ASYMPTOTIC EXPANSION OF THE HEAT KERNEL FOR A CLASS OF HYPOELLIPTIC OPERATORS

INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES

WARSZAWA 1992

ASYMPTOTIC EXPANSION OF THE HEAT KERNEL FOR A CLASS OF HYPOELLIPTIC OPERATORS

A L E X A N D E R L O P A T N I K O V

Space Research Institute, Russian Academy of Sciences Profsoyuznaya 84/32, Moscow, 117810 Russia

Introduction. Let M be a smooth compact manifold without boundary, let dx be a fixed positive smooth density on M , and let X

, . . . , X

be smooth real vector fields on M , i.e. in a local coordinate system, X

= P

i=1

a

∂

. We will consider operators A of the form (m is even)

(1) (−1)

l

X

j=1

X

+ X

|α|<m

a

(x)X

where α = (α

, . . . , α

), X

= X

. . . X

, |α| = k and the a

are smooth functions. The well-known example is the sum of the squares of vector fields,

(2) −

l

X

j=1

X

+ X

+ c(x) .

A result of H¨ ormander [7, 14] states that this operator is hypoelliptic if the vector fields X

, . . . , X

and all their commutators [X

, [X

. . . [X

, X

] . . .], s ≤ r, up to length r span the tangent space to M at each point. We recall that an operator A is said to be hypoelliptic on M if for any open set U ⊂ M and distributions u, f on U satisfying Au = f , f ∈ C

(U ) implies u ∈ C

(U ). In [17] for the operator (2) it was shown (with m = 2) that

kuk

≤ C(kAuk

+ kuk

) ,

for all u ∈ C

(M ), where k · k

denotes the norm in the usual Sobolev space H

(M ). For the operator (1) this estimate and hypoellipticity were proved in [6, 16].

We assume that the operator (1) is formally selfadjoint and positive, that is, (Au, v) = (u, Av) and (Au, u) ≥ 0 for all u, v ∈ C

(M ). It is easy to show that

[309]

under our assumption A is an unbounded selfadjoint operator on the domain D

= {u ∈ H

(M ) : Au ∈ L

(M )} and has discrete spectrum λ

→ ∞. Let U (x, y, t) be the kernel of the operator exp(−tA),

U (x, y, t) =

∞

X

j=1

e

ϕ

(x)ϕ

(y) .

Here ϕ

(x) is a complete orthonormal set of eigenfunctions of A with eigenvalues {λ

}. U (x, y, t) is a fundamental solution for the operator L = ∂

+ A and so it is called the heat kernel for L.

We denote by V

(x) the subspace of T

(M ) spanned by X

, . . . , X

and all their commutators of length ≤ k and let ν

(x) = dim V

(x) (ν

= 0). We say that H¨ ormander’s condition (of order r) holds if

ν

(x) = dim M = n for all x ∈ M . We will also use the condition introduced by M´ etivier [13]:

ν

(x) = ν

= const , 1 ≤ k ≤ r, for all x ∈ M . Our main results are the following.

Theorem 1. If H¨ ormander’s condition holds then the heat kernel for the op- erator (1) has the following asymptotic expansion as t → +0 :

(3) U (x, x, t) =

∞

X

j=−q(x)

c

(x)t

+

∞

X

j=0

d

(x)t

ln(t) where q(x) = P

k=1

(ν

(x)−ν

(x))k, and c

(x), d

(x) are some functions on M . We remark that in general this expansion is not uniform in x ∈ M and the functions c

, d

are not continuous.

Theorem 2. In the M´ etivier case the asymptotics in Theorem 1 is uniform in x ∈ M , c

(x), d

(x) ∈ C

(M ), and as t → +0,

(4) tr exp(−tA) =

∞

X

j=−q

c

t

+

∞

X

j=0

d

t

ln(t) where q = P

k=1

(ν

− ν

)k.

It is also possible to find the leading coefficient c

(x) explicitly (see below).

For elliptic operators this result is well known; in this case q = dim M and in

addition all d

= 0. For the operator (2) our results were obtained independently

by G. Ben Arous [2, 3], who used probabilistic methods. Formula (4) was also

proved in [19] for r = 2, and related results were obtained in [1, 18]. D. Jerison and

A. S´ anchez-Calle [8, 9, 15] estimated the kernel U (x, y, t) in terms of the metric

associated with the operator A. From the asymptotics of the heat kernel it is easy

to find the first term of the asymptotics of the spectral function of A [10–12] (for

second order operators in the M´ etivier case this was done by a different method

by G. M´ etivier [13]). To prove Theorems 1, 2 we use the method developed in [3, 7, 8].

