly on N Buch
"
L
(1 + Z
Il
- G)N e-(·-II) dz
.j(b-z)(z-G)
~
1
CN v'1+b-a'
Proor,
," (1
J.
+z -
a)N e-(·-II) dz
J(b - z)(z - a)
+z -
+ 1"
(1 + b - a)N e-("-Cl)/2 dz
Il
J(b - a)/2 (z - a)
JCllff,)/2
J(b - z )(z - a)
S
v'2 100 (1 + z - a)N e-(_-a) dz + (1 +b _ a)N e- C,,-a)/21"
da:
y'6::a CI
v'z - a
CI
J(b - z)(z - a)
S 1(11+.)/2 il
=
v'2 1
y'6::a Jo
00
a)N e-(·-II) dz
(1 + ,)N e-" d,
..fi
+!
(1 + 6 _ G)N e-("-a)/2 ~ ON
2
37
l
y'6::a
,
-
The result is clearly true if b assumed here that N
~
Oi if N
(J
is small (say, smaller than 1) sa we can conclude (we
< 0 the resuIt is a consequence
Lemma 2.5 Allume HE a+ Gnd / ù continuaul. We write
of the case N = 0).
II/IIB =
1/{eF)I. We
sup
FeDC·)(B)
M'lie:
1° Ig(n,/iH)1 ~ g(n,I/liH).
2° 1/6
= 1 then Ig(n, /i H)I ~ C n II/liB [6{H)t l / 2 where 0 < Cn < 00 depends uniquely
ann.
Prao!: l O is clear sinee g( n, /i H) corresponds ta an integration against a positive kernel.
Ta prove 2°, it is enough ta show that g( n, li H) ~ C n [5(H)]-1/2. This is true for n
with Cl
= 1. We assume true for n -
=1
1 (n ~ 2).
g(n,li H )
(2.2)
eH
Ifwe use bath sides of Lemma 2.3 and the rad that lei _ Hil =
{
i-
ifi
i
Hi - ei if i
~j
, we
<i
have
6(e) 5(8)
n'..;l ni=11 sinh(ei - Hi)1
~ C e- 2«(1-Hl) 1 + el - Hl
el - Hl
.........
iî
(1 + Hi-l - ei)(t + ei - Hi) ei=2
(Hi-l - ei)(ei - Hi)
38
2
H
«(i- i)
II
i<i~n-l
1
+ei -
ei -
Hi
Hi
(
(2.3)
This, if we refer to equation 2.2 and Lemma 2.4, aUows us to conclude.
The following lemma will be useful in the proof of the main result of the chapter. We will
explain its purpose in detail at the beginning of the next section.
Lemma 2.6 Assume H E a+ and
{Ha-2
W(H, z) =
J,
1 E Cc(A).
Let
{OO
... JI
8.-1
.1
n-1 n
Q(n -1, ftrHi e)
Hl
II II 1sinh(ei - HiW [S(e)]· tJe·
i=1 ;=1
1° W(H, z) i" an analytic function of z in the domain
~z
(2.4)
> (6/2 - 1) - 1/2.
2° The integrand in equation !.4 i.s 0 on the boundary of the domain of integration whenever Hz> 1.
Proor:
6 = 1: From Lemma 2.5, 2°, we know that Q( n - 1, ftrHÎ e) [6(~)]1/2 is uniformly bounded
for
eE A+ and vanishes outside a compact set. Clearly, the resuIt is true if Hz ~ O. Assume
-1
< Hz < O. H we proceed as in the proof of Lemma 2.5, we have
n-l
1
ft
II II 18inh(ei - HiW 61/ 2(e)1
i=l i=1
39
~
[6(H)]" [6(e)]"+1/ 2 [R(H,e)t lb
where R(H,e) is the term in equation 2.3. A careful examination shows that thi. proves our
assertion for 6 = 1.
-
6
=2: We fuat point out that Ü we set gel, fi H) = J(e H ) and
"(n, Ji H) =
1
8
1
00
..-'1
H.-l
...
Hl
- li H). We
"(n
- l, ArHi e) Il( then O(n, li H) = [6(H)t l 1:1 g(n,
can use this to write
g(71 -
l, Itr H,
e) has compact support as a function of ee A+i by considerations similar
to the previous case we can conclude.
6 = 4: We define g(l, li H)
-"(n, li H) = Jl1H
.-2
H.- l
= l(e H ) and
00
... 111 "(n
-l, ArHi e)
Hl
n-l n
II II 1slDh(ei
. -
Hi)1 [6(e)] -1/2 de·
i=l i=l
We find that
W(H, z) =
1Il.-2
lH..
_l
•••
lB
"(71 - l, lt.rHi e)
l
We note nat, that ü Rz
il!!
_ln
1'10 -
.•
1sinh(ei - Hj)1 [6(e)]
-1/2
Il(.
> 1/2,
n-l n
6(et
1/2
1
II II 1sinh(ei -
HiWI =
i=l i=l
n-2
.II sinh(ei -
n
Hn - 1 )
i=l
II sinh(ei i=l
n-l n
.II II 'sinh(ei -
-
i=l j=l
40
H;),"-l
Hn)
n-2
(
S
II .inh(e. -
.=1
n-2
Bn-l)
II .inh(e. -
Bn)
.=1
n-l n
.II II I.inh(e. -
B;)IIb-l .
•=1 ;=1
Ifwe use the same upper boundl, we find that lince 1 hu compact support, g(n - 1, t'rHi 0
il uniformly bounded for
eE A+ and vanishel outside a compact set, sa the resuIt follow.
Another useful item i. the following
Lemma 2.7 Il lm
-+
1 unilorml,l on A then g( n, Imi .) -+ g( n, li .) unilorml,l on compact
I"bleu 01 A+. We allume that lm E Cc(A).
Proor: 6 = 1: 1 being continuous and bounded, g(n,/i·) exists (an easy consequence of
Lemma 2.5). If E is a compact set of A+ then for BEE, Ig(n,l; B) - g(n, Imi H)I
~ g(n,l/- Imli H) S Cn III - lm lia [6(H)]-1/2 S CE,n III - lm 1100'
6 = 2 and 6 = 4: Same pro of with the estimate Ig(n, li H)I ~ lIeP
•
6 = 2 an<1 Ig(n,/;H)I ~
•
Ilia g(n, e-"j H) for
_ 3/2
2
Ile"3/2 ·/lIag(n,e"
;H) for 6 = 4 where r (F) =
n
~
2
L...JF;.
;=1
Remark: We can extend the resuIts in Lemma 2.6 and Lemma 2.7 by the same device. If
1 decreases at 00 fast enough (for example, if 6 = 4, faster than e-,,3/2), then the resuIts still
hold. We will make use of this observation.
2.3
ÂQ(n, Jj.) = Q(n, r(Â)J;.)
We :aow set Out ta prove the main result of the chapter, that is, âg(n, li·) = g(n,r(â)/i·)
41
,
1
~
We will firlt explain our strategy. In Proposition 2.1 of this section, we pave the way to
a proof by induction.
The step !rom n - 1 to n is done acc:ording to the following devic:e. In Lemma 2.6, we
showed that the function
[6(H)} t -1
lR_-2 loo g(n - 1, ArRi e)
...
R.-l
Hl
n-1 n
.II II 1sinh(ei i=1
Hi)la [6«()]t d(
(2.5)
i=1
is analytic as a function of z in the domain Rz > (6/2 -1) - 1/2 and that the integrand is 0
on the boundary of the domain of integration whenever Rz
> 1.
This last property allows us to use integration by parts and interchange the order of
integration and dift'erential operators without having to introduce new factors as long as
Rz is large enough. We then refer to the analycity of the function to extend the result to
Rz> (6/2 -1) - 1/2 and hence to z = 6/2 -1 whic:h corresponds to g(n, fi H).
We need fust to examine the eft'ect of various dift'erential operators on the "False Abel
Inverse Transform".
\\te will use the scalar product (F, H) = R tr H· F (a constant multiple of the Killing
form). In Chapter l, we define Pn{H) =
li- Et=l(n +1 -
-
2k)2 =
6- n(n
2
-1).
t Ei<i<n(Hi -
AllO,
42
Hi)' Hence,
1! =
(Hp., Hp.)
=
(
The Lemmas that follow involve the action of differential operators on the "False Abel
Inverse Transform". Nùte that theBe Lemmas and their respective proofs are also true for
the adjoint oC the Abel transCorm. We will write D" Cor the partial derivative with respect
to the k-th argument.
Lemma 2.8
d
1° dTg(n,lriH) = g(n, (Dnf)ri H).
2° D,,/r = (DIe/ - Dnl)r.
n 0
d
n
3° LOH.g(n,/i H )= -d 1_ g(n,/iH+'I)=g(n,L,D;/i H ).
;=1
,
• __ 0
;=1
Proof: l O is clear since
g(n'/ri H)
_ 1
JD(a)(B)
/(exp(diag[F,T-trF])0(H,F)dF.
2° is clear when we write /trH(X) = /(diag[X, T - tr X)).
The first equality of 3° is clear. We proye the second equality by induction. For n
= 1,
there is nothing to proye. Assume true for n -1 (n> 1):
Uaing the Lebesgue dominated convergence theorem and an obvious change oC variable,
we haye
=
lBa-2 . .• lOO -d
d 1
8._1
g(n - 1, /n_Hr8i
B I ' _=0
n-1 n
1Binh(e.
II II
i=1 ;=1
43
e+.1)
- Hi)1 6/ 2 - 1 [6(e)]1/6 de.
..,,-.
Now,
1
..).
:.I.=oQ(n - 1, In,HrBi E+ aI) = :.I.=og(n -1, l'l'Bi E+ ..1) + n :T Ir=trBQ(n - 1, fTi E)
n-l
v(n -1,
Di/'rBi E) +n g(n, (Dnf)'rBi H)
_=1
I:
=
(inductio'l\ hypotheBil)
n
-
v(n -1,
(I: D;f)'rBi E) (clear from 2°).
;=1
We will write Il. for the Laplacian of the Iymmetric space Pos(n, F) (in the variable H)
and dl for the Laplacian of the Bymmetric Bpace Pos(n - 1,F) (in the variable el. Recall
Lemma 2.9
(r(â)f)r = r(â1)/r + n(Dnn/)r + 2
n-l
E Di(Dnf)r -
b! - 'Y!-l)fT.
i=l
Prao!:
n-l
r(1l. 1 )Jr =
'L,(Diil- 2Din/ + Dnnf)r - 'Y!-dr
from Lemma 2.8,2°
i=1
=
(r(d)f)r + (n - 2)(Dnnf)T -
n-l
2(E Dinf)T + b! - 'Y!-l)lT
= (r(d)f)r + (n - 2)(Dnnf)r - 2
i=l
n-l
E D,(Dnf)r -
2(n -1)(Dnnf)r
i=1
from Lemma 2.8, 2°
n-l
= (r(â)f)r - n(Dnn/)T - 2
E D,(Dnf)r + b! - 'Y:-t)/T.
_=1
-
The nat result will permit us to use induction to prove Theorem 2.1.
44
(
Proposition 2.1
,p
= Q(n -
l, r(â1)JTi') + n dT2 Q(n - l, /ri')
,,-1 d
+2 D. dTQ(n -l, /ri') - (.,.! - "'!-1) g(n -l,!Ti .).
g('1l - l, (r(â)f)Ti .)
E
i=1
Proor: A consequence or Lemmas 2.8 and 2.9 •
We can now see that the use oC induction will allow us to handle the fust term oC the
right hand side oC the equation in Proposition
~.1.
H we Collow the strategy as explained at the beginning of the section we need to know the
eft'ect or some dift'erential operators on the other terms of the integrand oC the Cunction given
in equation 2.5.
Lemma 2.10
n-l n
[6(e)j1/6
II II Isinh(e. -
HjW
i=1 i=1
= z [6«()J1 /6
II rr 1sinh«(i -
n-l n
n-l n
HiW ~
~1~1
L: coth«(. - Hi)'
~1~1
Proor:
~ [6«()j1/1
n-l n
il II Isinh«(i -
HiW
,=1 ;=1
n-l n
= [6«()]1/6
II II 1S~h(ei -
Bi)la
i=1 i=1
(E coth«(. -
n
Eil + z
E coth(E. i=1
i~.
ni campanion result:
45
Hi» •
Lemma 2.11
Ei=l ~ [6(H)]I/I-l
ft-l ft
II II Ilinh«(. -
H;)I'
.=1 ;=1
ft-l
ft-l
ft
=Z [6(H)]1/I-l II TI Ilinh«(. •=1 ;=1
ft
L
H;)I- ~
coth(H; - (,) .
,=1 i=1
Proor:
ft-l
[6(H)]l/l-l
ft
II II Isinh«(i -
Hi)l-
i=l ;=1
ft-l ft
= [6(H)]1/6-1
II II Isinh«(i -
HiW
.=1 i=l
«1- 6) ~ coth«(; - (le)
ft-l
+ z E coth(Hi ,=1
Ie~;
(.».
The next lemma gives us a relation between the efFect of .d (in the variable H) and the
effect of .dl (in the variable (). The need for this is clear from Proposition 2.1.
Lemma 2.12
,,-1 ft
A
[6(H)]l/l-l [6«()P/6-1
II II 1sinh«(. -
H;W
i=l ;=1
ft-l
= .dl [6(H)]l/6-1 [6«()]I/I-l
ft
II II 1sinh(ei -
Hi)l-
i=1 ;=1
+[6(HW/6- 1 [6«()]1/I-l
ft-l "
II TI 1sinh(ei - H;)I-
i=l ;=1
.(2z(z - (6/2
" coth«(p -l»(EE
H;) coth«(, - H;)
;=lP<'
ft-l
- EL coth«(, -
H,,) coth«(, -
,=1"<'
....
46
H,» +n(n - 1)(z(1 - 6/2) + 1 - 6».
We recall that A is the Laplace-Beltrami operator for the space Pos(n, F) (varia.ble H)
(~
and that 61 is the Laplace-Beltrami operator for the space Pos(n -l, F) (variable
e).
Proof: As a consequence of Lemma B.2 in Appendix B, we have
"-1 "
II II 'Iinh«(i -
Al [6(e)j1/6-1
HiW
i=1 ;=1
= [6(e)]1/6-1
"-1 "
II II 'sinh(e, -
H;)I-
a=1 i=l
n-l
(z2
ft
E L coth'(e. -
Hi) + 2%2
EL sinh-'(e. -
L E coth(e. -
Hu) coth(e. - Hv)
.=1 u<v
.=1 i=l
n-l "
"-1
Hi) - %(~ - 6)
n
L E coth(e. -
Hi) coth(et - Hi)
.=1;=1
;=l.<t
- 2)
+(n -l)(n
6
(6zn+2(n-1)+2-6(3zn+2(n-l)+2»)
-z
and
"-1 "
II III sinh(ei -
A [6(H)j1/6-1
HiW
a=1 ;=1
"-1 n
=[6(H)]1/6-1 II III sinh(ei -
H;)I"
;=1 i=1
n n-1
(z'
LE
;=1.=1
n
(e. - Hi) +2Z2:E E coth«(1' -
coth2
n n-l
-z
E L sinh-'(e. -
n-l
Hi) - z(2 - 6)
i=1 .=1
+
Hi) coth(e, - Hi)
;=11'<'
E E coth(e. -
Hi) coth«(. - HII)
.=1 1I<i
n(n -1)
6 (6z(n-l)+2n+2-6(3z(n-l)+2n+2»).
