Heat kernel and Green function estimates on noncompact symmetric spaces

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noncompact symmetric spaces

Jean-Philippe Anker, Lizhen Ji

To cite this version:

Jean-Philippe Anker, Lizhen Ji. Heat kernel and Green function estimates on noncompact symmetric

spaces. Geometric And Functional Analysis, Springer Verlag, 1999, 9, pp.1035-1091. �hal-00022962�

ON NONCOMPACT SYMMETRIC SPACES

Jean–Philippe Anker & Lizhen Ji

In memory of Carl S. Herz (1930–1995)

1991 Mathematics Subject Classification. 22E30, 22E46, 31C12, 43A80, 43A85, 43A90, 58G11.

Key words and phrases. Green function, heat kernel, Iwasawa AN groups, Poisson semigoup, reduc- tive Lie groups, semisimple Lie groups, spherical functions, symmetric spaces (Riemannian, noncompact).

First author partially supported by the European Commission (HCM 1994–1997 Network Fourier Analysis and TMR 1998–2001 Network Harmonic Analysis). Second author partially supported by the U.S.A. National Science Foundation (postdoctoral fellowship DMS 9407427 and grant DMS 9704434)

Typeset by AMS-TEX

1

1. Introduction

Let X = G/K be a noncompact Riemannian symmetric space. Although basic har- monic analysis on X has been settled in the sixties (see [GV], [Hel2], [Hel3], for thorough presentations of this material), it is only recently that it has been used efficiently to pro- duce sharp and complete results comparable to the Euclidean or the compact case ([An3], [An5], [BOS], [CGM1], [CGM2], [MNS], [Str], ... ). The reason may be that time was needed to digest and refine the formidable work of Harish–Chandra and his followers.

Our main object of study in this paper is the heat kernel h _t (x, y) = h _t (y ^{− 1} x) on X , for which we produce optimal upper and lower bounds, as well as asymptotics at infinity, in what is arguably the most interesting case, namely when the time variable t is larger than (any constant times) the distance d(x, y) between the space variables. The upper bound was conjectured some years ago by the first author [An2] and proves amazingly to be a lower bound too. The restriction 1 + t ≥ const. d(x, y) in our results is due to a lack of control in the Trombi & Varadarajan expansion for spherical functions along the walls. Fortunately this is no problem in all applications since good upper bounds, with fast decaying Gaussian factors e ^{− const. d(x,y)}

^{/ t} , are known to hold in the remaining domain t d(x, y) .

Such heat kernel estimates have several important consequences. Among them let us mention the exact behavior of the Green function, which is fully obtained for the first time on general symmetric spaces X . Recall that this is precisely the analytic information needed for a complete description of the Martin boundary of X ([Gu], [GJT1], [GJT2]).

Other worth mentioning consequences are on one hand optimal kernel bounds for the Poisson semigroup e ^{− t √ − ∆} on X and on the other hand a delicate maximal inequality for a particular heat diffusion on the Iwasawa component S = (exp a) N of G , which was sought after in [CGGM].

Our paper is organized as follows. Section 2 contains some necessary recalls about symmetric spaces and spherical analysis, notably spherical function asymptotics. In Section 3 we establish our main result, namely the above mentioned bounds for the heat kernel on X . Section 4 is devoted to some applications: L ^p heat propagation on X , optimal bounds for the Bessel–Green–Riesz kernels (in particular for the Green function) and for the Poisson kernel on X , and finally the weak L ¹ → L ¹ boundedness alluded to above of the heat maximal operator associated to a distinguished Laplacian on S . In Section 5 we refine our previous results by obtaining asymptotics at infinity for the various kernels considered before. Historical comments will be made throughout the text.

The main results in this paper were announced in [AJ1] and in several talks during the

past years. Both authors would like to thank M. Babillot, Y. Guivarc’h and J. C. Taylor

for helpful discussions and indications. The first author enjoyed hospitality at the Uni-

versity of Wroc law, at the University of Wisconsin in Madison, and at the Mittag–Leffler

Institute, where substantial parts of this article were carried out.

2. Preliminaries

In this section we recall the basic material about noncompact Riemannian symmetric spaces and spherical analysis thereupon which will be used throughout the article. We shall seize the opportunity of setting up the notation. The book [GV] will serve as our main reference.

2.1. Noncompact Riemannian symmetric spaces. G will denote a noncompact reductive Lie group in Harish–Chandra’s class ¹ K a maximal compact subgroup, θ a cor- responding Cartan involution, g = k ⊕ p the resulting decomposition on the Lie algebra level, and X = G/K the associated Riemannian symmetric space with nonpositive cur- vature. It is understood that g is equipped with an admissible nondegenerate bilinear form B (coinciding with the Killing form on g ⁰ = [ g, g ] ) and X with the associated G–invariant metric. Using the inner product

(2.1.1) h X, Y i = − B(X, θ(Y )) ,

we shall systematically identify any subspace of g with its dual space.

Given a Cartan subspace a in p, A = exp a denotes the corresponding analytic sub- group of G, M its centralizer in K , M ⁰ its normalizer in K, Σ the restricted root system of (g, a), and W the associated Weyl group. Once a positive Weyl chamber a ⁺ has been selected, Σ ⁺ (resp. Σ ⁺⁺ or Σ ⁺⁺⁺ ) denotes the corresponding set of positive (resp. pos- itive indivisible or simple) roots, n the direct sum of all positive root subspaces g _α , N = exp n the corresponding analytic subgroup of G, and % the half sum of all positive roots α counted with their multiplicities m _α = dim g _α . Let n be the dimension of X,

` its rank (i.e. the dimension of a), and m = P

α ∈ Σ

m _α the dimension of N , so that n = ` + m. Recall the decompositions

G = K (exp a) N (Iwasawa ), G = K (exp a ⁺ ) K (Cartan).

Denote by H(x) ∈ a and x ⁺ ∈ a ⁺ the middle components of x ∈ G in these decompo- sitions, and by | x | = | x ⁺ | the distance to the origin.

Lemma 2.1.2. d (xK, yK) ≥ | x ⁺ − y ⁺ | ∀ x, y ∈ G.