1. Dilations and homogeneity. In this section we recall some definitions and propositions connected with homogeneous structures (see [4, 5, 15, 17] for details). Let e

, . . . , e

be a basis in R

and let 0 = ν

< ν

< . . . < ν

be integers.

We write [j] = k if ν

< j ≤ ν

. We define a group of linear automorphisms δ

of R

by

δ

(e

) = s

e

, 1 ≤ j ≤ n .

We also consider a homogeneous norm k · k with respect to δ

such that kuk = 0 ⇔ u = 0, kδ

(u)k = skuk .

For example we can take kuk = ( P

j=1

|u

|

)

. This norm satisfies the fol- lowing inequalities:

ku + vk ≤ C(kuk + kvk), C

|u| ≤ kuk ≤ C

|u| for |u| ≤ C , where | · | is the usual euclidean norm in R

. The number q = P

k=1

(ν

− ν

)k is called the homogeneous dimension of the space. It is easy to see that R

= L

k=1

V

, V

is spanned by the vectors e

for [j] = k, and q = P

k=1

k dim V

. A function f is homogeneous of degree λ if f ◦ δ

= s

f for all s > 0. A distribution v is homogeneous of degree λ if hv, ϕ ◦ δ

i = s

hv, ϕi. A function k(u) is said to be a kernel of type λ if it is smooth away from the origin and homogeneous of degree −Q + λ. A differential operator T is homogeneous of degree λ if T (f ◦ δ

) = s

(T f ) ◦ δ

for all s > 0. For example, the function u

= u

. . . u

is homogeneous of degree [α] = P

j=1

α

[j], the operator u

∂/∂u

is homogeneous of degree [j] − [α]. Let U be a neighborhood of the origin in R

. We define the function space

C

(U ) = {f (u) ∈ C

(U ) : |f (u)| = O(kuk

), u → 0} . A differential operator T = P

|α|≤k

a

(u)∂

from C

(U ) to C

(U ) is said to have degree at most p at 0 whenever T (C

(U )) ⊂ C

(U ) for all m ∈ N. For such an operator it is possible to define an operator b T ,

T = b X

|α|≤k

X

[β]≤[α]−p

(∂

a

(0)u

/β!)∂

,

which is homogeneous of degree p. The operator T − b T has degree at most p − 1 at 0.

Let g be a free nilpotent Lie algebra of step r with l generators, g = L

g

, and [g

, g

] = g

if i + j ≤ r, [g

, g

] = 0 if i + j > r. Using the exponen- tial mapping we can identify g and the corresponding Lie group G; the group multiplication in g will be given by the Campbell–Hausdorff formula

u · v = u + v + [u, v] + . . . , u, v ∈ g .

Following [15] we now define a function class F

: k ∈ F

if (i) k ∈ C

(g \ 0), k(u) = 0 for kuk > 1,

(ii) |P k(u)| ≤ C

(1 + kuk

), for all left-invariant differential operators P homogeneous of degree s.

We will also use another class HF

. A function k is in HF

if k ∈ F

and (i) if λ < Q then k(u) = b k(u) + g(u), where b k(u) is a kernel of type λ, g(u) ∈ C

(g),

(ii) if λ ≥ Q then (i) holds for the function P k for all left-invariant differential operators P of degree s, λ − s < Q.

Lemma 1. 1) If k is a kernel of type λ, ϕ ∈ C

(g) and ϕ(u) = 1 for kuk ≤ 1/2, then ϕk ∈ HF

and P ϕk ∈ HF

for P homogeneous of degree s.

2) If k ∈ HF

, h is a kernel of type β, 0 < β < Q, α > 0 and ϕ ∈ C

(g) with ϕ(u) = 1 for kuk ≤ 1/2 then ϕ(k ∗ h) ∈ HF

.

P r o o f. We have ϕk = b k + g with b k = k and g = (1 − ϕ)k(u) ∈ C

(g) since k(u) is a kernel of type λ, so ϕk ∈ HF

. For P ϕk we observe that since ϕ = 1 in a neighborhood of the origin, P ϕk − ϕP k ∈ C

; since P k is a kernel of type λ − s, we have P ϕk ∈ HF

and 1) is proved.