We will also need the following result:
Lemma 2.13
III
6(g(tr H) . /(H»
and 9 are lufficienUy differentiable then
=
ng"(tr H) . /(0) +2g'(tr H) .
n
L
i=1
(
(~
41
8
BH. /(H) +g(tr 0) . A/(H).
,
-
Proof: Straightforward.
l'
V.
fi
Finally, the main result of thÏl chapter:
Theorem 2.1 If f E C:,(A) then
Ag(n,/iH) = g(n,r(A)/i H).
Proof: In view of Proposition 2.1, Ü we use induction (fOT n = 1 there is nothing to prove),
it is enough to show that
A
[6(H)]t-1
l
n-l n
F(e, T)
[E:HJ
II II Isinh(e, -
H;)II-l [6(e)]t d(
,=1 ;=1
1 ', [Al + n :;'2 + 2 ï: :t, ~T - (-r! - 'Y!-1)]
J[E.HJ
,=1
T=tr
= [5(H)]t-
'0,
H
n-l n
FU,T)
II 111 sinh(ei -
H;)II-l [5(e)]t cIe
1=1 ;=1
where [e : H]
= {e: el > Hl> e2 > H2 > .,. > Hn-2 > en-l > Hn-1},providedF(e, T) that
is sufficiently difFerentiable and vanishes at
00
rapidly enough. Lemma
~,6
ascertains that
g(n -1'/ri') satisfies these conditions.
The strategy, again, is to compute
A [5(H)]t- 1
l
[f:BJ
n-l n
F(e, T)
II II 1sinh(e, -
H;W [5(e)]t de
,=1 ;=1
when R.z is large enough, using mainly integration by parts, Lemmas 2.10, 2.11, 2.12 and
2.13. We aim to arrive at a form which is analytic for Rz > (6/2 - 1) - 1/2. Byanalyticity,
our transformations will then be valid for z = 5/2 - 1.
.ft.
,
1
48
We Itart by uling the reluIts we mention above and the fact that 41 il lelf-adjoint with
relpect ta the measure 5(e) t:te.
!! Il,,inh(e. -
n-l n
A [5(H)]I-l i€:H] F(e, T)
,p
Hi)'· [5(e)]1 de
8 d
]
1. [â 1 + ft dT 2 + 2 ~ 8~. dT + n(n -1)(%(1- 5/2) + 1- 6)
J[t.H]
1=1
~I
= [5(H)]I-l
n-l
T=trH
n-l n
II II 'sinh(ei -
·F(e, T)
HiW [5(e)]1 t:te
.=1 i=1
[t E
+ 2%(% -
(6/2 -1»[5(H)]I-l'
-EE
coth(e. - HII) coth(e. - HI)]
J[t:H]
i=l,<q
coth(e, - Hi) coth(eq - Hi)
.=111<1
n-l n
·F(e, tr H)
II II 1sinh(ei -
H;W [6(e)]1 de
;=1 ;=1
= [5(H)]I-1 '.
J[(.H]
n-l n
·F(e, T)
[al + d~2 + ï: :~.~I ~ +
n
2
i=1
II II 'sinh(ei -
i=1 i=1
n(n -1)(%(1- 6/2) + 1- 6)]
Hi)'" [6(e)]1 t:te
+ 2z(z - (5/2 -1»[6(H)]I-l [,
tE coth(e, - Hi) coth(e
l[t:H] i=I,<1J
n-l
.F(e, tr H)
T=trH
lJ -
Hi)
n
II II 1sinh(ei -
HiW (6(e)]1 de
.=1 i=1
n-l
-,J[(:H] .=1
E I:
coth(e. - HII) coth(e. 11<1
·F(e.lr H)
ilfi
[(.H]
n-l n
·F(e, T)
1
Isinh(e, - Hj)l' [SUl]! dt
, [41
= [6(H)]I-l JrJ'
~ + 2 -18
d
+ n dT2
~ 8 t . dT
+ n(n -1)(%(1- 5/2) + 1- 6)]
1=1
('
~I
II II 'sinh(éi - Hi)I· [6(e)]! tJe
;=1 i=1
(
H,)
49
T=trH
-
+ 2z(z -
(6/2 -1»[6(H»)1-1 [(
J[€:B)
t 2)1 -
i=1 p<.
coth(e" - e.,)
n-1 n
.(coth(e" - Hi) - coth(e. - Hi»))F(e, tr H)
II II 'sinh(ei -
i=1
Hi)'" [6(e)]!
Ile
i=1
n-1
- J[€:B]
( E
E(t - {.oth(R, 11<' ,=1
il B
.FU, Ir Hl
= [6(H)]t-
loinh(e. -
l [
1
Al
•
[€.B]
n-1 n
tJ.2
Il(]
1
8 d
8 t . dT - n(n -1)(z2 +6 - 1)
,,-1
1=1
T=\rH
liil
Bi)'" [6(e)]l de
i=1
i=l
+ 2z(z -
H; JI' [6(eJ]l
+n dT2 +2 ~
II II 'sinh(ei -
·p(e,T)
H.)( coth(e, - B.) - coth(e, - H,»)
(6/2 - 1»[6(H)]1-1 [-
1
t ,,<.,
L coth(ep - e.,)
J[e:H] ;=1
n-1 n
·(coth(ep - Hi) - coth(e., - Bi»F(e, tr B)
II II 'sinh(ei -
Bi)l" [6(e)]l
i=l ;=1
+(
n-1
E Lcoth(B, -
,=1"<'
J[€:B]
e,
.F( Ir HJ
ilB
loinhU. - Hill' [6( Œt
l [
= [6(H)]1-1
B,,)(coth(e, - Bl) - coth(e, - B,»
Al
[e:B]
Il(]
+ntJ.2- + 2"-lad
E - - - n(n dT2
i=1
8ei dT
1
1)(z2 +6 - 1)
T=\rH
n-1 n
·p(e, T)
II II 1sinh(ei -
B;W [6(e)Jl de
i=1 i=l
+ 2z(z -
(6/2 - 1»[6(H)]l-1 [-
1
t
L coth(ep -
J[€:H) i=1 ",,".,
n-1 n
·F(e, tl H)
+
l
II II 'sinh(ei -
HjW [S(e)]! d(
i=1 i=1
n-l
LL
coth(B, ,,~,
[€:Hl ,=1
H,,) coth(e, -
50
n,,)
e.,) coth(e, -
Bi)
de
(
·F( (, Ir H)
= [6(H)]f- 1
ï( B1
,w,«(; - Hi )1' [S( ()Jt
'.
J[(.H]
n-1 n
.F(e, T)
,p
[ Al + n dT2
II II 1sinh(ei -
II(]
d
n-1 {J
+ 2 ~ (J~. dT - n(n -1)(z2 +6 1=1
\'1
]
1)
T=trH
H;W [6(e)]~ Il(
1=1 ;=1
+ 2(z - (6/2 - 1))[6(H))l-1
r~~q
E'
J[(:Hl
n-1 n
·II II 1sinh(ei -
H;W II( -
:t
L coth(HI -
~1~1
(F(e, t.r H) coth(ep - eq )[6(e)]J)
flOp
HIt)
.~
· :H. (fr/'H) F«(, Ir H)
TI lll'inh((; - H;ll' [S«(]]t d()] .
Both sides are equal and defined provided ft: is large enough (say ftz> 2). This is in
n-1 n
order to malee sure th&t F(e, T)
II II 1sinh(ei -
Hj)llI is zero on the boundary of [e: Hl to
i=l ;=1
allow our use of integration by parts.
Both sides are analyh!: function of z in the domain ft:
> (6/2 - 1) - 1/2. Rence both
sides are equal in that domain by analytic continuation. In particular, for z
have
n-l n
F(e, T)
11111 sinh(ei -
H;)I!-l [5(e)]l de
1=1 j=l
which allows us to conc:lude the proof oC the theorem.
(~
{
51
= 6/2 -l, we
Chapter 3
The heat kernel for pose n, R)
3.1
Pos(n, R)
We will concem ourselves with the real case hence assume that 6 = 1.
n
We recall that r
= r(H) = 11H11 = ( L Hl
) 1/2
.
Je=1
In tms sectioh we will use the "False Abel Inverse Transformn to define a "candidaten
for the fundamental solution to the heat equation in Pos(n,R). Any solution to the system LA -7 2
=
:t
will be transformed into a solution of the heat equation for the sym-
metric space provided it decreases rapidly enough. It would be natural to try the function
Qt(H)
=
e-'l'2t
t- n / 2 exp ( _r 2 j(4t». However it is plain that
g(n, Qt;') cannot be the fun-
damental solution for Pos(n,R) (for one thing, the theory tells us that g(n,Qt;') should
be a smooth function on a and it is easy to see that g(n, Qt;') will not be defined on the
boundary of a+). The next step is to t.-zy with g(n, DQti') where Dis a differential operator
--
~
--
'. '
52
with constant coeflicientlj it willitill be a lolution of the heat equation.
(
The remarks that follow are vaUd for aU n.
8
Propo.ition 3.1 Le' 8(11') = II(88 . - 8 H'> and 1I'(H)
a'
i<i
J
=II(Hi -
Hi)i
i<i
Next we show that the matrices ('Hi-l(Zj» and «2zj)i-l) are row equivalent. The
fust two rows are the same. Assume that, by using row operations on the fust m rows
of (Hi-l(Zj», the two matrices are the same up to the rn-th row (m ~ 2). 1tm(Zj) = (2zi)m
+Pm(z;), the degree of Pm being at most m -
2; Pm is then a linear combinat ion of the mst
m - 1 rowa. We can then transform the (m + l)-th row into the correct form.
We can now conclude the proof of the proposition:
8(11') exp(-r 2 /(4t»
= II(~
- ~) exp(-r2 /(4t»
H
i<j 8
8Hi
i
.
{}i-l
= det« _1)'-1 8H~.:ï) exp( _,2 /(4t»
,
-
det«l 1v'4ï)i-1 'H.i_l (Hi/Vii» exp (
-
(4t)-n(n-l)/4 det(14-1(Hil Vii
_,2 /(4t»
»exp(- -,2/(4t»
= (4t)-n(n-l)/4 det«H;/v'i)i-l) ~xp( _,2 1(4t»
(.
(
53
= (4t)-n(n-l)/4
=
II«Hi - Hill';;') exp ( -,.2/(4t)l)
i<i
(_2t)-n(n-l)/211'(H) exp( _,.2 1(4t».
One can find more information on the Hermite polynomial. in [13].
Proposition 3.2 I/we mte W,(H)
=C e--,2't-n/ 2 8(11') exp ( -,.2/(4t» (0 Il con.dllnt) Uaen
p,(e B ) = g(n, W,i H) ÏI Il .olution ta Uae hellt equlltion for Uae .ymmetric 'pace POI(n, R),
that ÏI, âP,
=
:t
Pt.
Proor: First note that Q,(H) = t- n / 2 exp ( _,.2 /( 4t» is the standard solution (modulo a con-
:t
stant) to the euclidean heat equation on A (LAQ, =
2
Q,). The factor e-'Y , takes eue of
the drift so that r(â)e-'Y"Q, = :te-'Y2'Q, (we recall that r(â) = LA _')'2). The à'ifferential operator 8(11') commutes with r(â) and
a
:t'
Hence, W, il a solution of the equl\tion
r(â) W, = 8t W,.
Now
= g(n,r(â)Wti H)
Ap,(eH ) = âg(n, Wti H)
IJ
=g(n, ItWti H)
= 8t g(n, Wti H)
= I;Pt(e H ).
We make use of
lim g(n, W, - Wti .)
, ..... t
-
-
, -
8
g(n, 8t Wti')
54
t
(see L~mma 2.7).
(~
It will more practical to conlider
We(e H ) = C e-,,'e t- ft / ' t-n(n-l)/'
II (Hi -
Hi) exp ( _1" 1(4t»
i<i
with C
> O. The dift"erential operator 8(11') hu played its role in the proof oC the relult above
and il no longer needed.
3.1.1
From Pos(n,R) to PosI(n,R)
We Ihow that the fundamental solution for the heat equation of one of poseni R) or PosI (ni R)
alloWI us to find the fundamental lolution for the heat equation oC the other.
First a lemma:
Lemma 3.1 Ifw(P) =
.!.n trlogP and â l
Itand" for the Laplace-Beltrami operator of
POSl(n, R), we have
â
Pro of: The operator
Jn :w
1 8'
= d 1 + - -,.
n 8w
is normal to the surface det = 1 and has norm 1.
Theorem 3.1
1° If Pt(e H ) il the fundamental.oltdionlor Posl(n,R) then
p,(e H ) =
2Jm: ~ exp( -(tr H)' 1(4nt» Pc(exp(H - (tr Hln) 1» u thefundamental.olution
for Pos(n, R).
2° If Pc (eH) il the fundamental "ol.tion 101' posen, R) "'en
p,(eH ) = n
(
{
1:
Pt(exp(H + wl» dw
u the fundamental "olution for PosI(n, R).
55
Proof: Using Lemma 3.1, we observe that the "induced" functions Pa and Pa are solutionl
of the respective heat equatkml.
1 E Cc(A)
10 : We will aslume that
and we will use the change of variable
F = H - (trHln) 1, w = (trH)/nj
lim
1
a-o+.+
l(e B ) ~exp( -(tr H)2 1(4nt» Pa(eB-(ll'B/n)') 6(H) dH
vt
1-oo..;t
1..t l(e )Pa(e
1 1..t I( v1 ) Pa(
1
1 l(e
1
/41.t v1) - I( »
1
/41.t
00
a_of
00
= n lim
a_Of
00
= n
e-nv2 /4
n fun
+
n fun
a_Of
F )6(F)dFdw
eF ) 6( F) dF d"
eF+v'C
w
=
..;t"
-00
e-n v'/4dv· lim
a_Of
-00
+
F +W1
..!...e- n w'/(4a)
= n lim
Jlvl~M
,-0+ Jlvl>M
Wefirst point out that P,
F
)Pa(eF )6(F)dF
At
eF
Pa( eF ) 6( F) dF dv
e-nv'
(J( eF+v'i
e-nv'
(J(eF+v'iv1 ) - l(e F)) p,(e F ) 6(F) dFdv.
~
0,1
.+
Ï'a(e F) 6(F) dF = 1 and If(eF+v'ivl ) - f(eF)1 is bbunded
(independently oC t, " and F). We choose t close to 0 such that the fuIt term is close to
2n f!./(I), then choose M large enough so that
V;;
1
Jlvl>M
e- nv2 / 4 dv is small (the choice of M
does not depend on the choice of t). Finally, picking a smaUer value of t ifnecessary, we have
I/(eF+v'"tvl) -/(eF)llmaU for aIl " S M (this il pOlsible since 1 is uniCormly continuouI).
The second and third terms are now arbitrarily close to 0, while the first is arbitrarily clole
to
2vmr 1(1).
20
:
We will assume that 1 E CcC Al) (Al stands for the Abelian component of the 1wuawa
decomposition of SL(n, R)). We will use the change oC variable F = H +œI, w = (tr F)/n
-
56
(tr H
= 0) and the auxiliary function g(e') = !(eF-(trF)/n)i
lim
t ...o+
1.t 1
(e B )jOO p,(exp(H+w/»dw6(H)dH =
-00
lim!.l
' ...0+ n
a+
g(e F )p,(eF )6(F)dF
= ;; g(/) = ;; 1(1).
For ft
=2 and =3, we can make lame explicit computations.
ft
The function Ph
&8
we willsee in the cases
ft
= 2 and
ft
= 3, can be written
&8
E(t,H) is of course the term of interest. In arder to prove that Pe(e B ) is indeed the
fundamental solution of the heat equation we will have to study the behaviour of E(t, H)
when t is close ta 0 and, more specffically, the behaviour of E(t, Vie) when t is dose to
o.