Let us give a proof of this elementary result, for lack of known reference (see [Cl; Lem- ma 6.3] for the analog in the compact Lie group setting). Since

d ( (exp X )K, (exp Y )K ) ≥ | X − Y | ∀ X, Y ∈ p

1

This class is the natural setting for Harish–Chandra’s theory. Actually, as long as the associated

symmetric spaces are concerned, the difference with the semisimple setting consists only in possible

additional Euclidean factors. Specifically let G

be the analytic subgroup of G corresponding to g

=

[ g, g ] , K

= K ∩G

, and V = exp (a ∩z) , where z denotes the center of g. Then G

is closed, semisimple,

with finite center, K

is a maximal compact subgroup of G

, V is a split component of G, and X splits as

the product of the Riemannian symmetric space of noncompact type X

= G

/K

times the Euclidean

space V ∼ = a ∩ z . Similarly M

/K

∼ = G

/(G

∩ K) × a

, where G

denotes the analytic subgroup

of G corresponding to [ m

, m

] = [ m

, m

] .

(see for instance [Hel1; Theorem I.13.1]), we can reduce to the corresponding inequality in the flat case, namely

| X − Y | ≥ | X ⁺ − Y ⁺ | ∀ X, Y ∈ p

with obvious notation. Writing X = Ad k 1 . X ⁺ , Y = Ad k 2 . Y ⁺ with k 1 , k 2 ∈ K , we can furthermore restrict to X ⁺ , Y ⁺ ∈ a ⁺ and k ₁ or k ₂ = e . Let us show that the minimum of f (k) = | Ad k . X ⁺ − Y ⁺ | is necessarily reached on M ⁰ . If k 0 is a local extremum of f , we have indeed

0 = _dt ^d

0 f (k ₀ exp tZ) ² = 2 h Ad k ₀ . X ⁺ − Y ⁺ , Ad k ₀ . [ Z, X ⁺ ] i = 2 h [ X ⁺ , Ad k ₀ ^{− 1} .Y ⁺ ] , Z i for every Z ∈ k , which implies successively [ X ⁺ , Ad k ₀ ^{− 1} .Y ⁺ ] = 0 , Ad k ⁻ ₀ ¹ .Y ⁺ ∈ a and k ₀ ∈ M ⁰ . Finally it is well known that min _{w ∈ W} | w. X ⁺ − Y ⁺ | = | X ⁺ − Y ⁺ | and this concludes the proof of Lemma 2.1.2 .

Let us fix the invariant measures on the groups and homogeneous spaces introduced so far. K is equipped with its normalized Haar measure, a, N , G/K and K/M with the invariant measures induced by the inner product (2.1.1), and G with

(2.1.3)

Z

G

dx f (x) = Z

G/K

d(xK) Z

K

dk f (xk) .

Thus the Haar measure on G writes (2.1.4)

Z

G

dx f (x) = 2 ⁻

Z

K

dk Z

a

dH e ^{2 h%,Hi} Z

N

dn f(k(exp H )n) in the Iwasawa decomposition and

(2.1.5)

Z

G

dx f (x) = | K/M | Z

K

dk 1

Z

a⁺

dH δ (H) Z

K

dk 2 f(k 1 (exp H)k 2 ) , with ²

δ(H) = Q

α∈Σ

sinh ^m

h α, H i n Q

α∈Σ

h α,H i 1 + h α,H i

m

o

e ^{2 h %,H i} ,

in the Cartan decomposition. The volume of K/M can be computed explicitly (see (2.2.3) and (2.2.4) below). Notice the differences with Harish–Chandra’s conventions, where the Lebesgue measure on a is divided by (2π)

, the Haar measure on N is nor- malized by the condition R

N dn e ^{− 2 h %,(H ◦ θ)(n) i} = 1 , and the Haar measure on G is given by dx = dk e ^{2 h %,H i} dH dn in the Iwasawa decomposition.

2

The symbol means precisely that there exist two constants 0 < C

≤ C

< +∞ such that C

˘

α∈Σ+

`

1 +hα,Hi

´

¯

e

≤ δ(H) ≤ C

˘

α∈Σ+

`

1 +hα,Hi

´

¯

e

∀ H ∈ a

.

Let us next describe the various faces of a ⁺ and the associated standard parabolic subgroups of G. They are in 1–1 correspondence with the subsets F in Σ ⁺⁺⁺ . Specifically denote by Σ ⁽⁺⁾ _F the (positive) root subsystem generated by F . Furthermore split

a = a _F ⊕ a ^F , n = n _F ⊕ n ^F and N = N _F N ^F accordingly , where a _F is the subspace generated by F , a ^F its orthogonal in a, n _F = ⊕ _{α ∈ Σ}

F

g _α , and n ^F = ⊕ _{α ∈ Σ}

F

g _α . Then the face of a ⁺ and the standard parabolic subgroup attached to F are respectively

(a ^F ) ⁺ =

H ∈ a ^F | h α, H i > 0 ∀ α ∈ Σ ⁺⁺⁺ r F and the normalizer P F of N ^F in G.

We shall write H = H _F + H ^F according to the decomposition a = a _F ⊕ a ^F . For example

% = % _F + % ^F , with % _F = ¹ ₂ P

α ∈ Σ

m _α α and % ^F = ¹ ₂ P

α ∈ Σ

m _α α . Similarly

` = ` _F + ` <sup>F</sup> , with ` _F = dim a _F and ` ^F = dim a ^F .

P _F has Langlands decomposition P _F = M _F (exp a ^F )N ^F . M _F and M ^F = M _F (exp a ^F ) are closed subgroups of G, which belong to the Harish–Chandra class and are θ–stable.

K _F = K ∩ M _F = K ∩ M ^F is a joint maximal compact subgroup. With the obvious notation, a _F and a are Cartan subspaces for m _F and m ^F = m _F ⊕ a ^F respectively, Σ F is the corresponding root system, and its Weyl group W _F is the stabilizer in W of any element H ∈ (a ^F ) ⁺ .

We conclude this subsection with some less standard results about the geometry of a.

Lemma 2.1.6. (i) |h α, H i| ≤ | H | ∀ α ∈ Σ, ∀ H ∈ a.

(ii) There exists a positive constant c ₁ such that

| H | ≤ c 1 max

α ∈ F |h α, H i| ∀ H ∈ a _F and ∀ F ⊂ Σ ⁺⁺⁺ . Proof . (i) Decompose H = H _Σ

+ H ^Σ

. Then

| H | ² ≥ | H _Σ

| ² = tr(ad ²

H _Σ

) = 2 X

α ∈ Σ

m _α h α, H _Σ

i ² =

= 2 X

α∈Σ

m _α h α, H i ² ≥ 2 h α, H i ² . (ii) Since F is a basis of a _F , H 7→ max α ∈ F |h α, H i| is a norm on a _F . Hence there exists a positive constant C _F such that

max α ∈ F |h α, H i| ≤ C F | H | ∀ H ∈ a _F .