For 2) we observe that if α + β < Q then by definition k(u) = b k(u) + g(u), so k ∗h(u) = R

k(v)h(v

u) dv = R

b k(v)h(v

u) dv + R

g(v)h(v

u) dv = I

+I

. I

is a kernel of type α + β by the result of Folland [4], I

(u) is a smooth function, and using the same arguments as in Lemma 3 of [15] one can see that g = I

satisfies the required estimate. In the case α + β ≥ Q we have P (k ∗ h) = (P k) ∗ h, P k is a kernel of type λ−s, and λ−s+β < Q, so as was shown before ϕP (k ∗h) ∈ HF

and by definition k ∗ h ∈ HF

.

We say that a function k is in SF

if for any s ∈ N with s > λ, k(u) =

s

X

j=0

k

(u) + q

(u) , where k

∈ HF

and q

∈ F

(g).

2. Lifting of vector fields. Let L(M ) be the Lie algebra of smooth real vector fields on M . There exists a partial homomorphism µ : g → L(M ), that is, µ is linear and for all a ∈ g

, b ∈ g

we have µ([a, b]) = [µ(a), µ(b)] if i + j ≤ r.

Write µ

(a) = µ(a)|

, x ∈ M . We now define

H

(x) = {a ∈ g

: µ

(a) ∈ V

(x)}, 1 ≤ k ≤ r , H(x) =

r

M

k=1

H

(x) . We select S

(x) such that g

= H

(x) ⊕ S

(x), and set S(x) = L

k=1

S

(x). As

was shown in [5], H(x) is a subalgebra in g, dim S

(x) = ν

(x) − ν

(x) and

dim S(x) = dim M . Obviously q(x) is the homogeneous dimension of S(x), and q(x) + β(x) = Q, where β(x) is the homogeneous dimension of H(x).

We now change the local coordinate system in a neighborhood of x ∈ M so that in the new coordinates the vector fields X

, . . . , X

have degree at most one.

It is easy to see that S(x) = g/H(x); let γ be a projection from g to S(x). The essential result in this situation is

Theorem 3 (Helffer–Nourrigat [5]). For any x ∈ M there exists a diffeo- morphism Θ

: U → ω, where U is a neighborhood of 0 in S(x) and ω is a neighborhood of x in M , so that if µ(a) = X then

1) ( d Θ

)

X = b X, Xf (u) = b d dt    

f (γ(u · ta));

2) (Θ

(0))

(0) = µ

|

.

In the M´ etivier case Θ

is smooth in x ∈ M .

We introduce coordinates (u, v) in g so that u ∈ S(x), v ∈ H(x). If µ(a) = X

(1 ≤ i ≤ l) we define a left-invariant vector field Y

on g by Y

f (u, v) = (d/dt)|

f ((u, v) · ta). Consequently,

Y

(f · γ) = d dt    

f (γ((u, v)) ◦ ta) = ( b X

f ) ◦ γ .

Let R

= X

− b X

, 1 ≤ i ≤ l. By Theorem 3 the vector fields R

have degree at most 0 at 0. If we now define e X

= Y

+ R

then we obtain

Lemma 2. e X

(f ◦ γ) = (X

f ) ◦ γ for 1 ≤ i ≤ l.

3. Construction of the fundamental solution. We will consider two dif- ferential operators connected with L:

L = (−1) e

l

X

j=1

X e

+ X

|α|<m

a

(x) e X

+ ∂

∂t , L = (−1) b

l

X

j=1

Y

+ ∂

∂t . Lemma 3. The operator b L is hypoelliptic.

P r o o f. For m = 2 this follows directly from H¨ ormander’s theorem. In the case m > 2 it can be shown by using a criterion of hypoellipticity by Helffer–Nourrigat [6] (see also [12]).

On g

= g × R

we define dilations by δ

(ξ, t) = (δ

(ξ), s

t), ξ ∈ g

, t ∈ R

.

Then Q

= Q + m is the homogeneous dimension of g

. For g

we can define the

spaces F

, HF

, SF

as in the previous section. It is clear that Lemma 1 is true

in this situation. The operator b L is homogeneous on g

of degree m. By a result

of G. B. Folland [4] we can find a kernel k(ξ, t) of type m which is a fundamental

solution for b L, that is,

(5) Lk = δ(ξ, t) b

in the sense of distributions, where δ is the delta distribution on g

.

We denote by U, U

neighborhoods of the origin in S(x), and by V, V

neigh- borhoods of the origin in H(x) which are sufficiently small and satisfy U b U

, V b V

. Let ϕ ∈ C

(U

), ϕ = 1 on U , ψ ∈ C

(V

), ψ = 1 on V , and

%(t) ∈ C

(−2, 2), %(t) = 1 for |t| < 1. We now define k

(ξ, t) = ϕψ%k(ξ, t) .