Indeed, if one is only interested in proving that Pt is the heat kemel, it is enough to show
that 0
< E(t, H) < C
and that lim E(t, ViH)
c...o+
=C' (C' 1: 0); the proofs of Theorems 3.2
(case n = 2) and 3.3 (case n = 3) are much the same and make use of Lebesgue dominated
convergence theorem.
3.2
Pos(2, R)
Although this can be found in the literature (for instance [11]) we will give the heat kemel
for Pos(2,R).
We will use the notation a = a(H) = Hl - H2.
(~
(
57
Using equation 2.1 we have:
Pe(e B )
"(2, Wei H)
-
= C e-~e t- 1 t-1 lBI
(00 (FI .[sinh(Fl
-
F2) exp( -,.2(F)/(4t»
Hl) sinh(Fl
-
H 2)]-1/2 dF1
= C e-,,2e t- 3/ 2 exp ( -r 2 (H)/(4t» E(t, H)
where
E(t,H)
.[sinh(Fl
-
t- 1/ 2
00
10
(2z
-
Hl) sinh(FI
-
H2 )t 1 / 2 dF1
+ a) exp ( -z(z + a)/(2t»
.[sinhz sinh(z + a)t 1/ 2 dz.
In the equations above and in what will follow, tr F = tr H.
We know that Pe satisfies the heat equation and that Pe(e B ) ~ 0 for aU H E a+. We
must now prove that Pe is the fundamental solution.
First a useful inequality:
Lemma 3.2 If z and a are non-negalille theflsinh(z +a)
"rict if a
> (z + a) Sinh4
4
-
(the iflequtJlily il
> 0 or z > 0).
Proor: The function g(u)
= .inh~
(g(O) = 1) is strictly increasing for u ~ o.
u
58
(
Lemma 3.3
o <
a
E( t, H) ~ y'2;' [ .inh 0
]1/' ~ .J2;.
Proof: Only the .econd inequality requirea a proof. USÎng Lemma 3.2,
[.
E(t, H)
~
t- l /'
00
1
lBa
·[(Fl
-
t-
(2Fl - Hl - H,) exp ( -(FI - Hl)(Fl
-
-
H,)/(2t»
Hl) (FI - H,) .inha/a]-t/' dFl
l/' [.0::..0] 1/2 L7 (2Ft -
Hl - Hd exp ( -(FI - Ht)(Fl - H 2 )/(2t»
·[(Fl - Hl) (Ft - H,)]-1/2 dFt
=
[~] 1/2 (OO e- c / 2 dz
sinha
.fi
Jo
= Vii [sU:af/2 •
Corollary 3.1 If H#:O then lime_ o+ Pe(e H ) = O.
Proof: An immediate consequence of Lemma 3.3.
Lemma 3.4
Proof:
00
lim E(t, H) =
'_0+
1 (2tz +a) exp ( -z(tz +0)/2)
'-0+ Jo
lim
.[sinh(tz) sinh(tz + oW/2dz
t
(~
(~
59
-
= 10
00
.r~
a exp ( -az/2) [z linhat 1/ 2 dz
Loo exp(-az/2) Z-I/2 dz
a
= v'iiiiiiO
sinha
= [ r/
0
_a_
sinh a
= v,;
[
2
-fl/2 -1/2 d
LOO
e
JI
11
0
a
sinha
r/
JI
= az
2
.
Our use of Lebesgue dominated convergence theorem i. justified by the following upper
bound: for t
~
1,
(2tz + a) exp ( -z(tz + a)/2) [SjD~(tZ) sinh(tz +a)J-I/2
S (2z + a) exp ( -az/2) [z sinha]-1/2
which is integrable over (0,00).
Note that Lemma 3.4 is not crucial to our purpose. It serves, along Lemma 3.3, to exhibit
the behaviour of E(t, H) as t nean O.
Lemma 3.5
1im E(t,../iH)
1-0+
=
~.
Proo!:
1im E(t, ../iH)
&-0+
-
lim t- I / 2
&-0+
1°O
VeIll
.[sinh(Fl
-
-
(2FI - ViHI - ../iH2 ) exp ( -(Ft - v'iH 1 )(Fl - v'iH2 )/(2t»
VtH t ) sinh(Ft
-
v'iH2 )J-l/2 dF1
60
-
ro (2z -
lim
lBl
1_0+
B'l - Hl) exp(-(z - Hl)(Z - H2)/2)
H1»/../i sinh(.;i(Z -
.[sinh(v'i(Z -
-
1 (2z - Hl - H
00
2)
Bl
-../f.i
H2»/v'it 1/2 dz
z =
Fl/../i
exp ( -(z - H1 )(z - H2)/2) [(z - Hl) (z - H 2 )t
112
dz
we computed this integral in the prooC oC Lemma 3.3.
The limit is justified by the Lebesgue monotone convergence theorem.
Theorem 3.2 P, û the fundamental ,ol,,'ion oJ the heal equalion Jor Pos(2, R) (witÎa the
appropriale cOMlant).
1.
ProoC: AlI that remains to prove is that lim
Pt(e B ) J(e B ) 6(H) dH = 1(1) (ü J is
1_0+ .+
"niee").
= C lim e--y2It-3/2!. exp(-,2(H)/(4t» E(t,H) J(e H) sinha(H)dH
1_0+
.+
= C lim 1 exp(-,.2(v'iF)/(4t»E(t,vtF)/(ev'iF) sinh(a(ViF»jv'i dF
1-0+ J.+
H = ViF
= C
1.
.+
exp(-,.2(F)/4)..fii I(I)a(F)dF
= ~ C J(I)
= y'2;: C 1(1)
2
L: Loo
1
00
-00
a exp( -T2/8 - a 2 /8) datIT
e- T2 / 1 tf1'.
00
1
Jo
T
=Ft + F2' a = a(F)
a e- Q2/1 da
= 87rC J(I).
Again we made use of Lebesgue dominated convergence theorem.
61
The only assumptionl on 1 used here are that / E LOO(A+) and that
1 ÏI continuous at
I.
With the previous computations in mind, we choose the conatant C to obtain
~ e--r e,-3/2 exp ( _r 2 /( 4t)) t- 1 / 2 (00 (2Fl
8~
JB,
. exp( -(FI - Hl)(FI
_
-
-
H2 )/(2t» [linh(Fl
~ e-.,.2& ,-3/2 exp ( _r2 /( 4t)) ,-1/2
8~
(3.1)
Hl - H2)
-
Hl) sinh(FI
-
H2 )t 1 / 2 dFl
ro (2z +a)
Jo
. exp( -z(z +a)/(2t» [sinhz sinh(z + a)]-1/2 dz.
3.3
Pos(3, R)
Even with Pos(3, R), the function We in the definition of Pe(e B ) = g(3, Wei H) ia not positive
on aU the domain of integration (which we called D(3)(H) in Chapter 2). Thia is the main
difficulty for n
~
3. However, in the case ft = 3, explicit computations are still possible.
We will use the notation a
=a(H) =Hl -
H2 and P = P(H) =H 2 - H3'
Deflnition 3.1 We will .ay liai" a junceion / on a U odd (willa re,ped to llae Weyl group
W)
il gitlen •
E W,
1(" H) = (detl)/(H) lor all H
the corre,pondin9 property
in a. We will u.e the 14me tema
lor
0/ funcf.iom on A.
We point out here that We(H) = C e--,2e t- n / 2 t-n(n-l)/2
II (Hi -
Hi) exp ( -,.2/(4t» i.
i<i
an odd function. Indeed, it is easy to show that if h i. a Weyl invariant function then 8(~)h
-
becomes an odd function. The reason behind thi. definition ÏI Propolition 3.3 where it il
62
shown that if' h i. an odd Cunction then g(3, hi') corresponds to the integration oC h on a
portion of a+ against a positive kerne1. This ensures that g(3, Wei')
Q(3 , /i H )
=
/Bl
JB.,
r»
> O.
l(exp(diag{(o'(l + (2 - (o,trH - (1- (2])
/00
JBl JEo
3
·[sinh«(o - (1) sinh«(o - (2)}-1/2
II Isinh«(l -
Hi) sinh«(2 - Hi)I- 1 / 2
;=1
sinh(El - (2) d(o d(l 11(2
_
4: /Bl
/00 /00 I( exp ( diag{(o, (1 + (2 JEo
(0, tr H -
JB2 JBl
3
.II 1cosh(2H; - (1 -
el - (2])
1
(2) - COSh«(l - (2)1- / 2 Sinh(El - (2) dei d(2.
i=l
g(3, li H) = 2
L
CuBu,·B
l
min{Fl-F2,FI+F.,-2B2}
1281-Fl -F., 1
l(eF ) (cosh(Fl - F2 )
-
coshzt1 / 2
3
. sinhz
II 1cosh(2H; -
Fl - F2) - coshzl-1/ 2 dzdF
(3.2)
;=1
=
(
JCUBU,.B
J(e F )0(H,F)dF.
•. B = {F: F2 S F3 S H3 S Hl S FI},. being theelement orthe Weyl group permuting the
second and third eigenvalues (see Figure 3.1 below). It is important to note that tr F
The elliptic integral K(m) is defined by the relation
1"/2
K(m) = Jo
(~
(
63
d8
';1 _ m2 sin2 9'
= tr H.
c
Figure 3.1: We use the following coordinates: (Fl! F2)'
In order to achieve some symmetry in our results we introduce the following modified
elliptic integral:
Defi nition 3.2
b-1/2 K( .ja/b) if 0
K:(a,b) =
~ a < b,
{ a- 1/2 K( .jb/a) if 0 ~ b < a.
le is smooth for a
K(a,b)
~
0, b ~ 0 exce1>t on the diagonal. Two immediate properties of K: are
= K(b,a) and X(ca,cb) =
C-
1/2 K(a,b).
Using equation 3.2, Lemma C.I in Appendix C and the function le (a..ld the change
of variable x = cosh z) , the kernel of g(3, fj H), 9(H, F), is then given by the following
equations:
3
9(H,F)
3
= 4K:(sinh,8 ITsinh(FJ-Ht},-sinha IIsinh(Fj-H3 )),
3=1
j=1
64
0(H, F) = 4 A:( - sinha
and if Fz
TI sinh(F; - H3 ), sinh,8 n sinh(F; 3
3
;=1
;=1
Hl»
< H3 (i.e. cosh(F1 - Fz) < cosh(2Hs - FI - Pz» then
0(H, F)
=
3
4X:( - sinha
11 sinh(F; -
3
Ha},sinh(a +,8)
;=1
II sinh(F; - H ».
;=1
2
(3.3)
Now,
Proposition 3.3 If f
Ù
odd the"
9(3,Ji H)
=
fa J(e
F
) 0(H, F)
dF.
Proof: This is clear from equation 3.3 (see also Figure 3.1).
In particular, we have
Pc(e H ) = 9(3, Wci H)
=
la
Wc(e F )0(H,F)dF.
We know that Pt satisfies the heat equation and that pte eH) ~ 0 for aU H E a+. We
must now prove that Pt is the Cundamental solution.
We follow the same program as in the previous section (or as explained at the end of section
3.1). The situation being rather more complicated, we need to prove several preliminary
lemmas.
65
--------------------..............
3
1°
-SÎIÙIar
:1
II smh(Pj -
Ils) - siDh,8
j ... 1
II tiDh(P, -lIt.)
pl
3
::: -1Iinh(a +.8)
3
-a
II (Fj -
II 1iDh(F; -
Hl),
S
:1
Hs ) - ~ n{Fj
jst
-
Hl)
= -(0 + 11) ll(Pz - Hz),
pl
J=<1
,-2(F) - ,.'(H) = 2 rrbl(F.. - Hp) - fIf:t(F. - Hq} (p:/; Il)
Hp - Hq
::: (FI -
Hd' + 2(01 +fi + F,.- H,)(F1 - Hd + 2(Fl -
H2 )(F2
-
H3)'
proftde4 tMt tr F = tr H.
Proof: To prove 1<\ one ean multiply both tida by ~trB and upreu the result as StmlS of
exponential
~.
R.eplaeing, in 1G , Hi by tH" Fj by tF;, dividing both sides by t 4 and letting t tends to 0
3° ean be proven," in I G , by direct eomputationse
The nat few Lemmaa ate to acquaint us a bit more with the modified elliptic integral /C.
Lemma 3.1
! /00
du
2 Jo v'(u2 +1/1 - 61){u2 + max{a,6}"
,(0,6) =
Prool.
AIIUII1e 4
< 6;
,((11,6)
1
=
rI'
ï6 Jo
dB
JI - (a/6) liD.' fi
66
.,.
c.
LemmaS.8
n
;=1
s
l'
i
ï
1°
- liDha
n
;=1
a
liDh(F; - Ha) - liDhll
IiDh(F; - Hl)
a
=- linh(a + Il) II liDh(Fj ;=1
a
1
;=1
;=1
HI)'
a
-a n(F; - Ha) - fJ n(Fj - BI) = -(a +fJ) n(F; - BI)'
j=1
rl(F) - r 2(B) = 2 nl=l(F.
- Bp) - nl=l(F. - H,)
(P:F 9)
Hp-H,
= (FI - Hl)1 + 2(0 + Il +FI - BI)(FI - BI) + 2(FI - BI)(FI - Ha).
provilled
Ua", tr F = tr H.
Proo!. To prove 10 , one can muItiply both lidel by ellr B and exprell the reluIt
al
lums of
exponential tenDI.
Replacing, in 10 , H; by tH;, Fi by tFi, dividing both lide. by t4 and letting t tends to 0
we obtain 20 •
3° can be proven, u in 10 t by direct computationl.
The nut few Lemmu are to acquaint ua a bit more with the modified elliptic integralIC.
Lemma S.T
00
A:(",6) Proo!. Allume 4
l
Jo
du
..j(u2 +14- 61)(u2 +max{4,6}}'
< 6i
=
(.
!2 10
1
rll
v'6 Jo
dB
JI - (4/6) lin 8
2
66
The following two lemmu are going to be u.eful in finding lower and upper bounds.
Lemma 3.8 If mu{a,6} S mu{A,B} arulla - 61 S lA - BI Uaen ,t(A,B) ~ K(a, 6) (~
can he replacttl6,
<J.
We cuI.me here Uaat (a, 6) and (A,B) are in the domain of,t.
Proof: A direct cODlequence of Lemma 3.7.
Lemma 3.9 If a ~ 0 and 6 ~ 0 (a
1: 6J
the"
Proof: With the ob.ervation that mu{a,6} ~ 16 - al, thi. become. an immediate consequence of Lemma 3.7.
Another l'onu of the fonction 1C:
Lemma3.10
~(CI, 6)
67
Proof. Il CI
r
< 6,
~
K:(a, 6)
"
=
1"/2
Jo
1"/2
dl
"'6 - cnin' 8
=Jo
v'6 cot 9 cie 9 d8
V6cot' 9 (6 cie' 9 - CI)
v
=v'6 clc9.
Lemma3.11
Proof. Ilwe refer to Lemma 3.10, the left hand lide il equal to
J.œo 1." L" ap(-(...,+ 62,)/2) .j(v2 - zd(v'
1
dz, dz, cl.