We obtain (ii) by taking for c ₁ the maximum of all constants C _F .

Assume temporarily that X has no split component (i.e. Σ generates a) and consider the cones

a ⁺ (F, δ, ε) = n

H ∈ a ⁺ | h α, H i ≤ δ | H | ∀ α ∈ F and h α, H i ≥ ε | H | ∀ α ∈ Σ ⁺⁺⁺ r F o , where F is a subset of Σ ⁺⁺⁺ and 0 < δ ≤ ε < + ∞ .

Lemma 2.1.7. (i) | H _F | ≤ c ₁ δ | H | if H ∈ a ⁺ (F, δ, ε) .

(ii) a ⁺ can be covered with 2 ^` − 1 subcones a ⁺ (F, δ F , ε F ), where F ranges through the proper subsets of Σ ⁺⁺⁺ and δ _F , ^δ _ε

F

are arbitrarily small.

Proof . The statement (i) follows from Lemma 2.1.6.ii and the definition of a ⁺ (F, δ, ε) :

| H _F | ≤ c ₁ max

α∈F |h α, H _F i| = c ₁ max

α∈F |h α, H i| ≤ c ₁ δ | H | .

(ii) Observe first that one can restrict by homogeneity to the unit–sphere S(a) =

= { H ∈ a | | H | = 1 } . For ε > 0 small enough, the ` hyperplanes h α, H i = ε (α ∈ Σ <sup>+++</sup> ) divide the (spherical) domain S(a <sup>+</sup> ) = { H ∈ a <sup>+</sup> | | H | = 1 } into 2 <sup>`</sup> − 1 (spherical) subdomains, defined by the conditions

h α, H i ≤ ε ∀ α ∈ F

h α, H i ≥ ε ∀ α ∈ Σ ⁺⁺⁺ r F

and indexed by the proper subsets F of Σ ⁺⁺⁺ . Given such an ε and 0 < γ < 1 , set δ F = γ ^{` −| F |</sup> ε and ε F = γ <sup>` −| F |− 1} ε . Then ^δ _ε

F

= γ and one shows by (backward) induc- tion on k = ` − 1, ` − 2, ... , 1, 0 that S

k ≤ | F | < ` S(a ⁺ )(F, δ _F , ε _F ) (obvious notation) contains n

H ∈ S(a ⁺ ) | h α, H i ≤ γ ^{` − k} ε for at least k simple roots α o . For k = 0 this amounts to S

F

Σ

a ⁺ (F, δ _F , ε _F ) = a ⁺ .

2.2 Spherical analysis. The role played by exponentials in Euclidean Fourier analysis is played by the (elementary ) spherical functions in the Fourier analysis of bi–K –invariant functions on G. Recall Harish–Chandra’s integral formula

(2.2.1) ϕ λ (x) =

Z

K

dk e ^{h iλ − %, H(xk) i}

for these functions and the definition of the spherical Fourier transform H f (λ) =

Z

G

dx f (x) ϕ ₋ λ (x) .

Among its mapping properties we shall use the following result.

Theorem 2.2.2. ([GV; Theorem 6.4.1], see also [An4].)

(i) H is a (topological) isomorphism between the Schwartz spaces S (G) ^\ and S (a) ^W . (ii) Moreover we have the inversion formula

f (x) = _{| W} ^c

_| Z

a

dλ

|

(λ) |

H f (λ) ϕ _λ (x) with c ₂ = _(2π)

² |

K/M | .

This statement requires some explanations. The (L ² ) Schwartz space S (G) consists of all functions f ∈ C ^∞ (G) satisfying

sup

k

∈ K, H ∈

, k

∈ K

(1 + | H | ) ^N e ^{h %,H i} | f (D 1 : k 1 (exp H)k 2 : D 2 ) | < + ∞

for every D ₁ , D ₂ ∈ U(g) and N ≥ 0, where f (D ₁ : x : D ₂ ) denotes left and right dif- ferentiation of f at x ∈ G with respect to the elements D ₁ and D ₂ in the universal enveloping algebra U(g). The superscript \ denotes the subspace of bi–K –invariant func- tions. Similarly S (a) ^W is the subspace of W –invariants in the classical Schwartz space on a.

c(λ) =

₍

₋ ^(λ) _i%) , where I(λ) = Z

N

dn e −hiλ+%,(H◦θ)(n)i when λ ∈ a − i a ⁺ , is the famous c–function of Harish–Chandra, which was determined explicitly by Gindikin

& Karpeleviˇc. By resuming their computation carefully as in [DKV; pp. 43–50] or in [Mn], one obtains for our choice of invariant measures

I(λ) = Q

α ∈ Σ

I _α

h α,λ i h α,α i

, (2.2.3)

with I α (ν) = √ 2π

| α |

m

Γ(iν )

Γ(iν+

m

) × √ 2π 2 | α |

m

Γ(

ν+

m

) Γ(

ν+

m

+

m

) . As a consequence

c(λ) = Q

α ∈ Σ

c _α

hα,λi h α,α i

,

where c _α (ν) = ^Γ

hα,%i hα,αi

+

m

Γ

^Γ

α,%i

hα,αi

+

m

+

m

Γ

+

m

_Γ(iν ^Γ(iν) ₊

m

)

Γ(

ν+

m

) Γ(

ν+

m

+

m

) . Formula (2.2.3) is also useful to evaluate

(2.2.4) | K/M | = 2

I( − i%) .

We shall often consider the expression

b (λ) = π (i λ) c (λ) = Q

α∈Σ

b α

h α,λ i h α,α i

,

where π (i λ) = Q

α ∈ Σ

h α, λ i and b _α (ν) = | α | ² i ν c _α (ν) =

= | α | ^{2 Γ}

hα,%i hα,αi

+

m

Γ

^Γ

⁺

^m

⁺

^m

Γ

+

m

_Γ(iν+ ^Γ(iν+1)

m

)

Γ(

ν+

m

) Γ(

ν+

m

+

m

) . Notice that b _α ( − i ν) ^±1 is a holomorphic function for Im ν > − ¹ ₂ , with ³

Γ(iν+1) Γ(iν+

m

)

Γ(

ν +

m

)

Γ(

ν+

m

+

m

) ∼ 2

ν ¹⁻

⁻

as | ν | → + ∞ .