From the definitions of the operators b L and e L we see that L = b e L + R ,

where R has degree at most m − 1. Consequently, for any s ∈ N, the operator e L can be written in the form

L = b e L +

s

X

i=1

R

+ Q

,

where the R

are homogeneous operators of degree m − i and Q

has degree at most m − s − 1 at 0. Using (5) and Lemma 1 we obtain

(6) Lk e

(ξ, t) = ϕψ% · δ +

s

X

i=1

ϕψ%kr

+ q

for ξ ∈ U

× V

, t ∈ (−2, 2), where r

∈ HF

, q

∈ F

.

Lemma 4. Given s ∈ N there exists a function K

(ξ, t) ∈ SF

such that LK e

= ϕψ% · δ + H

, H

∈ SF

.

P r o o f. We use induction on s. For s = 0 we set K

(ξ, t) = k

(ξ, t), and the statement of the lemma follows from (6). Assume that it is true for s − 1; then we have

LK e

= ϕψ% · δ + H

, H

∈ SF

.

We now define K

(ξ, t) by K

= K

− a(ξ, t)k

∗ H

, where a(ξ, t) ∈ C

(g

), supp a ⊂ U

× V

× (−2, 2) and a ≡ 1 in supp H

. We have

LK e

= ϕψ% · δ + H

− aH

+ H

,

where H

= a(ξ, t) b Lk

∗ H

− e L(a(ξ, t)k

∗ H

). By Lemma 1 it is clear that K

(ξ, t) ∈ SF

, H

∈ SF

and the proof is finished.

By Sobolev’s embedding theorem for any p ∈ N there exists s so that SF

⊂ C

(g). From this fact and the previous lemma

LK e

= ϕψ% · δ + H

, H

∈ C

(g) .

We now want to construct a fundamental solution for the original operator L. Set p

(u, t) = R

K

(u, v, t) dv, h

(u, t) = R

H

(u, v, t) dv .

Lemma 5. Lp

= ϕ% · δ + h

. P r o o f. Let R = L − e L. Then

Lp

= ϕ% · δ + h

− R

RK

dv + R

RH

dv .

The operator R is selfadjoint and acts only in the v variables so it is easy to see that R RK

dv = 0 and R RH

dv = 0, and the lemma is proved.

Using the second property of the map Θ from Theorem 3 one can show that in the original coordinate system (y) in some small neighborhood ω of the point x ∈ M ,

Lv(x)p

= δ + h

,

where v(x) = |det (µ

|

)|. This formula and Lemma 5 imply that (7) U (x, x, t) = v(x)p

(0, t) + g(t) ,

where g(t) ∈ C

(ω × (−1, 1)). By construction, p

(0, t) =

s

X

j=1

R k

(0, v, t) dv + . . .

If k

∈ HF

for λ < β(x) then by definition of this class

R k

(0, v, t) dv = R

b k

(0, v, t) dv + R

g

(0, v, t) dv .

Consequently, p

(0, t) = p b

(0, t) + g(t), where p b

(0, t) is homogeneous of degree λ − β(x) and g ∈ C

(−1, 1).

If λ > β(x) then we have

∂

p

(0, t) = R

∂

k

(0, v, t) dv + R

∂

c(0, v, t) dv = p b

(0, t) + g(t) . The function p b

(0, t) is homogeneous of degree (j −Q)/m−a. If (j −Q)/m−a 6∈ Z then b p

= c

t

, and after integrating over t we obtain

(8) p

(0, t) = c

t

+ g(t) ,

g ∈ C

(−1, 1). If (j − Q)/m − a ∈ Z then p b

= c

t

and so in this case (9) p

(0, t) = c

t

ln(t) + d

t

+ g(t) ,

g ∈ C

(−1, 1). From (7)–(9) it follows that for any s ∈ N 

 U (x, x, t) −

s

X

j=−q(x)

c

(x)t

−

s

X

j=0

d

(x)t

ln(t)  

 < C

t

and the proof of Theorem 1 is finished.

From the proof of Theorem 1 one can find the leading coefficient c

ex- plicitly. It is clear that

c

= v(x) · R

k(0, v, 1) dv ,

where v(x) = |det (µ

|

)| and k(u, v, t) is a fundamental solution for the oper- ator b L.

For the proof of Theorem 2 we observe that in the M´ etivier case Θ

is smooth in x ∈ M and so the asymptotic formula of Theorem 1 is uniform in x. Conse- quently, to obtain the statement of Theorem 2 we just integrate this formula over the manifold M .

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