- zi)
0
0
00
10
exp( -(a + 6)v2/2)
fov 10" exp(avfj2 +6v1;2) dVl dV2dv
-
4
=
4 a- 1/ 2 6- 1 / 2 (a + 6)-1/2 Jo
=
4 a- 1/ 2 6- 1/ 2 (a + 6)-1/2
ro e-
li
v
r";a/(aH) r.Jb/(a+b)
2
/2
Jo
Jo
Vi = ../v2 - zi
exp«U~ + ul)/2) dUl dU2 du
(11' - arclin( V6/(CI + 6» - arclin( va/(a + 6»)
= (~)a a- 1/ 2 6- 1/ 2 (a+6)-1/2,
We now will write Pe(e N ) = Ce- 3 e--,2e exp(-r 2 /(4e»E(e,H) (the constant C
lorbing any previoUi cOllJtants) where
E(e,H) =
(.
68
> 0 ab-
-----_.
u--
__... _----
- ._-
----------------
(we used Lemma 3.6, 3°).
As in the previous section, E(t, H) il the term we need to inveltigate.
Lemma 3.12
(~)3
0< E(t H) <
,
-
2
[ ni<i(Hi - Hi) ]1/2 < ($)3.
ni<i sinh(Hi - Hi)
2
Proor: Only the second inequality requires a proof.
E(t,H) S
This is a consequence of Lemmas 3.2, 3.6 (1° and 2°) and 3.8. Hence,
E(t, H) S t- 3 / 2
[
ni<~(Hi -
Hj)
ni<iSlnh(Hi - Hi)
.e
xp
]1/2
·!C(P
[
(Hl
lmax{H3,Hl+H2-Fl}
II (Fi -
Hl)' -a
II (Fi -
H3» dF1 dF2
i=1
ni<~(Hi - Hi) ]1/2 ({HHï
ni<i linh(Hi - Hi)
i<i
_ Hi»-l
. (00 (00 exp(-(I1(Hi _ Hi»-1[ az l
Jo Jo
i<i
'!C(ZI' Z2) dZ 2 dZ I
= (~)3 [ ni<i(Hi - Hi) ]1/2
--
2
11(1;, - Fi)
i<i
3
i=l
t- 3 / 2
00
(_nt-1(F,. - Hl) - nt=1(F,. - H 3 »
2(a +P)t
3
=
1
lH1
ni<i sinh(Hi - Hi)
69
+{Jz2]/(2t))
(
a
The change oC variables uled above wu Zl = Il II (Fi - Hl), z2 =
i=l
a
-0
II (Fi - Ha).
i=l
This bound is going to be used with the Lebesgue dominated convergence theorem.
Corollary 3.2 If H -:F 0 Ihen limt-o+ Pt(e B ) = O.
Proof: An immediate consequence oC Lemma 3.12.
Lemma 3.13
Proof: We use the change of variables zi = Fi - Hi (Z3
= -Zl -
Z2) folIowed by zi
(Jl3 = -YI - Y2)' Again, we reCer to Lemma 3.6, 3°j
lim E(t, H)
t .... o+
=
lim t- 3 / 2
t .... O+
00
n(Zi - Zi + Hi - Hi)
1 101
max{ -~,-Zl}
Jo
i<;
.exp«zf + 2z1(Z2 +0 + P) + 2z2 (Z2 + P»/(4t))
a
'~(sinhp
II sinh(z; + H; -
Hl),
i=1
a
- sinha
00
=
lim
1
1....0+ Jo
1
II sinh(z; + Hi -
;=1
01/t
max{ -lJ/t,-1II}
II(tri - tri
i<i
Ha» dZ 2 dZ 1
+ Hi -
Hi)
.exp«trl + 2rl(tJl2 + a +P) + 2r2(tY2 + fJ»/4)
'~(sinh,8 sinh(trl)/t
II sinh(tri + Hi #;1
- sinha sinh(tY3)/t
nsinh(tri + Hi - Ha» dY2 drl
#3
= II(Hi - Hi)
i<i
[II sinh(Hi i<i
70
Hl),
Hi)t 1 / 2
=tJli
= II(B, - Hj) [II sinh(H, -
Hj)t 1/ 2
i<i
'<j
00
00
00
.10 10 10
exp ( -[a'l
+(J (-'3)1/2)
'~('1I -'3) d( -'3) d'l
=
(v'2i)S [ n,<~(Hi - Hj)
2
ni<j linh(H, - Hj)
)1/2.
The last equality il a consequence of Lemma 3.11.
conve!(~ence
The use of Lebesgue dominated
theorem is justified by the following upper
bound: if t $ 1,
lI( t
. - t . + H. _ B.)
i<i'a
'II,
a
,exp
(_ n~-l(tYli
'~(sinh{J sinh(tyt}/t
+ HII -
Ht)- n:-t(t'II + BII - H 3 »
2(a + {J)t
II sinh(tYi + Hj -
Htl,
#1
- sinha sinh(t1t3)/t
II sinh(t'j + Hi -
H3
#3
$
II(I" - 'jl + H, -
»
Hi) exp(-[a'l - (J'31!2)
'<j
.~(sinh{J'l
II sinh(Hj - Hl)' - sinha'3 II sinh(Hj i~l
= [c5(H)t 1 / 2 11(1"
i<j
j~3
- Yjl + Hi -
Hi) ~(Y1t -Y3).
The last term (without the factor) has integral
-
-
......
71
B3
»
=
{OO
(OO exp(-[OJI1 _ pJla]/2)
k k
II(I'i - 'JI + Hi - HJ)
ici
',('1, -'3) d( -'a) dJll
~ LOO Loo exp(-[O'l o
PJla]/2)
0
II(I'i - 'il + Hi - HJ)
ici
·1'1 - (_,a)I- 1/ 2 d( -,s) d'l
<
ooe
AI in the previoui lection, Lemma 3.13 i. not crucial to our purpole. Its &Ïm, along
Lemma 3.12, il to exhibit the behaviour of E(t, H) as t nearl O.
Lemma 3.14
lim E(t,VtH)
&_0+
Proor:
lim E(t,ViH) =
&-0+
lim t- 3/ 2
&-00+
B
1°O
l./i l
II(Fï - FJ)
./iBl max{v'iB,.v'iBl+./iB2- Fl} ic;
.exp(_n2=1(F; -
ViHl ) - nf=l(Fï - v'iHa»
2(.;iHl
-
y'iHa)t
3
.,(sinh(VtP)
II ainh(F; - -liH
l ),
;=1
3
- sinh( v'ia)
=
lim
t'0
&-00+ JBI
exp
l
II ainh( Fi -
;=1
v'iHa» dF, dFl
BI
mu{B,.Ba+ B 2-L.}
II(Li - Li)
ici
- n2=1(L, - Hs »
(_nf-l(L, - Hl)
2(0 + P)
3
.,(sinh(VtP)/v'i
II ainh(Vt(L; - H1»/v'i,
;=1
72
a
II IÏDh(v'ï(Li -
-linll(v'ïa)/v'ï
;=1
H:t»/v'ï)dL,dL l
Fi = ..fiLi
00
=
1
JB
1
l
BI
II(Li - Li)
-{BI.Bl+Ba-Ll} i<i
.
exp
(_n:=l(Li - Hl) - n:-l(Li - Ha»
2(a +~)
a
s
i=1
;=1
.~(~ II(L; - Hl),-a II(L; - Hs»dL,dLl
-
(IT(Hi - Bi»-l
i<i
00
. 1"0 1
Jo Jo
exp(-(II(Bi - H;»-I[a%l + 11.1,1/2)
i<i
'~(Zh%2)dz2dZI
=
(v'21r)S
2
3
The change of variables uled ab ove was
%1
=P
II (Li - Hl) and
3
Z2
= -a IT (Li - H 3 ).
;=1
;=1
The last equality is again a consequence of Lemma 3.11. The limit is justiJied by Lebesgue
monotone convergence theorem.
Lemma3.15
3
Proor: We mite T =
E B".
"=1
1+
=
11<.(Hi - Hi) exp ( _,,2/4) dB
i L:
, J
Ta 12
e- /
00
1 J!OO Q~(a +
P) exp ( _[a2 + (a +P)2
73
+P2]f12) dadP dT
(-,
=
2J3;" 1"0 1"0 aP(a + P) exp( _[a 2 + (a + p)2 + P2]/12) dadP
3
Jo Jo
= - 4~ LOO LOO ap(:a + :p) exp(_[a2 + (a+p)2 + P2]/12) da dtJ
= - 8~ LOO LOO atJ :a exp( -[a2 + (a + tJ)2 + tJ21/12) dadtJ b,. Iymmetry
= 8~ LOO 10
= 8~ 10 10
00
00
00
tJ exp ( _[a 2 + (a + p)2 + tJ2]/12) dadtJ
00
tJexp(-(a+tJ/2)2/6-{J2/8)dadp
00
1 1 tJ exp ( _z2 /6 - {J2/8) th d{J
= 8J3r
3 Jo JfJ /2
1 e-.2/ 8 r· {J e-fJ2/1 d{J th
= sJ3;"
3
Jo
Jo
00
=
32v'31r
3
00
1
Jo
Theorem 3.3 Pe
e-.2/8(1_ e-e2/2)dz = SV21re
u the fundamental
.olution 01 the heat equation for Pos(3, R) (with the
appropriale con.'ant).
Proo!: AIl that remains ta prove il that 1im /. Pe(e B ) l(e H ) 6(H)dH = 1(1) (if
e...o+ .+
f
il
"nice").
1.
1im
Pe(e H )/(eB )6(H)dH
a... o+ .+
1.
-
C lim e-.,2, t- S
exp ( -,,2(H)/(4t» E(t, H) l(e H ) 6(H) dB
e...o+
.+
-
C lim /. exp(-r 2 (VtH)/(4t» E(t,v'iH) f(ev'iB)
-
C
, ... 0+
.+
1.+ exp ( -r2(H)/4) (~)S 1(1) Jl<,,(Hi -
.
6(~~)
dH
t
Hi) dH
by the Lebesgue dominated convergence theorem
-
1611'5/2 C 1(1).
74
replacing Nb,. v'iH
....
-
,
The only assumptions on / used here are that /
~
'
e LOO(A+) and that /
is continuous at
1.
With the previous computations in mind, we choole the conltant C to obtain
1& 2 t- 3 e-"'C exp ( _,.2 /( 4t)) t- 3 / 2
16r /
/00
lBI
.
exp
l
BI
mu{B3.BI+B,-FI}
II(Fi -
i<;
F;)
(_n~=l(Fi - Hl) - n:-l(Fi - H 3 »
2( a +,8)t
3
·K:(sinh,8
II sinh(F; -
3
Hl), - sinh 0
;=1
II linh(F; ;=1
Hs» dF:a dFl .
(3.4)
We can aIso write
PC( eH) = 16!5/2 t- 3 e-"'C exp ( _,.2/( 4t»
.t-3/:a [
/00 /Bl II(1'i- F;) exp( _ n~-l(Fi - Hl) - n~-l(Fi 2(0 + ,B)t
i<;
lBI lB,
3
.,(sinh,8
1
+
B3
-00
(3.5)
II sinh(F; -
3
Hl)'- sinha
;=1
/B, II(Fi
JB3 •'<'1
;=1
»
H3 dF:a dFl
_ F;) exp(_n~=l(.Fï - Hl) - n~-l(Fi - H 3 »
2(0 + ,8)t
3
.K:(sinh,8
II sinh(F; -
H3 »
II ainh(F; -
3
Hl)'- sinha
;=1
II sinh(F; -
H3» dF3 dFl ].
;=1
If we use the change of variable Zl = FI - Hl, z~ = F:a - H2 (H3 - F3 = Zl
+Z2)
the mat integraI of equation 3.5 and the change of variable Zl = H3 - F3 , Z:a = H2
(FI - Hl = Zl
+ z:a) in the second, we see that Pt il a symmetric function of the roots a
,8.
....
75
-
in
F:a
and
Chapter 4
The asy:mptotic expansion of the
heat kernel of Pos(n,R)
We have leen in Chapter 3 that, for t near 0 and H clole to the origin, the behaviour of
the heat kemels for Pos(2, R) and Pos(3, R) is much the same as that of the euclidean heat
kemel, as we should have expected. We will now investigate what happens when t
~ 00.
More exactly, we give the asymptotic expansion of E(t, H) in negative powers of..;ï (we recall
that Pe(e H ) = C e-,.2e t-(climX)/2 exp ( -r 2/(4t» E(t,H». The relativeease in obtaining these
asymptotic expansions is a credit to the usefulness of our formulas.
4.1
About asymptotic expansions
Sorne standard terminology (see for example [13] and [12]).
(.
(
76
00
Deftnition 4.1 We will mte I(t) x ~ Gm t- m G.t t
JI
lim tM
•
1-00
l/(t) - ""
LJ
il, lor each fized M,
-+ 00
m=O
Ga
Gm t-m )
=
m=O
o. ""
tin. t- m
LJ
ù then ,aid to be the G.tymptotic ,erie, ezpcln-
m=O
,ion al 1 G.t t tencü to infinity.
This definition is also used in the case where t is a complex variable in a given wtbounded
domain R.
00
Note that if I(t)
=L
tln.t- m in the interval
m=O
(C,oo) with C ~ 0 (or in an wtbounded
00
domain R orthe complex plane) then I(t):::::
L
tin. t- m
as t
-+ 00.
m=O
The expression I(t) ::::: h(t) + g(t)
I(t} - h(t} x
g(t)
t
00
L
am t- m as t
-+ 00
will be interpreted as
m=O
tin. t- m
as t
-+ 00.
m=O
00
Theorem 4.1 Il c E R, I(t) :::::
00
E tin. t- m and g(t) x L
m=O
I(t) + c· g(t) :::::
1°
6m t-'" G.t t
then
-+ 00,
",=0
00
E (élm +
C· 6m )
t-
m
a, t
-+ 00.
m=O
00
I(t) . g(t):::::
2°
m
E E a"6",_,, r
m
tu
t -+
00.
m=O.=O
3°
If I(t) ÏI analytic in a neighbourhood
01 infinity le.. the point at infinit y the)"
Ga
I( t) =
L
Gm t- m , that ù, the ,erie, converge, in the given regÎon and I( t) ù equal ta the
m=O
,um al the ,erie, in that region.
Proo!: This is rather standard (see for instance [12]) •
00
Corollary 4.1 If 1 ÏI G.t in 3° al the theorem then l'(t) :::::
E (-m) tin. t- m m=1
--
-
the gÎven region.
77
1
tu t -+ 00
in
Proor: Straightforward.
00
Lemma 4.1 If f(a,t) x ~ Cm(a)t- m tu t
00
-+ 00
and A/(a,t) ~
m=O
L
dm(a)t- m tu t
-+ 00,
m=O
uniformly on compact aubaeu of A+, fhen Aem = dm for all m. We allume here that the
functionl f(a,t), A/(a,t), Cm(a), ACm(a) and dm(a) are continuoul on A+.
Uniformly on compact subsets of A+, in this context, means that for any given compact
subset of A+, the limit in Definition 4.1 will be uniform on that compact subset.
Proor: Fix M ~ O. Pick any he Cc(A+). We recall that A is a self-adjoint operator with
respect to the measure 6'(a) da. The convergence being uniform on the support of h, we have
=
1 limtMCE
+
L+ t~ t
=
t~~
M
o
A+ t_oo
h+
+ ,1!..~L+
m=O
dm(a)r m -8f(a,t»h(a)5(a)da
M
M
(I(a, t) -
~o Cm(a) t- m) Ah(a) 6'(a) da
M
t
M
(~o dm(a) t- m -
8/(a, t)) h(a) 6(a) da
M
tM(I(a,t)-
~o cm(a)t- m)Ah(a)6'(a)da
M
=
+
t~~h+ tM(~o dm(a)t-m_Af(a,t))h(a)6(a)da
lim
1
t_oo A+
t_oo
J1A +
lim
L
= lim
M
=
t_oo
m=O
t M (Af(a, t) -
M
(tM(~ dm(a)t- m - ~ ACm(a)t-m))h(a)6(a)da
L.J
L.J
m=O
m=O
(1J + (dm(a) - ACm(a» h(a) 6'(a) da) tM-m.