Hence b( − λ) ^{± 1} is a holomorphic function for λ ∈ a + i a ⁺ (actually in a neighborhood of a + i a ⁺ defined by ω(Im λ) = min _α∈Σ

h α, Im λ i > − η for some small η > 0 ), which has the following behavior :

(2.2.5) | b( − λ) | ^{± 1} Q

α ∈ Σ

( 1 + |h α, λ i| ) ^{± 1 ∓}

,

and whose derivatives can be estimated as follows, using Cauchy’s formula : (2.2.6) p( _∂λ ^∂ ) b( − λ) ^{± 1} = O ( | b( − λ) | ^{± 1} ) .

For an effective use of the inversion formula (2.2.2.ii), one needs precise information about spherical functions. Their behavior away from the walls is well described by the following converging expansion of Harish–Chandra (& Gangolli ).

Theorem 2.2.7. We have a converging expansion ϕ λ (exp H) = e ^{−h %,H i} P

q ∈ 2Q

e ^{−h q,H i} P

w ∈ W

c(w.λ) γ q (w.λ) e ^{i h w.λ,H i} for all λ ∈ a regular and H ∈ a ⁺ . Here,

(i) Q = P

α ∈ Σ

N α is the positive root lattice, (ii) the leading coefficient γ ₀ is equal to 1,

(iii) the other coefficients γ _q (λ) are rational functions in λ ∈ a

, which have no poles for λ ∈ a + i a ⁺ (actually in a neighborhood of a + i a ⁺ defined by ω(Im λ) > − η for some small η > 0) and satisfy there

| γ q (λ) | ≤ C (1 + | q | ) ^d ,

for some nonnegative constants C and d (independent of q ∈ Q and λ ∈ a + i a ⁺ ).

Moreover,

(iv) all derivatives of ϕ _λ (exp H) in H have corresponding expansions p( _∂H ^∂ ) ϕ λ (exp H) =

= e ^{−h %,H i} P

q ∈ 2Q

e ^{−h q,H i} P

w ∈ W

c(w.λ) γ q (w.λ) p(i w.λ − % − q) e ^{i h w.λ,H i} . Theorem 2.2.7 is proved for instance in [GV; Ch. 4] or [Hel2; Ch. IV]. The crucial point is the estimate in (iii), which is due to Gangolli and which provides quite a good control of convergence.

3

The symbol ∼ means precisely that

Γ(iν+¹₂mα)

Γ(₂ⁱν+¹₄mα)

Γ(ⁱ₂ν+¹₄mα+¹₂m_2α)

ν

−→ 2

as |ν| → +∞ .

As far as the behavior of spherical functions along faces is concerned, the best in- formation available is provided by the asymptotic expansion of Trombi & Varadarajan.

Before stating this result, let us introduce the height in Q : κ(q) = P

α ∈ Σ

q α if q = P

α ∈ Σ

q α α . Moreover, given a subset F in Σ ⁺⁺⁺ , let us decompose

Q = Q _F + Q ^F and

c(λ) = c _F (λ) c ^F (λ) , π (λ) = π _F (λ) π ^F (λ) , b(λ) = b _F (λ) b ^F (λ) , where Q _F = P

α∈F N α [ resp. Q ^F = P

α∈Σ

N α ] , and c _F (λ) [ resp. c ^F (λ) ] , π _F (λ) [ resp. π ^F (λ) ] , b _F (λ) [ resp. b ^F (λ) ] denote the products over α ∈ Σ ⁺⁺ _F [ resp.

over α ∈ Σ ⁺⁺ r Σ ⁺⁺ _F ] of c α h α,λ i h α,α i

, h α, λ i , b α h α,λ i h α,α i

. Finally let ω ^F (H ) = min _α∈Σ

h α, H i ∀ H ∈ a .

Theorem 2.2.8. Let F be a nontrivial subset of Σ ⁺⁺⁺ . Then we have an asymptotic expansion

ϕ _λ (exp H) ∼ e ^{−h %}

^{,H i} P

q ∈ 2Q

P

w ∈ W

\ W

c ^F (w.λ) ϕ ^F _w.λ,q (exp H) for all λ ∈ a regular and H ∈ a ⁺ with ω ^F (H) > 0 . Specifically:

(i) ϕ ^F _λ,0 (y exp H) = ϕ ^F _λ

(y) e ^ihλ

^,Hi is the spherical function of index λ on M ^F =

= M F exp a ^F .

(ii) The other terms ϕ ^F _λ,q (x) are bi–K _F –invariant C ^∞ functions in the variable x ∈ M ^F and W _F –invariant holomorphic functions in the variable λ ∈ a + i (a ^F ) ⁺ (actually λ can be taken in a neighborhood of a + i (a ^F ) ⁺ defined by | Im λ _F | < η and ω ^F (Im λ ^F ) > − η for some small η > 0), which satisfy

ϕ ^F _λ,q (x) = ϕ ^F _λ,q (y) e ^{h iλ − q,H i} ∀ x = y exp H ∈ M ^F = M F exp a ^F .

(iii) For every q ∈ Q ^F and D ∈ U(m ^F ), there exist a constant d ≥ 0 and, for every η > 0, another constant C ≥ 0 such that

| ϕ ^F _λ,q (exp H : D) | ≤ C e ^{η | H}

^| (1 + | λ | ) ^d e ^{−h Im λ+%}

^{+q,H i} for all λ ∈ a + i (a ^F ) ⁺ and H ∈ a ⁺ .

(iv) For every integer N > 0 and every D ∈ U(m ^F ), there exist a constant d ≥ 0 and, for every η > 0, another constant C ≥ 0 such that

| ϕ _λ (exp H : D) − e ^{−h %}

^{,H i} P

q ∈ 2Q

P

w ∈ W

\ W

c ^F (w.λ) ϕ ^F _w.λ,q (exp H : e ^%

◦ D ◦ e ^{− %}

) | ≤

≤ C (1 + | λ | ) ^d (1 + | H | ) ^d e ^{−h %,H i− N ω}

^(H)

for all λ ∈ a regular and H ∈ a ⁺ with ω ^F (H) > η .

Remark 2.2.9 . (i) This result refines Harish–Chandra’s constant term theory (see for instance [GV; Ch. 5]) by expanding spherical functions along a face beyond the leading term.

(ii) Here is the relation between the expansions in Theorem 2.2.7 and in Theorem 2.2.8 : ϕ ^F _λ,q (exp H) = e ^−h%

^,Hi P

q

∈ 2Q

e ^−hq+q

^,Hi P

w ∈ W

c _F (w.λ) γ _q+q

(w.λ) e ^ihw.λ,Hi for all λ ∈ a (Σ _F –)regular and H ∈ a ⁺ .

(iii) Expansions along faces actually converge [CM], like expansions away from the walls.

But we miss a control comparable to the Gangolli estimates in Theorem 2.2.7.iii .

Theorem 2.2.8 is essentially a restatement adapted to our needs of the main results in [TV], which are reproduced in [GV; Ch. 7]. Let us elaborate on (ii) and (iii).

Consider the algebra A ^F of functions on M ^F , which is generated by 1 and by the matrix entries of a ^F (x) = Ad

{ x ^{− 1} θ(x) } , and which is graded by A ^F = ⊕ q ∈ Q

A ^F q , where A ^F q = { f ∈ A ^F | f (x exp H ) = f (x) e ^{−hq,H i} ∀ x ∈ M ^F , ∀ H ∈ a ^F } . We know from [loc. cit.] that P _q (λ) ϕ ^F _λ,q belongs to {A ^F q ⊗ U(m ^F ) } ^K

ϕ ^F _λ , where P _q is a W _F –invariant polynomial and U(m ^F ) acts on ϕ ^F _λ by differentiation on the right. Hence P _q (λ) ϕ ^F _λ,q (x) is a W _F –invariant holomorphic function in λ ∈ a

and a bi–K _F –invariant C ^∞ function in x ∈ M ^F , which has the required homogeneity

P q (λ) ϕ ^F _λ,q (x exp H ) = P q (λ) ϕ ^F _λ,q (x) e ^{h iλ − q,H i} ∀ x ∈ M F , ∀ H ∈ a ^F , and which can be estimated by

(2.2.10) | P q (λ) ϕ ^F _λ,q (exp H) | ≤ C q (1 + | λ | ) ^d

(1 + | H F | ) ^d e ^{| Im λ}

^{|| H}

^{|−h Im λ}

^+%

^{+q,H i} for H ∈ a ⁺ (C _q , d _q and d are nonnegative constants, the first two depending on q).

Moreover similar estimates hold for all derivatives P _q (λ) ϕ ^F _λ,q (exp H : D) with respect to D ∈ U(m ^F ), since U(m ^F ) preserves A ^F q . Thus the main problem consists in getting rid of the factor P q (λ). Since P q (λ) is a product of (non necessarily distinct) factors f (λ) = h µ, λ i + iν , with µ ∈ a r { 0 } and ν ∈ R r { 0 } , this will be achieved by repeated application of the following elementary lemma.

Lemma 2.2.11. Let U be an open subset in R ^{`</sup> , T (U ) = U + iR <sup>`} the open tube over U in C ^{`</sup> , Ξ a C <sup>∞</sup> manifold, F = F (z; ξ) a holomorphic function in z ∈ T (U ) depending smoothly on ξ ∈ Ξ, and f (z) = h µ, z i + ν an affine function on C <sup>`} with µ ∈ R ^` r { 0 } . (i) Assume that F (z; ξ) vanishes whenever f (z) does (in T (U )). Then G(z; ξ) = ^F _f(z) ^(z;ξ) is a holomorphic function in z ∈ T (U ) depending smoothly on ξ ∈ Ξ .

(ii) Moreover, for any η > 0, there exists a nonnegative constant C such that

| G(z; ξ) | ≤ C sup

| ζ − z | ≤ η | F (ζ; ξ) |

for all z ∈ T (U ) at distance > η from C ^` r T (U ) and for all ξ ∈ Ξ .

Proof of Lemma 2.2.11 . One can reduce to the case f(z) = z 1 . (i) Let ζ ∈ T (U ) . If ζ ₁ 6 = 0, then G(z; ξ) = ^{F (z;ξ)} _z

1

defines obviously a holomorphic function in z, for z close to ζ, which depends smoothly in ξ. If ζ ₁ = 0, we have

F (z; ξ) = F (0, z ₂ , . . . , z _` ; ξ) + Z 1

0

dt _∂t ^∂ F (tz ₁ , z ₂ , . . . , z _` ; ξ)

= z ₁ Z 1

0

dt (∂ ₁ F )(tz ₁ , z ₂ , . . . , z _` ; ξ) , for z close to ζ , hence also in this case

G(z; ξ) = F (z; ξ) z ₁ =

Z 1 0

dt (∂ ₁ F )(tz ₁ , z ₂ , . . . , z _` ; ξ) is holomorphic in z and C ^∞ in ξ.

(ii) Let z ∈ T (U ) . If | z 1 | ≥ ^η ₃ , then | G(z; ξ) | ≤ ³ _η | F (z; ξ) | . Otherwise G(z; ξ) = _2πi ¹

Z

| ζ

| =

dζ ₁ Z 1

0

dt ^{F (ζ} _(ζ

^,z

₋

^,...,z _tz

₎

^;ξ)

by Cauchy’s formula, hence | G(z; ξ) | ≤ ³ _η sup _{| ζ − z | ≤ η} | F (ζ; ξ) | . This concludes the proof of Lemma 2.2.11.

Let us now complete the proof of (ii) and (iii) in Theorem 2.2.8 . Restrict temporarily to H ∈ a ⁺ . We know (see Remark 2.2.9) that λ 7→ π _F (λ) ϕ ^F _λ,q (exp H) is a holomorphic function in a tubular neighborhood T _η = { λ ∈ a

| | Im λ _F | < η and ω ^F (Im λ ^F ) > − η } of a + i (a ^F ) ⁺ in a

. Since π _F (λ) and P _q (λ) have no common factors, we deduce that P q (λ) ϕ ^F _λ,q (exp H) vanishes in T η whenever P q (λ) does. By combining this observation with Lemma 2.2.11.i, we can eliminate successively all factors in P q (λ) and obtain as a first conclusion that ϕ ^F _λ,q (x) is holomorphic in λ ∈ T η and C ^∞ in x ∈ exp a ⁺ . This result extends to x ∈ K _F exp { (a ^F ) ⁺ +a ^F } K _F by bi–K _F –invariance and (exp a ^F )–homogeneity.