A
1 (dm(a) - Acm(a» h(a) 6'(a) da = 0 for 0 :5 m :5 M.
JA+
h is arbitrary, this implies dm = ACm for 0
(
L
Aem(a) t- m ) h(a) 6(a) da
m=O
M
This cannot happen unless
(
M
:5 m :5 M. Since this is true for every M
78
~
Since
0 the
-
conclusion Collows.
Our asymptotic expansions will be rather in terms of ..fi.
We want the asymptotic expansion of the heat kemels for Pos(2, R) and Pos(3,R). We
will proceed the following way:
We will prove
1°
Pe(t: H ) X C e-'Y2t (-/i)-nCnH)/2 exp( _,2 I( 4t»
00
L
(_l)m bm(H)( Vt)-2m-n(n-l)/2,
m=O
uniformlyon A+ (n = 2 or n = 3). From now on, it will be understood that the asym~totic
expansions are "as t ..... 00".
n
We recall that , == r(H) =
11H11 = ( E Hl
) 1/2
.
JI=1
.
We note that dimPos(n,R)
1) .
= n(n+
2
Since Pt satisfies the heat equation, 2° gives us an asymptotic expansion Cor APt. This
allows us to use Lemma 4.1.
-
79
4.2
Pos(2, R)
We have (see equation 3.1)
-.!...e--Y"t- 3/ 2 exp(_r2 /(4t»t-1 / 2
811'
/00 (2Fl - H1 -
lBs
. exp( -(FI - H 1)(F1 - H 2 )/(2t» [sinh(Fl
Again, we will use the notation a
= a(H) = Hl -
-
H2)
Hl) Sinh(Fl - H2)]-1/2 dF1.
H2.
First a lemma:
Lemm.a 4.2
Proo!: Dnly the second inequality requires a proof;
{OO (2F1
lBs
_
00
Hl - H2 ) «FI - Hl )(Fl - H 2»N [sinh(Fl - Hl) sinh(Fl - H 2)t 1 / 2 dFl
+ a»N [sinhz sinh(z + a)t 1/ 2dz
=
10
:5
.f2 e-a:/2 Loo (2z + a) zN (z + a)N-l/2 Vi + z + a e-~/2 [sinhzt l / 2dz
:5
.f2 (1 + a)N1"l e- a / 2 Loo (2a: + 1) zN (1 + z)N e-~/2 [sinhZ]-1/2 dz.
(2z + a) (z(z
1
We made use of Lemma 2.3: fi u > 0 then -2 eU - U1 < sinh u
-
+u -
< eU _u_
-
The following asymptotic expansions are intuitively easy to obtain.
(
(
80
l+u
•
.
oC>
Theorem 4.2
8~ e--r2 a(...ji)-3 exp(-,.2/(4t»
......
Pa(eH ) .....
Ë (-1)mbm(H)(v'i)-2m-t,
m=O
~ e--r2 t (...jit 3 exp ( _,.21 (4t»
H
......
:tPa(e ) ......
811'
00
.L
(_l)m ( --y 2 bm (H) + (m + 1)bm- 1(H)
m=O
where
(6-2
uniformly over A+. Furthermore, we have 0 $ hm $ Cm (1 +r)m (1
= b_ 1 = 0),
+a) e-0l/ 2 •
Prao!: We will start with the first expansion. Naively, this cornes to expanding the term
More precisely, we need to show that for each M
tends ta 0 as t tends ta
~
0,
00.
Now,
(...ji)2M+1
$
--
-
~
811"
'l e'Y2t (...ji)3 exp(r 2 j(4t» Pt(e H ) - ~
M
L
(_l)m bm (H)(v'tt 2m - 11
811' m=O
(00 (2Fl _ Hl - H2 )( yt)2M+1lt- 1/ 2 exp ( -(FI - H 1 )(F1
lB
l
81
-
H2 )/(2t»
(
·[sinh(FI
Hl) sinh(FI
-
-
H 2 )t1 / 2 dFI
~ 8~ bm+t(H)(Jit 2 •
The last term tend. to 0
le-· -
M
E
m=O
(-l)mzm
,1 ~
m.
&1
zM+l
(M
t -+
00.
Note that we used the fact that for z
~
0,
)' (from the Lagrange remainder theorem).
+1 .
That the convergence above is uniform over A + is a direct consequence of Lemma 4.2
where we give bounds on the coefficients bm (we make use of the fact that 1 + r and 1 + Ct
1
are orthe same order).
The second part of the theorem is a consequence of the fust part and of the corollary of
Theorem 4.1. Indeed, if R is the region {t : Rt > C} (C > 0) then, for t E R,
~ C- 1/2. In view of our upper bounds in Lemma 4.2, this implies that Pt (eH) is analytic
as a funetion of t in the domain R (actually, this shows that Pt ( eH) is analytic in t in the
domain Rt
> 0).
As a consequence of Lemma 4.1 and of the remarks at the end of Section 4.1, we know
that the coefficient bo(H) ofthe asymptotic expansion of Pt(e H ) in the statement of Theorem
4.2 satisfies the equation
4[~
(.
82
-
Furthermore,
1im bo(H) =
lim
B-O
H-O
l'JO (
JBI
2Fl - Hl - H,) [sinh(Fl - Hl) sinh(Fl - H2 )t l / 2 dFl
lim (00 (2z + a) [sinhz sinh(z + a)]-1/2 dz
=
CII-O+
1"0
2 Jo
=
Jo
Z
.,2
.inh z dz
= 2"'
Our use of Lebesgue dominated convergence theorem is justified by the upper bound
(2z + a) [sinhz sinh(z + a)r l / 2
~ 2 (z + a) [sinhz sinh(z + a)t l / 2
=
2 (z
+ a)J/2 [sinhz
sinh(z
+ a)/(z + a)t 1/ 2
(we may assume a < 1).
Renee,
tPo is ealled the Legendre function.
We now state
Theorem 4.3
Pt(e H)
~ 8~ e-..,2 t (v'i)-3 exp(-r 2/(4t»
2
.(~ (v'i)-l tPo(e B ) +
00
E (-1)mbm(H)(v'it 2m - l ),
m=l
Pt(e H) ~
--
~ e-..,2
t
(v'it 3 - 1
e:xp(-r2/(4t»q,o(e B ).
83
Proof: The fust part i. proven in the comment. preceding the theorem and the second part
i. clear Crom the expre••ion for Pc and 60 (u exp ( -(Ft - Hl)(F2 - H 2)/(2t» $ 1).
4.3
Pos(3, R)
We will proceed along the lame linel u in the previous section.
Again, we will use the notation a = a(H)
= Hl -
H2
and p
=/3(H) =H2 -
Ha.
We have (see equation 3.4)
\
2
1611' 1
e-"'t-3 exp(-r2 /(4t»t- 3/ 2 }H
(00
l
.e
xp
lm.x{B~.Hl+B2-Fl} II(F. BI
i<i
Fi)
(_n:-l(Fi - Hl) - nf=I(Fi - H3 »
2(a +P)t
3
·K:(sinhf3
II sinh(F; -
;=1
3
Hl), - sinha
II sinh(F; -
;=1
»
H3 dF2dFt.
We want to prove a lenuna corresponding ta Lemma 4.2. The situation heing somewhat
more involved, we give a preliminary lemma.
n:-l(Fï - Htl- n:=I(Fi - H3 ) There e:r.istl a polynomial
Lemma.
4 3 Let p (H , F) __
2(a+f3)
.
q =q«Fï - Hi)isn) luch that p(H,F) $ (1 + r) Iq«Fi - Hi)iSn)l.
Proof: Using 3.6, 3° we write:
The resuIt is then an easy consequence oC the fact that a S r and f3 $ r (actually r and
the largest of a and
f3 are of the same order).
84
Lemma 4.4 Le' p( H, F) be 41 in LemmCl4.3. There eN" a con.dan' 0 < Cft
< 00 depending
f
f
onl, on n ,uch Cha'
'-lB.
JB.
II(Fi - F;)[P(H,F)]"
max{B,.B.+B3-F.} i<;
S
.X:(sinhP
II sinh(F; -
3
Hl), - sinha
;=1
~
II sinh(F; -
H3 )) dF2 dF1
;=1
Cn (1 + r)n (1 + a) (1 +P) (1 + a
+P) exp ( -(a +P)).
Proo!: It will be convenient to use the form given in equation 3.5. Also, because of the
symmetry involved, it suffites to consider only one of the terms. We will use Lemma 2.3
repeatedly.
fOC fHI
11 JI
Hl
II(Fi -
B, i<;
F;) [P(H, F)]n
3
·K:(sinh,8
II sinh(F; -
3
Hl), - sinha
1['
f- fHI
'2 JB JB2
H3 )) dF2 dF1
;=1
~1
~
II sinh(F; -
[ 3Il sinh(F; - 9 ]-1/2'
dF dF1
;Q(Fi - F;) [P(H, F)]n - sinh(a +,8)
I
2)
from Lemmas 3.6 and 3.9
i (1 + r)n 1 1'" (Zl - Z2 +a) (Zl +2Z2 +P) (2z1+ Z2 +a +P) Ig(z)l"
00
=
·[sinh(a + P) Sinh(Zl
-
+ a) sinhz 2 sinh(Zl + :1:2 +,8)t 1/ 2 dZ 2dZ l
85
2
{
s
C(l+f')ftexp(-(a+p))Jl::;P
.Loo L'" ";ZI + 0 (ZI +2z 2 + P) (2~;1 + Z2 + 0 + P)
'Vl +ZI + 0 Ig(z)1 exp ( -ZI $
C(l t.)"
ZI + Z2 ] -1/2
.
z2/2) [ sÏDhz2 1
dZ2 dZ I
+ZI + Z2
_(-(QtP»)(ltQ)(1+P)(1+QtP)~
.Loo 1'" ";ZI + 1 (ZI +2Z2 + 1) (2Z1 + Z2 + 1)";2 + Zl ..;'1 + ZI +Z2
'lg(z)1 exp(-ZI - Z2/2)[Sinhz2(ZI + Z2)t 1/ 2 dz 2 dz 1.
Ha ~ 1,
tt
a+{J
a + P
Jor~ Jof·
";ZI
+1 (ZI + 2Z 2 + 1)(2z1 + Z2 +
'lq(z)1 exp ( -ZI - z2/2) [sinhz 2 (ZI
v'2
$
Loo 1
00
1)";2 + ZI VI
+ZI +
Z2
+ Z2)t1/ 2 dZ 2 dZ I
..;'ZI + 1 (ZI + 2z 2 + 1) (2Z1 + l:2
+1)..;'2 + ZI ..;'1 + Zl +Z2
'lq(z)1 exp ( -Zl - z2/2) [sinhz 2 (ZI + Z2)t 1/ 2 dZ 2 dZ l
$ C.
Ha$I,
JI +
a +P
a +P
Jo1 Jora ";ZI + 1 (ZI +2Z2 +
00
1)(2:1:1
+Z2 + 1) ..;'2 +ZI ";1 + Zl + Z2
'Ig( z)1 exp ( -ZI - z2/2) [sinh Z2 (ZI + Z2)t 1/ 2 dZ 2 dz 1
$
t ! : ; ..ra f J.'
P
v'i1TI (,,1+ 3)(201 t
2) v'2 t
·li(zl)1 e- 1I1 [sinh( OZ2)/0 (ZI + aZ2)]-1/2 dZ 2 dZ l
S
v'2 LOO
LI ";Zl +1 (ZI + 3) (2z1 + 2) (2 + ZI)
86
"1
"'2 +01
-
s c.
This allows us to conclude.
Agam, the asymptotic expansion. are intuitively easy to obtam
Theorem 4.4
(
,-3 exp( -,.2/(4t» f: (_l)m 6 (H)(v'it2m- a ,
1 e-"Y't ,-3 exp( ,.2/(4t»
16",5/2
-
Pt(e B ) :::: 16!5/2 e--y
2
t
m
m=O
8atRt(eB).....
.......
00
. }:(_l)m( --y 2 6m(H)
m=O
2
+:
7
+ (m+ 2)6m - 1 (H)
6m _. 2(H)) (v'it 2m - 3
where
hm(H) =
2-71
m.
B1
max{H3.Hl+H3-Fl}
n(fi - F;)(_n~-l(Fï - Ht) - ~~-l(F. - H3 »m
,<;
a + f3
3
·A:(sinh,8
II sinh(F; -
3
Hl), - sinha
;=1
(6-2
II sinh(F; -
H 3 ») dF2 dF1
;=1
= 6_ 1 = 0),
uniformly over A+. Furt.hermore, we have
o ~ hm
~ Cm (1
+r)m (1 + a) (1 + (3) (1 +a + (3) e-(Q+t').
Proo!: The proof is much the same as that of Theorem 4.2. The difficult part wu to prove
Lemma4.4.
.....
Again, as a consequence of Lemma 4.1 and of the remarks at the end of Section 4.1,
87
.(
we know that the coefficient 60 (H) of the asymptotic expansion of Pt in the statement of
Theorem 4.4 satisfies the equation
Lemma 4.5
Proor: The fundamentalsolution Pt or the heat equation is smooth on A if we define it to
be Weyl invariant (see for instance [9]). This implies that 60 (H) has to be smooth and Weyl
invariant on
9..
Thus, 60 (H) is a constant multiple of the Legendre function. In particular it
is continuous. Hence, üwe write 60 (H)
= 60(a,fJ), then 60 (0) = a lim
(lim
.... O+ 13 .... 0+
60 (a,,8».
The manipulations and the use of Lebesgue dominated convergence theorem that follow
are much the same as in the proof of the previous result. We use the change of
suggested at the end of Chapter 3.
60 (0) =
sinha sinh(Zl + a +,8) sinh(Z2 + fJ) sinh(ZI + Z2» dZ 2 dZ l
+
lim lim
a_Of
13.... 0+
Joro JoI~ (Zl -
r
0 .... 0+
(.
(
+ a)
sinh,8 sinh(Zl
+fJ + a) sinh(z2 + a) sinh(Zl + :1:2» dZ 2 dZ l
Jo1 Jora (Zl -
:1:2 + a) (:1:1 + 2Z2) (2z 1 + :1:2 + a)
00
= !2 lim
Z2 +,8)(ZI + 2Z2 + a)(2z1 + Z2 + fJ
88
var~ables
-
.[sinha sinh(Zl
+
+a) sinhz2Sinh(Zl + Z2)t1/ 2 dZ 2 dZ l
lim..;p Joro Jo/1 (Zl -
lim
a-O+ 11-0+
PZ2 + P) (Zl + 2PZ2
Z
·K:(sinha sinhz 1 sinh(fJP- P 2) sinh(Zl
sm;fJ sinh(Zl
Joro Jor
= !.2 lim
a_O+
(Zl - aZ2
Jo/00 Jor
lim O·
a_O+
sinh(Zl
=
11'
foOO
+ PZ2 + {J +a),
+fJ + a) sinh({Jz2 + a) sinh(Zl +PZ2» dz 2dz 1
+ a) (Zl + 2a2:2) (2Z1 + aZ2 +a)
sinha .
sin(aZ2) .
. [ - a - sinh(Zl + a)
a
sinh(Zl
+
+ a) (2Z1 + fJZ2 +P+ a)
Zt{Zl
+ a)(2Z1 + a)
+ a) sinh a
sinh:r:t} dZ 2 dZ 1
fol Z~ [Z2 sinh2:r:l}-1/2 dZ l dZ 2 + 0 =
Note that if b > 0 then K:(O, b) =
+ aZ2)]-1/2 dZ 2 dZ 1
i
11'
.2.
~ ",4 = :5.
b- 1 / 2 •
Theorem 4.5
1 e-'Y2t t- 3 exp(-r2j(4t»
1611'5/2
Pt (eH) ....,
,....
5
.(: (.Jt)-3 t/>o(e H) +
00
E (_l)m bm (H)(Jit 2m - 3),
m=l
11'5/2
2
64 e-'Y t(VI)-9 exp(-r 2 j(4t»t/>o(e H).
Pt(e H ) S
Prool: The fust part is proven in the remarks preceding the theorem and the second part is
.
t
••
n1c_ (F; - Hl) 3
clear from the expression for P, and b (as exp(0
89
-1
2( a
n~-l(Fï - H3)
+ {J)t-
)
< 1).
-
(
Chapter 5
Estimates
We mention in the Introduction our intention to prove that Philippe Anker's upper bound,
C e-..,'t t-(dimX)/2 exp ( _r 2 j(4t» 4>o(H)
II
(1 +Q
+ t)(m
Cl
+m
2C1
)/2-\
(5.1)
exEEt
is vaUd for the symmetric spaces Pos(2, R) and Pos(3, R) and that a constant multiple of
that upper bound will serve as lower bound (as usuaI, r
= r(H) = 11H11). We will endeavour
to fulfill this program in this chapter. We stress that while various upper bounds for have
been given for the heat kemel of Pos(3, R), there were no good lower bound existing (to our
knowledge). In that respect, Theorem 5.4 is the most important result ofthis chapter.
For each of the spaces under consideration in this chapter we have a subsection titled
"Final estimates". We write
with q and C chosen in luch a way that
limt-+oo Y,(H) =
90
1. This is possible due to the
asymptotic expansion of e.,2t exp(r21.(4t» P,(H) (as t _ (0) whic:h has C t-" ~ as fust term
(as shown in Theorem 4.3 and 4.5).
There we express our resuIts concerning the asymptotic expansion and estimates on Pt ( eH)
in terms of Ve(H) and add some further estimates to our collection.
5.1
Pos(2. R)
For the space of positive definite matrices, Anker's bound (equation 5.1) can be written in
simpler terms. Indeed, the roots of the spaces pose n, R) are aU indivisible and of multiplicity
one.
For the space Pos(2, R), the bound is given by
C e-,.2t t- 3/2 exp(-r 2 j(4t» tPo(H) (1 + Q
As before,
Q
= Q(H) = Hl -
+t)-1/2.
H2 •
This section will consist of two parts. We will fust prove that with the appropriate choice
of positive constant C, the above bound constitutes a lower bound for Pt(e B ). We will then
show t nat, with a difFerent choice of positive constant C, the abov~ also can serve as an upper
5.1.1
-
Lower bound
91
Theorem 5.1 There eN"
Il
con.dan' C
> 0 luch tha'
Proo!: We will give bounds on the function E defined by the relation
Pt(e B ) =
:1r e-.,2t t- 3/ 2 exp( -,,2/(4t)) E(t, H).
We will fust show that
(5.2)
The following lemma about the sinh function will be used:
Lemma 5.1 Ha> 0 and 0 < z
J
-
<
1 then
-
sinh(z + a)
sinha
<
4--.
z+a
a
Pro of: Simple calculus.
We have
E(t,H)
'[sinh(Fl - HI) sinh(Fl - Hl
~
J
Q
inh t- 1/ 2 fB1+t (2Fl - Hl - H2 ) exp ( -(FI - Ht)(Fl - H2 )/(2t))
s Q
JBl
·[(FI
-
Ht
,
(
(
-
Hl) 4 (FI - Hl
+ a)t 1 / 2 dFl
1 ~ l(Ha)/e -1/2 -œ/2
-=---nh
z
e
dz
SI
Q
0
-2
:S 1 + a we conclude from equation 5.3 that
E(t H) > !
- 2
+ a)t 1/ 2 dFl
Jar
sinh a Jo
z-1/2 e- z / 2 dz
.
92
from Lemma 5.1
(5.3)
..
I!. t
~
1 + Q we eonclude !rom equation 5.3 that
e- l / 2
E(t,H) ~ -2-
-
e-
We now recall Lemma 2.3: il u
1 2
/
{ifi
la(1+ )/e z
inh
CII
a
1
V.:na
~ 0 then
i
-1/2
0
d:r:
J(1 + a)/t.
eU 1 : u
~ linh u ~ eU 1 ~ u'
AllO, if 0 S u
~
U
l, then u ~ aiDhu ~ 3u and, if u ~ l, then e ~ ainhu ~ eU.
4
Uaing Lemma 2.3, the two lower bounds Cor E(t, H) given above prove equation 5.2.
In Chapter 4, we saw that E(t,H) ~ Ct- 1/ 2 60 (8) (see Theorem 4.3). In Theorem 4.2,
we state that bo{H) ~ C' e- a / 2 (1 + a).
If we put aU this together, we have:
C e- a / 2 (1 + a) (1 + a
+ tt 1/ 2 ~ E(t, H) ~ C' t- 1/ 2 e- a / 2 (1 +a).
Ifwe multiply these inequalitiea by t l / 2 and take the limit as t tends to infinity, the resuit
is
o e-a / 2 (1 + a)
$ bo(H) ~
C'e- a / 2 (1 +a).
We know that bo is a constant multiple of the Legendre function tPo,
the proof of the theorem if we refer to equation 5.2.
We have also prov"!d the following corollary:
Corollary 5.1 There ezilta Cl
> 0 and O2 > 0 4uell that
93
80
we can conclude
for "" H E a+.
We mention tbis result in the Introduction (equation 0.13). It wu proven in the general
case
by Phlippe Anker.
5.1.2
Upper bound
Theorem 5.2 There ezilu a cORltant C
Proo!: The case t
5.1. The case t
5.1.3
~
~
< 00 ,uch thut
1 + Q is a consequence of Lemma 3.3, Lemma 2.3 and of the Corollary
1 + a is easily derived !rom Theorem 4.3 •
Final estimates
We now write the heat kernel the following way:
(5.4)
This formulation is chosen in IUch a way that lim Ve = 1. Tbis il made possible by the
t-oo
asymptotic expansion of p,(e B ) (more precisely, the fust term ofthat expansion).
We now give our results concerning the asymptotic expansio~ and estimates on pte eH) in
term.s of Vi(H) together with some new estimates.
Theorem 5.3 Let Vi be
1°
Vi(H) xl +
(JI
in equation 5.-1.
00
I: (_l)m em(H) t-m where 0 ~
m=l
po,Îtive cORltant depending on m "'one).
--
94
em(H) ~ Am(l
+ r)m
(Am il a
(
+ ,,)/t
ilt ~ 1 +,. where 0 < A < 00.
2°
0 S 1- V,(B") S A(l
3°
B Qe(H) S Ve(H) S C Qe(H) ilt S 1 +" where 0 < B
Q,(H) =
< 00, 0 < C < 00
and
«1 + a)/t)-1/2.
Proof:
1° This is a direct consequence of Theorem 4.2, Theorem 4.3 and Corollary 5.1.
2° We can write
so Ve(H) S 1 sinee ~o(H)
= 22 JBl
ro (2F1 -
Hl - H2 ) [sinh(F1 - Hl) sinh(F1
'Ir
-
H2 )t 1 / 2 dF1
as we proved in Theorem 4.3.
We introduce the function ,peu) =
1-e-u
u
. It is easy to observe that for u
~
0, 0 :5
t/1(u) S 1. Now,
[1 - Vi(H)] ~o(H) = (22 /(2t»
1
'Ir
·",«Fl
1
,
rIO (2Fl -
lHl
-
Hl - 9 2) (FI - Hl )(F1 - H2)
H1 )(F2 - H2 )/(2t» [sinh(Fl - Hl) sinh(F1 - H2 )t l / 2 dFl
S (22 /(2t» CtC1 + a)2 e- a / 2.
11'
ci
The last inequality is a consequence of Lemma 4.2. To conclude the proof of part 2°, it
is eno\4gh to remember that the values of
(1
+ a) e- a / 2
~o(H)
(see Corollary 5.1). Note that the upper bound is true for all t but is of little
interest if t is close ta O.
{
are between two constant multiples of
95
This derivation was shown to
UI
by Carl S. Herz in a personal communication.
3° is a straightforward consequence of TheoretnJ 5.1 and 5.2.
5.2
Pos(3, R)
As in the previous section, we can state Anker's bound (equation 5.1) in simpler terma.
For the space Pos(3, R), the bound is then given by
As before, cr
= o(H) = Hl -
H 2 and fj
=fj(H) = H2 -
H3'
The strategy is the same as in the previous section. We will first prove that with the
appropriate choice of positive constant C, the above bound constitutes a lower bound for
Pe(e H ). We will then show that, with a different choice of positive constant C, the above
also can serve as an upper bound for Pt(e H ).
5.2.1
Lower bound
Theorem 5.4
Pe(e H ) ~ C e-oft t- 3 exp ( _r 2 j(4t)) tPo(H)
.(1 + 0
+ t)-1/2 (1 + p +tt 1/ 2 (1 +cr + fj +t)-1/2.
Proof: We will give bounds on the function E defined by the relation
-
96
We will mst show that
(
E(t,H) ~ C e-(Q+~) (1 + a) (1 + fj) (1 + ct +~)
·(1 + ct + t)-1/2 (1 + ~ + t)-1/2 (1 + ct + fj
Since the problem is symmetric in a and
that
p :5
+ ttl/2.
(5.5)
f3, we may assume without 10ss of generality
a. We will make frequent use of the tact that a and a
+ fj are of the same 01'deT
(fj $ a).
'l'he constant C that follows is generic and can be computed.
As one might ha.ve expected, the method we will use to fmd the lower bounds will depend
on how large
/l,
{3 and tare with respect to each other. We will proceed case by ca.se.
Suppose mst that 1 :5 a and 1 $ p.
t
3/
2 E(t, H)
2:
{oo r/2
Jo Jo
.10
(Zl - Z2
00
+ a)(Zl + 2z2 +(3)( 2Z1 +Z2 + a + P)
exp ( -(Z~
+ 2z 1 (Z2 + a +(3) + 2Z2(Z2 + (3))/(2t))
·[u2 + sinha sinh(Zl + a
+ fj) sinh(z2 +P) sinh(Zl + :1:2)]-1 / 2
.[u2 + sinh(a + P) sinh(Zl + a) sinh Z2 sinh(Zl + Z2 + ~)]-1/2 dud:l: 2 dZl
00
2:
10 fo
Q/2
.10
00
(a/2)f3(ct + P) exp (
-(:I:~ + 2z 1(:l:2 + a + f3) +
[u2 + C exp(2a + 2p +:1:1 + Z2) Sinh(Zl
·ru2 + C exp(2a + 2f3 + Zl + Z2) e
= C exp ( -(a + P)) ap (a + P)
97
21
2:1:2(Z2 + f3»j(2t»
+ :1:2)]-1/2
sinh Z2]-1/2 du dZ 2dZl
.10 100./2 exp(-(z~ + 2Z1(Z2 + a +~) + 2z 2(Z2 + ~»/(2t))
00
• exp ( -(Zl
+ z2)/2)
We us~d Lemmas 3.6 and 3.7, equation 3.5 and the change of variables suggested at the
end of Chapter 3.
Case 1 S IJ
~
~ t:
a
10 100./2 exp ( -(z~ + 2Z1(Z2 + a
00
+ fJ)
+ 2Z2(Z2 + ~»/(2t)) exp ( -(Z1 + z2)/2)
. fooo[(u 2 + sinh(Zl + Z2))(U2 + elll1 sinhz2)rl/2dudz2dz1
~ c 11 10 /
1 2
exp(-(1 + 2(1
. Loo [( u 2 + sinh(Zl
~
C
Case 1 S fJ
10
00
LI
+ 2t) + 2(1 + t»/(2t»
+ Z2»)( u 2 + ellli
sinh Z2)r 1/ 2 du dZ 2 dZ 1
fo1/21°O[(u2+Sinh(zl+z2»(u2+ellll sinhZ2)]-1/2dudz2dz1'
~ t ~
a:
10./2 exp ( -(z~
+ 2Z1(Z2 + Il: +~) + 2Z2(Z2 + fJ»/(2t» exp ( -(Zl + z2)/2)
.Loo [(u + sinh(Zl + Z2»(U
2
2
+ elll1 sinhz 2)r 1/ 2 dudz 2 dZ l
~ ~ (00 (Q/2 exp ( _(Y~ + 2Y1Y2 + 2tYl + 2Y2(Y2 + ~»/(2t» exp ( -(YI + Y2)/2)
a +,.., Jo Jo
. LOO [(u2 + Sinh(Yl + Y2»(U2 + ell1
YI
-{
.
= -a+~
t- Z},
lai L
a+,..,
~ C - -t a
1 2
/
0
0
Y2
sinhY2)r 1/ 2 du dY2 dYl
= Z2
exp( -(y~ +2Y1Y2 + 2Y1
98
+ 2Y2(Y2 + 1»/2) exp ( -(Yl +Y2)/2)
{
LOO
p:
~
Case t
L
a/2
exp(
-(z~ + 2211(212 +a + P) +2z 2(Z2 +P»/(2t)) exp ( -(211 + 212)/2)
oo
.L [(u2 + Sinh(ZI + Z2))(U2+e
12
2! C 101
10 /
.10
>
c
00
exp (
lllt
SinhZ2)tl/2dudz2dzl
-(2:~ + 2z1 (2:2 + + P) + 22:2(Z2 +,8»/(2t»
Q
[(u2 + 211 + Z2)(U2 +
Z2)tl/2d~dz2dzl
t2
l(a+IJ)/t 1{J/(2t)
~(a +~) Jo
Jo
exp(-(~: + 2Yl(Y2 + 1) + 2Y2(Y2 + 1»/2)
.Loo [Cu' + t(a~P + ~»(u2 + t ~)rl/2 dudY2dYl
a+p
YI = -t-2:1, Y2
t2
,8
= ïZ2
Il 11/2 1 [2 t
] -1/2
2 t
> C /3(0+/3) Jo Jo Jo (u +~(rl +Y2»(U t ~Y2)
du dY2 dYl
=
00
Cv'i1 r~: Il) 1.' 1.'1' 1.œ [(v' +(y, + y,))(u' + y,) J-'/' du dy, dy,
u=
#
v,
Suppose now that 0 < p ~ 1 ~ a.
J
J
t 3 / 2 E(t, H)
~
Loo foa (211 -
Z2 + a) (Zl + 2Z2 + P) (2211 + 212 + a
· exp ( -(Z~ + 2z 1(Z2 + a
· foOO [u2 + sinh a
+P)
+ P) + 2z 2(Z2 + P»/(2t»
Sifu't(Zl + a
+P) Sinh(Z2 +P) sinh(Zl + Z2)]-1/2
.[u2 + sinh(a + P) Sinh(ZI + a) sinhz 2 Sinh(Zl + 212 +P)]-1/2dudz 2dz 1
~
C e- a (a/2) (a +P)
· exp( -(Z~
(
(
r
l
foOO fo1/2 (Zl + 2Z2 +P)
+ 2211(212 + a +,8) + 2:1:2(:1:2 + ,8»/(2t» e- lIIt / 2
99
.10
00
-
,
[U2 + sinh(Z2 + (J) Sinh(ZI
= C e- (a/2) (a + (J)
Q
+ Z2 +{J)1- 1 dudz 2 dZ I
10 L1/2 (ZI + 2Z2 +(J)
00
. exp ( -(zt + 2z 1 (Z2 + a +~) + 2Z2(Z2 + (8»/(2t» e-e1/2
Case aS t:
fooo fol/2 (ZI + 2Z2 + P) exp ( -(zt + 2Z1(Z2 + a +P) + 2Z2(Z2 + fJ»/(:lt»
·e- 21/2 [sinh(Z2 +P} sinh(ZI
+Z2 + {J)t 1/ 2 dZ 2dZ I
~ fooo fo /'l (ZI + 2z 2) exp ( -(z~ + 2z 1(Z2 +2) +2Z2(Z2 +1»/2) e- Z1 / 2
1
Loo [sinh(Z2 + 1) sinh(ZI + Z2 + l}t 1/ 2 dZ 2 dz 1 .