Further extension to x ∈ M ^F is achieved by density and by reapplying Lemma 2.2.11.i to P _q (λ) ϕ ^F _λ,q (x) . Finally the estimate (iii) in Theorem 2.2.8 is obtained by applying Lemma 2.2.11.ii to the expression P _q (λ) ϕ ^F _λ,q (exp H : D) e ^{− i h λ}

^{,H i} and using (2.2.10) for derivatives of ϕ ^F _λ,q .

Finally let us recall the particular behavior of the basic spherical function ϕ 0 . Proposition 2.2.12. (i) Global estimate ⁴ :

ϕ ₀ (exp H ) n Q

α ∈ Σ

( 1 + h α, H i ) o

e ^−h%,Hi ∀ H ∈ a ⁺ .

4

The symbol , let us recall, means precisely that there exist two constants 0 < C

≤ C

< +∞

such that

C

≤ ˘

α∈Σ++(1+hα,Hi)

¯

e^−h%,Hi

≤ C

∀ H ∈ a

.

(ii) Asymptotics ⁵ :

ϕ ₀ (exp H) ∼ c ^F ₃ π ^F (H) ϕ ^F ₀ (exp H) e ^−h%

^,Hi when

 



 

H ∈ a ⁺ ,

h α, H i = o ( ω ^F (H ) ) ∀ α ∈ F ,

ω ^F (H) = min _{α ∈ Σ}

h α, H i → + ∞ .

Here F is a proper subset of Σ ⁺⁺⁺ (possibly empty ) and c ^F ₃ = π ^F ( % e ) ⁻¹ b ^F (0) a positive constant, with % e = ¹ ₂ P

α ∈ Σ

α .

Remark 2.2.13 . (i) Recall that ϕ ₀ controls all spherical functions ϕ _λ with parameter λ ∈ a . More precisely, for every D ∈ U(g) ,

ϕ _λ (x : D) = O ( (1 + | λ | ) ^{deg D} ϕ ₀ (x) ) ∀ λ ∈ a , ∀ x ∈ G . This follows easily from (2.2.1) (see for instance [GV; Proposition 4.6.2]).

(ii) We can disregard Euclidean factors in X = G/K , which don’t contribute to ϕ 0 , and will do so in the rest of this subsection.

(iii) The asymptotics in Proposition 2.2.12.ii hold in particular when H ∈ a ⁺ tends to infinity in either of the following ways, which are most often considered :

h α, H i remains bounded ∀ α ∈ F , h α, H i → + ∞ ∀ α ∈ Σ ⁺⁺⁺ r F , (a)

h α, H i = o ( | H | ) ∀ α ∈ F , h α, H i | H | ∀ α ∈ Σ ⁺⁺⁺ r F . (b)

These particular asymptotics, for the various functions ϕ ^F ₀ , are actually equivalent to the general asymptotics stated in Proposition 2.2.12.ii . Let us for instance deduce the general case from the particular case (a). In the proof of Proposition 2.2.12.ii , we shall do the same with (b). Assume that, for all subsets F ₁ ⊂ F ₂ in Σ ⁺⁺⁺ ,

ϕ ^F ₀

(exp H ) ∼ Q

α ∈ Σ

h α, % e i ⁻¹ b _α (0) h α, H i ϕ ^F ₀

(exp H) e ^{− h %}

^−%

^{, H i} when

 



 

H ∈ a ⁺ _F

2

,

h α, H i remains bounded ∀ α ∈ F 1 , h α, H i → + ∞ ∀ α ∈ F 2 r F 1 ,

but that the asymptotics in Proposition 2.2.12.ii fail to hold. Thus there exists a proper subset F 2 of Σ ⁺⁺⁺ and a sequence H j ∈ a ⁺ such that

 

 

 



ω ^F

(H j ) → + ∞ ,

h α, H _j i = o ( ω ^F

(H _j ) ) ∀ α ∈ F ₂ , inf j

^ϕ

^{(exp H}

⁾

π

(H

) ϕ

(exp H

) e

− π ^F

( % e ) ^{− 1} b ^F

(0)

> 0 .

5

The symbol ∼ , let us recall, means precisely that

π^F(H)ϕ^F₀(expH)e^{−h%F ,Hi}

→ c

under the

indicated assumptions.

By passing to a subsequence, we can assume that sup _j h α, H _j i < + ∞ ∀ α ∈ F ₂ ,

h α, H _j i → + ∞ ∀ α ∈ F ₂ r F ₁ ( actually ∀ α ∈ Σ ⁺⁺⁺ r F ₁ ) , for a subset F ₁ of F ₂ . According to our assumptions,

π(Hj

)

e

ϕ

(exp H

)

πF2

(H

)

e

ϕ

(exp H

) −→

^%

⁾

⁽⁰⁾

F2

(e % )

2

(0) . Hence a contradiction.

(iv) ⁶ The correct asymptotics

ϕ ₀ (exp H) ∼ c _F π ^F (H) ϕ ^F ₀ (exp H) e ^{−h %}

^{,H i} were announced in [Ol2] under the assumptions

( H ∈ a ⁺ tends to infinity i.e. | H | → + ∞ ,

the a _F – component H _F tends to a vector in (a _F ) ⁺ .

But the analysis along faces relied on a misuse of Harish–Chandra’s expansion (recalled in Theorem 2.2.7), as came out in the preprint version of [Ol3]. The final version contains the weaker result [Ol3; Proposition 2.6], where asymptotics of ϕ 0 (exp H ) are stated with a nonexplicit polynomial factor and under the additional assumptions

( lim H _F 6 = 0 ,

H

| H | i.e. _|H ^H

| tends to a unit vector in (a ^F ) ⁺ .

But the proof requires actually more, namely H ∈ a ⁺ and lim H _F ∈ (a _F ) ⁺ . Besides, notice that error terms, involving derivatives of f , are missing in [Ol3; Lemma 2.5].

Thus the gap in Olshanetsky’s results about ϕ ₀ (exp H ) consists essentially in tangential asymptotics along faces.