Case 1 S t
~
a:
·e-21/2 [sinh(z2 +P) sinh(ZI
~ !a Jo1 Jor
00
/2
(2,12) exp ( -Cyl
.[sinh(Y2 + 1} sinh(,Il
Case P S 4t, t
1 10
00
--
!.,
112
~
+ Z2 + P)J-l/2 dZ 2 dZ l
+ 2,11 (112 + 2) +2112(112 + 1»/2} e-J/1/2
+ 1/2 + l)tl/~ d1/2 d1/1
1:
(ZI +2Z2
+ P) exp ( -(Z~ + 2z1(Z2 +a + P) + 2Z2(Z2 +fJ»/(2t»
. e-e1/2[sinh(Z2 + P) sinh(ZI
+ z2 +fJ}t 1/ 2 dZ 2 dZ I
100
(~
Case 4t $
f3:
We Collow the same pattern u in the proof oC Lemma 3.12 except that we restrict the
integration to the domain D
E{t,H) ~ t- 3 / 2
= {F: IFi -
Hil $ 1} and we make use oC Lemma 5.1.
1 II(Fi _ F;) exp ( _ n~=l(F,. - Hd - n~=l(FII - H3 » ,
JD
2(a + .8)t
i<i
3
.K:(64 6(H)/(o (0
+ f3»
II (Fi -
Hl),
;=1
3
- 64 6(H)/(8 (a +
fJ» ;=1
II (F; - H »dF dF
3
1
2•
Hence,
E(t,H)
~
C t-3/2 [
ni<i(Hi - H;) ]1/2
ni<isinh(Hi - Hi)
. 1 II(Fi _ Fi) exp(_flLl(F,. -
JD i<;
3
Hl) - nl-l(FII - H 3 »
2(a +fJ)t
3
·K:(fJ II (Fi - Hl), - 0
II (Fi -
i=l
;=1
101
H3» dFl dF2
.....
=
[ ni<!(H, - Hi) ]1/2 (IT(Hi _ Hj»1/2
ni<i Sinh(Hi - Hi)
i<;
.Jf) exp( -(P Zl +
Q
The change of variables used above wu Zl
z2)/2) K:(Zl' Z2) dZ 2 dz l ·
- Hl)
= Pt TITI~-l(F·
~-.( H~ _ H .) ,
a<,
l
Z2
TIj=l(Fj - H 3 )
•
t TIi<i(Hi - Hi}
-Q
=:
It i8 not difficult to Bee that {F : 0 S
Zi
Si} c jj.
Hellce,
.....
102
,
(
=
l.JifE
+
L..fofE
o
e··
>
o
2
e- u 21 2 L'
0 (-exp(u (cosB/cost/l)2/2)-1)dt/ldu
u2
12
L~/2-'
0
(exp(u2 (sin8/ sin t/l)2j:i.)-1)dt/l du
max{1.../o/ .-.,/, 1.' exp(U' - ' 1/2)d"du- If; l,
1.../0/
[12-'
o
eo·
exp(u sin2 C/2)dt/ldu- Ii
2(1""/~
max{ 10 ~,
L exp( -u:l 8in2 B/2) dt/l du - lii 9,
u2
t
/
2
2
C)}
t
{93A f"-'
l..ftfE l'
o
= max{-.1&lu9
1
cose
= max{-.Bsme
.iD,
0
0
1va;IJ co., L
0
0
0
'K/2 - B .
l.ff
2
fi
·-('K/2-0)}
2
(JI
e- v2 / 2 dv
0
'2('K/2-B)}
f.K
y-O,
2
e- v/2 dt/ldv-
e- v2/2 dv - " 2
sin('Ir /2 - 9)
~
e- v2 / 2 dt/ldfJ -
fr/2 -'
LVf
Ii
exp(-u2 cos 2B/2)dt/ldu-
0
t
-Ii
2
('Ir/2 - O)}
0>0
The last inequality is due to the fa ct that if 0 :5 , ~ 'Ir / 4 then
2
v2/2
etlfJ
c > o. This is clearly true if ( = 0 (we set
s~, 1
1
2
and -.1sm'
-Ii' ~
e- v 2 12 dv -
0
.
fi-2 is decreasing to sm(w"/4)
1 e2
1
Remark: Actually, we proved that whenever 4 t 5
E(t, H) ~ C (1 + a)t/2 (1
v2
12 dfJ -
0
P Oir 4 t :5 a
s~, = 1, if' = 0)
li > o.
2
we have
+fj)1/2 (1 +a + fj)1/2 exp( -(a + fj».
Finally, we assume that 0 < P :5 a :5 li we have to show that E(t, H) ~ 0 (1
103
+ tt 3/2 •
1ft ~ 1, then. t 3/ 2 E(t,H) ~ E(l,H). E(l, H) il bounded away from 0 since it is continUOUI and str:ctly positive on the compact set {(a,,8) : 0
~
,8
~
a
~
l} (E depends ol'Jy 011 a
and 13).
Case 0
< .IJ ~
t3 / 2 E(t, H)
a
~
LI LO&
~
4t, t
~
1:
(Zl - Z2 + a) (Zl
+ 2~2 +,8) (2Z1 + Z2 + a + {3)
· exp ( -(z~ + ~Zl(Z2 + a +,8) + 2": 2(Z2 + {3»/(2t»
·Loo [u
l
+ sinha Sinh(Zl + a + {3) Sinh(Z2 +,8) Sinh(Zl + Z2)t1/ 2
.[u2 + sinh(a +13) sinh(Zl
=
fol
la
(Zl - Z2 + a)(ZI
+ a) sinh Z2 sir.h(Zl + Z2 + {J)t l / 2 dudz 2 dZ 1
+ 2z 2 +,8)(2Z1 + Z2 +
a +
f3)
· t:xp( -,8(Zl + Z2)/t) exp ( -aZl/t) exp( -(z~ + 2Z1Z2 + 2z~)/(2t»
·L [u +
uo
2
sinha Sinh(Zl
+ Il: + f3)
sinh(z2 + (3) sinh(ZI
+ Z2)t1/ 2
·[u2 + sinh(a +(J) sinh(Zl + a) sinh Z2 sinh(ZI + Z2 + {J)t 1/2 d.c2 dZ 1
~
C
fol 10& (Zl · exp( -(z~
00
·10
Z2
+ a)(ZI + 2z 2 + (J)(2~! + Z2 + a
+ ~ZlZ2 + 2z~)/{2t»
[u2 + a (Zl + a) (Z2 + P)(Zl + Z2 + {3)t 1 / 2
.[u2 + a (Zl + a) (Z2 +
= C
f3) (ZI + Z2
+ f3)t 1/ 2 dZ 2 dZ l
va 11 . ( (YI - Q1I2 + a) (YI + 2aY2 + 13) (2711 + aY2 + a + 13)
· exp ( -(Y~
....
+ P)
+ 2aYIY2 + 2a2Yn/(2t)
[(YI
104
+ a) (aY2 + 13)(711 + aY2 +13)]-1/2 dY2 dYl
(
~
C
ra fol fol hl1 - ClY2 +a) (YI + 2aY2 +f3) (2Yl +0112 +
·e-.,f/(2t) [(YI
C fol
1
(YI - 0Y2
·e -~ /(2t) [(YI
/1/";;
~ c..fi Jo
r/";; Jor/
·e- Ill /2 [Y2
11 10
+ a) (Y2 +1)(Yl + a(Y2 +1»] -1/2 dY2 dYl
r (v'iYl + a(1- Y2» (v'iYl +2aY2)(2vtYl + 0(Y2 + 1»
> c..fi Jo
C t3/ 2
+ a) (YI +2aY2 +{3) (2Yl + aY2 + 0 +{3)
J'l
·e-.,f/2 [(v'iYl
>
+{3)
+ a) (aY2 + a)(YI + aY2 + a)t l / 2 dY2 dYl
1
>
0
/
2
(.;t'YI
+ c:j1/2 (v'iYl +2aY2) (2v'iYI + a(Y2 +1»1/2
+ 1]-1/2 dY2 dYI
2
1
+ a)(Y2 +1)(v'iYI + a(Y2 + 1»t l / 2 dY2 dYl
y~ e- IIU2 [Y2 +1]-1/2 dY2 dYl.
The case 4 t $ {3 or 4 t $
0
has already been settled.
This concludes the proof of equation 5.5.
In Chapter 4, we saw that E(t,H) ~ Ct- 3 / 2 bo(H) (see Theorem4.5). In Lemma4.4, we
found that bo(H) $ C' exp ( -(a +{3» (1 + a) (1 +P) (1 +0
+ P).
If we put all this together, we have:
C e-(a+#9) (1 + 0) (1 +{3) (1 + a
·(1 + a
$
+(3)
+tt 1/ 2 (1 +{3 + t)-1/2 (1 + ct + {3 +t)-1/2
E(t, H) $ C' t- 3 / 2 e-(a+~) (1 + a) (l + (3) (1 + ct + (3).
If you multiply these inequalities by t 3 / 2 and take the limit as t tends to infinity, the
(
(
105
._ .. _--_._----_._-----------,
'
.....
-
result is
C e-(a+,g) (1
+ a) (1 + P) (1 + a +~)
$
bo(H) $ C' e-(a+,g) (1
+ a) (1 + P) (1 + a + P).
We know that bo is a constant multiple of the Legendre function
q,o, so we can conclu de
the proof of the theorem if we refer to equation 5.5.
We have also proved the following corollary:
Corollary 5.2 There ezist& Cl > 0 and C 2 > 0 s'Uch that
Cl e-(a+,g) (1 + a) (1 + P) (1 + a
+~)
$ q,o(H) $
C 2 e-(a+,g) (1 + a) (1 + (J) (1 + a
+ ~)
for all H E a+ .
5.2.2
Upper bound
Theorem 5.5
'(1 + a
+ t)-1/2 (1 + ~ + t)-1/2 (1 + a + ~ + t)-1/2.
Proo!: As a consequence of Corollary 5.2, it is enaugh ta show that
E(t, H) $
C e-(a+,g) (1 + a) (1 + P) (1 + a +~)
·(1 + a
+ t)-1/2
(1
+ ~ + t t 1/: (1 + a + ~ + t)-1/2.
As in the previous subsection, we may assume without 10ss of geneI'ality that {J $ a.
-
4..1-
106
The eue t S 1 + fJ i. a cOlllequence oC Lemmu 3.12 and 2.3.
We DOW prove the eue 1 +fJ S t S 1 + a. The eue 0 < fJ SaS 1 ÏI also cODsequence of
Lemmu 3.12 and 2.3.
Suppose 1 < {J and 1 < a. A. Doted in the prooC oC Lemma 4.4, if we Ule the Conn given
in equatioD 3.5, we Deed to comider oui,. ODe oC the terma (we will Dot make use here of the
U.umptiOD that (J Sa). We will Collow the lame approach.
E(t, H)
=t-
+12) where
00
10 10° (Zl -
S
Il
3 2
/ (Il
z2
+a)(zl + 2Z2 +fJ)( 2Z1 + Z2 + a +fJ)
· exp( -(zf + 2z1(Z2 + a
s
C (1 +a)(1 + fJ)(l + a
+fJ) +2Z2(Z2 + fJ»/(2t»
+P) exp (-(a +P»
00
·10 loG (Zl +1) (Zl + 2Z2 + 1) (2z1 + Z2 + 1)
· exp( -(z~ + 2z1(Z2 + a
+fJ) +2Z2(Z2 +(J»/(2t»
Now,
00
00
10 10
(Zl
+1) (Zl +2Z2 + 1) (2Z1 + Z2 + 1)
. exp( -(zf + 2Z1(Z2 + a
S C
OO
Lo
+(J) +2z2(Z2 + fJ»/(2t» e-·' e-·.,/2 [sÏDhz2]-1/2 dZ 2 dZ 1
Loo op( -Zl(a +(Jl/t) [.inhz2]-1/2 dZ2 dz1 < C --a.
t
0
.
-
107
a+,..,
Suppo:se 0 < {J S 1 ~ a.
Il
S
a
fooo Io (Zl -
Z2
. exp ( -(zf
+ a)(zl + 2Z2 +P)(2z1 + Z2 + a + P)
+ 2z 1(Z2 + Q +(J) + 2Z2(Z2 + (J»/(2t»
.[sinh( a -1- 13) sinh( ZI
a
S e- (1 + a) (1 + Q
+ a) sinh z2 sinh( ZI + Z2 + p)t 1 / 2 dZ 2 dZ I
+ 13) Loo foa (ZI + 1) ";ZI + 2z 2 + P
+ Z2 + 1) exp( -(z~ +2z 1(Z2 +a + 13) +2Z2(Z2 +(J»/(2t» esinh(ZI +Z2 + Pl] -1/2 d d
• [S inh Z2
2
Z2 ZI'
ZI + Z2 +
.(2Z1
Z1
/
2
Q
#J
Now,
00
10 foa (ZI + 1) ";ZI + 2z2 +13 (2Z1 + Z2 + 1)
, exp ( -(z~
'nh
'S
[I
< C
00
(00
Jo
1
Jo
Z2
+ 2z 1 (Z2 + a +13) + 2Z2(Z2 + 13»/(2t»
e- œ1 / 2
Sinh(ZI + Z2 + (J)] -1/2 d d
2
Q
Z2 ZI
Zl + Z2 + #J
exp ( -zl(a + fJ)/(2t))
.J2Z
:t.
1 (Z2 + 1) dZ dZ S
2 I
sinhz 2
C~.
a
+
#J
The situation is unfortunately not symmetric in a and {J sinee we make use of a
~ 10
00
12
1
11
(Zl - Z2
• exp( -(z~
+(J)(ZI +2z2 + a)(2Z1 + Z2 + + P)
Q
+ 2z1(Z2 + a +P) + 2Z2(Z2 + a))j(2t))
.[sinh(a + P) Sinh(ZI
~
C e- a (1 + a)(l + a
+(J) sinh Z2 sinh(ZI + Z2 + a)rl/2 dZ 2 dZ I
11
+13) .0loo Jo1
108
v'Zl
+ 13 (ZI + 2Z2 + 1)
~
l,
(
Now,
10 10" ';Zl + P(Zl + 2z2 + 1) (2Z1 +Z2 + 1)
00
. exp ( -(z~ + 2z 1(Z2
linh(Zl + P) inh ] -1/2 d d
A
1 Z2
Z2 Zl
Zl + #J
00
1 /1 exp( -Zl (0 +P) /t)[sinh Z2r 1/ 2 dZ 2 dZ l S C
•
S
+ a +P) + 2Z2(Z2 + 0»/(2t» exp ( -(Zl + z2)/2)
[
c Jo Jo
~.
a + #J
These estimates finish the proof of the case 1 +PSt S 1 + a.