Proof of Proposition 2.2.12 . The global estimate (i) was established in [An1], essen- tially as a consequence of the Harish–Chandra converging expansion away from the walls (Theorem 2.2.7). The same expansion yields the asymptotics (ii) away from the walls i.e. when F = ∅ . We refer to the proof of [GV; Theorem 4.6.6] for details. In addition we use the identity ∂(π) π = | W | π( % e ) , which is obtained by applying π( _∂H ^∂ )

H=0 to the Weyl denominator formula

(2.2.14) Q

α ∈ Σ

2 sinh ^hα,Hi ₂ = X

w∈W

(det w) e ^{h w.e % ,H i}

(see for instance [Hel2; Proposition I.5.15]). Let us turn to the asymptotics (ii) along a face i.e. relatively to a nontrivial subset F of Σ ⁺⁺⁺ and let us first consider the

6

See also Remark 5.2.2.iii .

special assumptions (2.2.13.iii.b). In this case we use the Trombi–Varadarajan asymp- totic expansion along faces (Theorem 2.2.8), or the simpler Harish–Chandra constant term theory (see for instance [GV; Ch. 5]), and more precisely the resulting asymptotics [GV; Theorem 5.9.5]

ϕ ₀ (exp H) = e ^{−h %}

^{,H i} p ^F − i _∂λ ^∂

λ=0 ϕ ^F _λ (exp H) (2.2.15)

+ O ( (1 + | H | ) ^d e ^{− h %,H i − 2 ω}

^(H) ) ,

which holds for H ∈ a ⁺ with ω ^F (H) bounded below. Here p ^F is a polynomial on a , which is W – harmonic and W _F – invariant, and which is uniquely determined by (2.2.15).

Recall that p ^∅ = c ^∅ ₃ π + derivatives of π (see the proof of [GV; Theorem 4.6.6] ). We need a similar information about the other polynomials p ^F . First of all, the space H _W (a) ^W

of W – harmonic W _F – invariant polynomials on a can be described as follows :

H _W (a) ^W

= ∂ ( P(a) ^W

) ∂ (π _F ) π = ∂ ( H _W (a) ^W

) ∂ (π _F ) π .

This is easily deduced from the case F = ∅ , which is well–known (see for instance [Hel2; Theorem III.3.6.i ] ). Thus ∂( π _F ) π has maximal degree in H _W (a) ^W

and all other elements are derivatives thereof. In particular

p ^F = c ^F ∂ ( π _F ) π + ∂ (q ^F ) ∂ ( π _F ) π = c ^F { ∂ ( π _F ) π _F } π ^F + . . . ,

where c ^F is a constant, q ^F is a W _F – invariant polynomial on a with no constant term,

∂(π _F )π _F = | W _F | π _F ( % e _F ) = | W _F | π _F ( % e ) is a positive constant, and the dots stand for a sum of terms obtained by suppressing some α in the product π ^F = Q

α ∈ Σ

Σ

α or by replacing them by some other α ∈ Σ ⁺⁺ _F . Next, let us consider the expression

p ^F − i _∂λ ^∂

λ=0 ϕ ^F _λ (exp H ) = Z

K

dk p ^F (H { (exp H _F ) k } + H ^F ) e ^{−h %}

^{, H { (exp H}

^{) k }i} , which is a consequence of Harish–Chandra’s integral formula (2.2.1) for ϕ ^F _λ . By expand- ing p ^F (H((exp H F )k) + H ^F ) and estimating

| H { (exp H _F ) k } | ≤ | H _F | max

α ∈ F h α, H i = o ( | H | ) , we obtain

p ^F − i _∂λ ^∂

λ=0 ϕ ^F _λ (x) = c ^F | W _F | π _F ( % e ) π ^F (H ^F ) ϕ ^F ₀ (exp H) + o ( | H | ^|Σ

^{| − |Σ}

^| ϕ ^F ₀ (exp H) ) , hence

ϕ ₀ (exp H ) = c ^F | W _F | π _F ( % e ) π ^F (H ) ϕ ^F ₀ (exp H) e ^{−h %}

^{,H i} (2.2.16)

+ o ( | H | ^{| Σ}

^{| − | Σ}

^| ϕ ^F ₀ (exp H ) e ^{−h %}

^{,H i} )

∼ c ^F | W _F | π _F ( % e ) π ^F (H ) ϕ ^F ₀ (exp H) e ^{−h %}

^{,H i}

Finally the constant c ^F = | W F | ^{− 1} π ( % e ) ^{− 1} b ^F (0) is determined by considering a partic- ular sequence H j going to infinity in a ⁺

with

h α, H _j i | H _j |

∀ α ∈ F

h α, H _j i | H _j | ∀ α ∈ Σ ⁺⁺⁺ r F

and by comparing (2.2.16) with the asymptotics away from the walls of both ϕ ₀ (exp H _j ) and ϕ ^F ₀ (exp H j ) . Now that we have established the desired asymptotics in the special case (2.2.13.iii.b), let us extend our result to the general case. Arguing by contradiction, assume that there exist a subset F ₀ ( Σ ⁺⁺⁺ and a sequence H ⁽⁰⁾ _j ∈ a ⁺ such that

 

 

 



 

 

 

ω ^F

(H ⁽⁰⁾ _j ) → + ∞ ,

h α , H ⁽⁰⁾ _j i = o ( ω ^F

(H ⁽⁰⁾ _j ) ) ∀ α ∈ F ₀ , inf j

ϕ

(exp H

)

π

(H

) ϕ

(exp H

) e

(0)

j i

− π ^F

( % e ) ^{− 1} b ^F

(0) > 0 .

By passing to a subsequence, we can assume that ^H

(0) j

|H

| tends to a unit vector H _∞ ⁽⁰⁾ ∈ a ⁺ . Then F ₁ = { α ∈ Σ ⁺⁺⁺ | h α , H _∞ ⁽⁰⁾ i = 0 } is a proper subset of Σ ⁺⁺⁺ containing F ₀

ϕ ₀ (exp H ⁽⁰⁾ _j ) ∼ π ^F

( % e ) ⁻¹ b ^F

(0) π ^F

(H ⁽⁰⁾ _j ) ϕ ^F ₀

(exp H ⁽⁰⁾ _j ) e ^{− h %}

^{, H}

ⁱ ,

and (

h α , H ⁽⁰⁾ _j i = o ( | H ⁽⁰⁾ _j | ) ∀ α ∈ F 1 , h α , H ⁽⁰⁾ _j i | H ⁽⁰⁾ _j | ∀ α ∈ Σ ⁺⁺⁺ r F ₁ . since

If F 1 = F 0 , we have a contradiction. Otherwise F 1 ) F 0 and in this case H ⁽¹⁾ _j = (H j ) F

is a sequence in a ⁺ _F

1

verifying

 

 

 

 

 

 

 

 

 

 

 



min _{α ∈ F}

h α , H ⁽¹⁾ _j i → + ∞ ,

max _{α ∈ F}

h α , H ⁽¹⁾ _j i = o min _{α ∈ F}

h α , H ⁽¹⁾ _j i , lim inf

j → + ∞

ϕ

(exp H

)

Q

α ∈ Σ

1rΣ⁺⁺_F0

h α , H

i ϕ

(exp H

) e

(1) j i

−

− Q

α ∈ Σ

h α , % e i ⁻¹ b _α (0) > 0 ,

i.e. violating Proposition 2.2.12.ii for ϕ ^F ₀

. Repeating the same reasoning, we end up

with a contradiction, after a finite number of steps. This concludes the proof of Propo-

sition 2.2.12 .

3. Heat kernel bounds

During the last decades heat kernels have become a theme of extensive research in differential geometry, global analysis and probability. Among the vast literature we shall cite [Chav], [Da], [Gr], [Ro], [Va], [VSC] (and the bibliographies therein) as general references for the (scalar) heat kernel on Riemannian manifolds or Lie groups, and [An2], [An5], [CGM1], [Lu] for the particular case of noncompact symmetric spaces.

It is well known that the heat kernel on X = G/K is given by (3.1) h _t (x) = _{| W} ^c

_|

Z

a

dλ

|

(λ) |

e ^{−t ( |λ|}

^{+ |%|}

⁾ ϕ _λ (x)

(see for instance the pioneer work [Ga]). In [An2] the first author conjectured the global upper estimate

(3.2) h t (exp H) ≤

≤ C t ⁻

n Q

α ∈ Σ

(1 + h α, H i ) (1 + t + h α, H i )

^{− 1} o

e ^{− | % |}

^{t − h %,H i −}

2 4t

for t > 0 and H ∈ a ⁺ . This guess was based on some particular cases where spe- cific expressions were available for the heat kernel, namely when G is complex, when rank X = 1 , or when G = SU(p, q) . Around the same time the global behavior of the heat kernel on real hyperbolic spaces was determined in [DM]:

h _t (r) t ⁻

(1 + r) (1 + t + r)

e ^{− (}

⁾

^{t −}

^{r −}

for an appropriate normalization of the Riemannian structure, where r denotes the geo- desic distance to the origin. Thus the right hand side in (3.2) proved to be not only an upper bound but also a lower bound in this case. This result was extended later on to all hyperbolic spaces [GM], to the larger class of Damek–Ricci harmonic spaces [ADY], and actually to a much wider family of radial Laplacians in [LR]. Let us turn to the higher rank case. In [An5] the first author obtained global upper bounds of the form

(3.3) t ^{− d}

(1 + | H | ) ^d

e ^{− | % |}

^{t − h %,H i −}

2 4t

,

where d ₁ and d ₂ are positive constants depending on the position of H ∈ a ⁺ with respect to the walls and on the relative size of t > 0 and 1 + | H | . Although quite general and rather precise, this result was clearly not optimal. On the contrary (3.2) was established by specific computations for G = SL(n, R) , SL(n, H) and “ SL(3, O) ” in a series of papers by P. Sawyer ([Saw1], [Saw2], [Saw3], [Saw4]); moreover the right hand side of (3.2) was shown in [Saw1] to be also a lower bound in the particular case G = SL(3, R) . All these results has lead us to update (3.2) as follows.

Conjecture 3.4.

h t (exp H) t ⁻

n Q

α∈Σ

(1 + h α, H i ) (1 + t + h α, H i )

^{− 1} o

e ^{− | % |}

^{t − h %,H i −}

2 4t

for all t > 0 and H ∈ a ⁺ .

Remark 3.5 . Let us make some comments about the various factors in this conjectural estimate. The whole expression reduces as expected to t ⁻

e ⁻

in the Euclidean case or for small t and H in the general case. The exponential decay e ^{−| % |}

^t in t is connected with the bottom of the L ² spectrum [ | % | ² , + ∞ ) of − ∆ , while the expo- nential decay e ^−h%,Hi in H is related both to the exponential growth of the volume (see (2.1.5)) and to the temperedness of the spherical functions ϕ _λ entering (3.1) (see Remark 2.2.13.i). Eventually the expression between braces is related to the behavior of the c –function. More precisely, after dividing by t

,

Q

α ∈ Σ

1+ h α,H i

t 1 + ^{1+ h α,H} _t ⁱ

₋ 1

c − i ^H _κ

^+H

1

t

₋ 1

| c − ^H

_κ ^+iH

_t

| ^{− 1} ,

where H ₁ (resp. H ₂ ) is any fixed element in a ⁺ (resp. any fixed regular element in a+i a ⁺ ) and κ 1 , κ 2 are any fixed positive constants.

Beside the heat kernel itself, we have also a conjectural estimate for its derivatives.

Conjecture 3.6. Let D ∈ U(g). Then h _t (x : D) = O

t ^{− deg D} ( √

t + t + | x | ) ^{deg D} h _t (x) for all t > 0 and x ∈ G.

We shall establish these conjectures for general noncompact symmetric spaces but under some restrictions on the time and space variables, which prove fortunately to be unessential in all applications (see Section 4).

Theorem 3.7. (i) Estimates away from the walls:

Conjectures 3.4 and 3.6 hold when ω(H ) = min _α∈Σ

h α, H i is large. This means precisely that there exist positive constants κ, C 1 , C 2 and C 3 (the last one depending on D) such that

C 1 ≤ h _t (exp H)

t ⁻

Q

α ∈ Σ

h α, H i (t + h α, H i )

^{− 1} e ^{− | % |}

^{t − h %,H i −}

≤ C 2

and

| h _t (exp H : D) | ≤ C ₃ t ⁻

^{− deg D} n Q

α ∈ Σ

h α, H i (t + h α, H i )

^{− 1} o

×

× (t + | H | ) ^{deg D} e ^{− | % |}

^{t − h %,H i −}

2 4t

for all t > 0 and H ∈ a with ω(H ) ≥ κ . (ii) Global estimates:

Conjectures 3.4 and 3.6 hold when | H | ≤ κ (1 + t), κ being an arbitrary positive constant.