The case t
5.2.3
~
1 + 0 il easily derived from Theorem 4.5 •
Final estimates
As before, we follow closely what we did in the section 5.1 for the case n = 2. The difficruties
inherent to the case n = 3 have, in this situation, aIready been dealt with.
We write the heat kemel the following way:
(5.6)
Again, this formulation is chosen in such a way that lim V,(H)
' ..... 00
We will now give estimates in terms of the Cunction
Theorem 5.6 'Let vt be al in equation 5.6.
109
v,.
= 1.
00
V,(H) xl + L (-l)mem(H)t-m where 0 S em(H) S Am{l
+ ,,)m
(Am ia
Cl
m=-l
po.ititJe con.dGnt depending on m Glone).
s A(1 + ,,)/t
i/t? 1 + r where 0 < A < 00.
2°
0 S 1- l'teH)
3°
B Q,(H) S Vi(H) S C Q,(H) ift S 1 +" where 0 < B <
Q,(H) =
00,
0<C
< 00
Gnd
Il«l +fJ)/t)-1/2, the product heing ttJken otJer the roob "1 auch thGt 1 +"1 > t.
Proo!:
1° This is a direct consequence of Theorem 4.4, Theorem
4.~
and Corollary 5.2.
2° The proof is exactly the lame as for the part 2° of Theorem 5.3 except that we use
the corresponding resuIts for n = 3. Namely, instead of using Theorem 4.3, Lemma 4.2 and
Corollary 5.1 respectively, we use Theorem 4.5, Lemma 4.4 and Corollary 5.2 respectively.
Our contribution here is providing these results.
3° The values oC Vi(H) are between two constant multiples oC
II (1 + (1 + fJ)/tt 1!2
'1>0
(Theorem 5.4 and 5.5). The resuIt Collows readily Crom this.
,
110
,~,
(-,
Appendix A
Integration on the sphere s6n-l
We fust note that dv(z) = dZ l
ï~~~m-l
is a rotation invariant measure on 5 m -
l
(see [7]).
In Chapter 1, we use a peculiar parametrization to integrate over 5 6n- 1 • We derive here
the corresponding integration formula.
Lemma A.l Let z E s6n-l and fiz an orthonormal ba4i4 {VIJ ... tin} o/Fn. We can write
z = :Eh=1
v'fi Wlc Vi,
where t" ~ 0 and
56n- 1 and 1 depenü onty on t
,
JS I . - l
1
,
where Bn i5
:1
J(z)dv(z)
=
W"
E 5 6 - 1 (50 = {-l, 1}J. II 1
u a function on
= (th ... , tn) then
Bn , ..
J~
L"l=1
l(t)(tl ... tn)6/2-ldtl ... dtn_l
tL=1
, tL>O
__
po&itive cowtant which depenu only on n.
f
Proo!: We use induction. The result is clear when n
z=
(ZIl"" Zn6-1)
'S'''-1
J~
I(z) dv(z)
in such a way that
=
(Zl,"" Z6)
1 /(z) dz 1 .• • dZ6n-1
JS'''-I
IZ6nl
111
= 1.
Assume n
~
2. We write
is a parametrization of.;t; Wl'
r (
lez) t~/2-1 dZ6+1" , dZ6n-1 dtl
Jo JEi>.-1=1-el
I 6nl
BI r (
I(tl i ~,) t~/2.. 1 (1 - tl)6(n-l)/2-1
= BI
Z
=
Jo JEi>.Iit=l
, dll6+1 ' •• d1l6n-l dt!
1'6nl
.~
r
z, = vI - tl ,., i ~ v
= Bl'Bn-l Jo(1 JE.=2
( •
l(tli(1-tl)('2"""n»t~/2-1
_.=1",~0
,(1 - tl )6(n-l)/2-1 ('2. , "n)6/2-1 dtl d'2. ,d'n-l
induction hypothesis
We need to know
Lemma A.2 If N
~
2,
=
(r{p+ l»N
r«p+ l)N)
where r(z) is the Gamma function,
ProoC: We will call the left hand term
fi
Jo
B~,
The resuIt is true for N -= 2:
tr-l (1- t)_-l dt = r(r) r(,), Assume true for N - l, N > 2,
r(r +,)
p
BN
-
-
112
=
(r(p+l»N-l
r«p+ l)(N -1»
r f.+l-l(l_t
Jo
1
)(P+1)(N-l)-ldt
1
induction hypothesis
=
(r(p+ l»N-l r(p+ l)r«(p+ l)(N -1».
r«(p + l)(N - 1»
r«p +l)N)
113
1
-
Appendix B
Technicalities
Sorne preliminaries:
Lemma B.l In what follow.t, 1 S r SN, 1 S, SN, r'l q and,
'1 q.
N
EEcoth(Fq - F,,)coth{Fq - F.) = N{N -l){N - 2)/6.
q=l"<'
Proof: By induction. To simplify notation, we calI the value of the equation at the left hand
side DN. The result is vacuously true for N
DN =
N-l
E E coth(F
II -
q=l ,,<.
= 1, N =2. Assume true for N -1 (N ~ 3).
F,,) coth(Fq - F,) +
E
coth(FN - F,,) coth(FN - F.)
,,<.<N
N-l
+ E Lcoth(Fq -
Fr)coth(Fq
-
FN)
q=l ">/:11
= DN-l +
E
E
coth(FN - Fr) coth(FN - F.)
r<.<N
+
coth(F, - F,,)[coth(Fq - FN) - coth(F" - FN)]
,,<q<N
= DN-l +
coth(FN - Fr) coth{FN - F,)
r<.<N
E
---
-
114
(~
+ :E
coth(F, _ F,.)l- coth(F, - FN)coth(F,. - FN)
,.<,<N
coth(F, - Fp )
= DN-l
+ :E
coth(FN - F,.) coth(FN - F,)
,.<,<N
+ :E
[1- roth(F, - FN)coth(F,. - FN)]
,.<,,<N
= DN-l +
coth(FN - F,.) coth(FN - F,)
,.<,<N
:E
- :E coth(F" -
FN) coth(F,. - FN) + (N -l)(N - 2)/2
,.<,,<N
= (N -l)(N - 2)(N - 3)/6 + (N -l)(N - 2)/2 = N(N -l)(N - 2)/6e
In what rollows, the independent variables are {F"hs"SM and
M
b.
= 'E
;=1
We need first
~o
82
8F2
;
Al
+ 6 L E coth(F; ;=1 ":F;
0
Fil) OF..•
,
compute
~ [6(F)Jl/6-1
NAl
II II 1sinh(-y, -
F;W
i=I;=1
NM
= [6(F)]1/ 6
n II sinh(1i - F;W«l- 6) liN
1: coth(F; - F,,)
1
bl~1
N
- z E coth(-y, - Fi»,
,=1
~
[6(F)]1/6-1
81';
NM
II II 1sinh(-y, -
F;W
;=1;=1
N
= [6(F)]1/6-1
M
Il II 1 sinh(-y; -
F;)IA'
;:::li=1
N
.([(1- 6)
E coth(Fj - Fil) - L coth(1, Z
,=1
":Fi
(.
(
115
F;)]2
-
-
N
- (1- 6) Elinh-2(F; - F,) -
te Elinh- 2(-y. -
"f.;
F;».
.=1
We are now ready to prove
Lemma B.Z
NM
A
[6(F)]1/6-1
II II Ilinh(7i -
Fi)l·
i=I;=1
NM
= (6(F)]1/6-1 Il II IlinhC7, - FiW
;=1;=1
MN
I: I: coth2(-y. -
M
F;) + 2z2 I:
L
coth(-yp - F;) coth(-yv - F;)
;=1.=1
i=IP<4
JI N
N
2
- z E Esinh- (-y. - F;) - z(2 - 6)
toth(7. - F;)coth(-y. - Fil)
;=1.=1
.=1"<;
·(Z2
EE
+ M(~ -
1) (6zN
+2M + 2 -
6(3zN
+ 2M +2»).
Proof: Thls is just a matter of differentiating and thE!n simplifying as much as possible.
NM
A [c5(F)P/6-1
Il II 1sinh(-Yi -
F;W
i=I;=1
N
M
=[6(F)P/6-1 II 111 sinh('Yi -
F;W
'=1;=1
N
JI
·(E[(l- 6) E coth(F; - F,,) - z E coth(-y. - Fi)]2
"#:i
;=1
,=1
M
- (1 - 6)
E E sinh- (F; 2
MN
F,,) - z
L E sinh- (-y, 2
M
+ 6E E coth(F, -
N
F,,)«l- 6) Ecoth(F; - Fi) -
i=1 '#:;
N
= [6(F)]1/6-1
If.;
JI
II I1l linh('Yi -
Fi)l·
;=1;=1
-
F;)
;=1.=1
;=l''f.;
116
Z
Lcoth(-y. - Fi)))
.=1
JI
·«1- 6)2 :~:)E coth(F; - F,»2
i=1
,~,
M
N
JI
N
- 2%(1 - 6) EL E coth(Fi - F,) coth(-y. - Fi) + %2 L(E coth(7. - F;»2
;=1 '~i .=1
i=1 .=1
JI
LE
- (1 - 6)
linh- 2(Fi - F,) -
M
%
i=I'~i
E L coth(F; -
6)
F,)
i=I'~i
M
- %5 E
N
E linh- 2 (-y. - Fi)
i=I.=1
M
+ 5(1 -
L
E coth(F; -
F,)
l~;
IV
L
coth(F, - F;)
;=1 "~i
L
coth(7. - Fi»
.=1
NM
=[5(F)]1/6-1 JI II 'Sinh(7i -
Fi)'·
i=I;=1
JI
N
JI
·«1- 5)2 L(E coth(F; - F,»2
i=1
+ %2 L(Lcoth(-y. -
Fi»2
i=1 .=1
,,~;
JI
JI N
- (1 - 5) E E sinh- 2 (F; - F,,) -
%
L
E sinh- 2 (-y. - Fi)
;=1.=1
;=IIc~;
JI
+ 5(1- 6) ~)E coth(F; ;=1
F,»2
Ic~;
N
JI
- z(2 - 6)
E E coth(F; -
F,,) E coth(7. - Fi»
.=1
;=1'~;
NM
=[5(F)]1/6-1 II II 'Sinh(-yi -
Fi)'·
i=I;=1
JI
JI
.«1- 5)2 L(E coth(F; - F,»2
i=1
N
+ %2 E(E coth(-y. -
F;»2
i=1 .=1
,~;
JI N
JI
- (1- 6) E E sinh- 2 (F; - F,,) - z E Esinh- 2 (-y. - Fi)
i=I.=1
,=IIc~;
JI
+ 6(1 -
6)
E(E coth(F; -
F,»2
i=1 ':#i
(
(
111
-
M
'E L coth(F; -
- z(2 - 6)
;=I11<j
N
Fil) L(coth(-y, - F;) - coth(7, - Fil»)
,=1
NM
={6(FW/6 - 1 TI II 1sinh(-yi -
Fj)l·
i=lj=1
M
·«1- 6)
2)E coth(Fi -
M
F,,»2
N
+ z2 E L coth2b', -
j=1 Ir"'i
M
+ 2Z2 E 'E cothb'p -
M
F;) coth(-y, - Fj) - (1- 6)
;=IP<9
M
Fj)
j=I,=1
E E sinh- (Fj 2
Fil}
j=I11"';
N
- Z EEsinh- 2 (")',
;=1'=1
LM E
-- z(2 - 6)
-
Fj)
coth(F; - Fit)
;=111<;
N
= [6(F)]1/6-1
N
'E
1- coth(-y, -
Fj} coth(-y. - FJc})
coth(Fli - F;)
,=1
M
TI II 1Sinh(-yi -
F,;),-
i=l;=l
M
·«1 - 5) E E coth2 (F; - Fil)
;=1
""'i
M
+ 2(1- 5) E
E
coth(F; - Fil) coth(Fj - Ft) + z~
M N
L E coth (-y. 2
F;)
j=l ,=1
;=1 ,11<1)"';
M
+ 2Z2 E L coth(7p -
M
F;) coth(7q - F;) - (1- 6)
;=lp<q
EE
sinh- 2 (F; - Fil)
;=111"';
MN
M
N
E L
- ZEE sinh- 2 {.y, - F;) - z(2 - 6)
E
coth(")', - Fj) coth(-y, - Fil)
j=1,=1
j=l II<j .=1
M
+ z(2 - 6) E
E
N)
j=I1c<jSM
NM
= [6(FW/6- 1 II II 1sinhb'i - Fj)li=lj=l
-
118
(~
M
·«1- 6) ~
LI +2(1- 6)M(M -
l)(M - 2)/6 +z2
M
N
L 1: coth2b'. -
Fi)
;=1.=1
i=1"~;
M
M
+ 2Z2 L
~ coth(-y, - F;) coth(-yq - Fi) i=lp<q
Z
N
L ~ sinh- 2('Y. -
Fi)
i=I.=1
N
- z(2 - 6)
EL coth('Y. -
Fi) coth('Y. - F,,)
.=1 Ic<i
+ z(2 -
6)N M(M -1)/2)
N
M
II II 1sinh(-y, -
= [6(F)]1/6-1
FiW
i=I;=1
MN
'(Z2
L
M
~coth2(-y. - Fi)
;=1.=1
M
+ 2Z2 L
Lcoth('Yp - F;)cC\th(-yq - Fi)
;=lp<q
N
- Z:E ~ sinh- 2(')', -
N
F;) - z(2 - 6)
.=1"<i
;=1 .=1
+ M(~ -1) (6zN +2M + 2 -
(
(
E:E coth(')'. -
119
6(3zN + 2M + 2»).
Fi) coth(')'. - Ffi)
----------------------,
-Appendix C
The elliptic integral
Lemma C.l lIa> b> c> d then
r
dz
Jd y'{a - z)(b - z)(c - z)(z -
d)
= 2 [(a _ c)(b _ d)t1 / 2 K (
(c - d)(a - b»)
(a - c)(b - d)
where
K(m) =
Proo!: Let J =
e
L
cl
dO
1(/2
1
o
J1 - m 2 sin2 0
dz
.
(c - d)bu + d(b - c)
. We WItte z = -,
i
J(a-z)(b-z)(c-z)(z-d)
{c-d)u+b-c
dz
= _(c -
d)(b - c)(b - d) du
d)u + b - c)2
1
(c - d)(b - a)u + (d - a)(b - c)
(c - d)u + b - c
(d- b)(b- c)
(c - d)u +b - c'
(c- d)(b- c)u+ (d- c)(b- c) and
(c-d)u+b-c
(c - d)(b - d)u
(c - d)u + b - c·
«c -
z
--a
z-b
=
=
z_c =
z-d =
-....
--....
.
120
We then have
r
du
-
2 (1
dt
-
2
J -
-
Jo J(c - d)(a- 6)u + (a- d)(6 - c)v'f=Uv'u
Jo v'(c - d)(4- 6)t 2 + (4- d)(6 - c)v'I- t 2
dIJ
11/2
1
2 Jo
v'(a- d)(6 - c) + (c - d)(a- 6) - (c - d)(a - 6) sin2 9
(11/2
d8
v(a- c)(6 - d) - (c - d)(a- 6) sin2 9
2 Jo
_
2 [(a- c)(6 - d)]-1/2
dIJ
11/2
1
o
2 [(G _ e)(6 _ d)]-1/2 K (
"'1- sin2 9 (c - d)(c - 6)/((a- c)(6 - d»
(c - d)(a- 6)
(a - c)(6 - d)
(
(
= Vu
t == cos 9
o J(a- d)(6 - c) + (c - d)(a- 6) cos2 9
(11/2
d9
-
_
t
121
•
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123