The estimation of certain kernels - Introduction Contents Balls and metrics defined by vector

We begin by studying the behavior of our families of balls under suitable mappings of the underlying space. The situation we study is motivated by the process of freeing of vector fields in Rothschild and Stein [RS].

Suppose as before that { Yj}, 1

<~j<~q

is a family of C | vector fields on a connected open set f ~ c R N. Assume that the vector field Yj has formal degree

dj,

that for a n y j and k we can write

[YJ'Yk] = E

cJ'k(x) Y~' cJ kEC|

(1)

dl<~dj+d k

and that the vector fields span the tangent space at each point of Q.

Suppose in addition there is a connected open set

V c R M

(with coordinates

sl ... SM) and C |

vector fields I7"1 ... IT"q on f~ • V with the following properties

(1) For each j ,

l<~j<~q,

M a

s) = rp)+ aj (x, s) as "

k=l

(2) The formal degree of ~ is

dj

and we can write for any j and k

d,<-aj+dk

where c~k E C| • V).

(3) The vector fields I7"1 ... IT"q are linearly independent, and span the tangent space at each point of • • V.

The vector fields Y1 ...

Yq

give rise to a metric p and a family of balls

{B(x,

6)} on [2. In the same way, the vector fields I7"1 ... IT'q give rise to a metric ~ and a family of balls

(B((x,

s), 6)} on f2x V, We want to study the relationship between these metrics.

Let :r and :r2 denote the projections from f~ x V to f~ and V

:r(x, s ) = x :r2(x, s ) = s .

If d:r is the differential of this mapping, then condition (1) on the vector fields }7"1 .. . . . ]~'q is equivalent to

rj,

BALLS AND METRICS DEFINED BY VECTOR FIELDS I: BASIC PROPERTIES

This implies that if q3: [0, 1]--.s V is a curve satisfying

~'(t) = ~ aj(t)

~(q~(t))

j=l

and if we let q0:

[0, 1]---~f~

be the composite qo(t) = zr(q~(t)) then

rp'(t) = ~ aj(O

Yj(qg(t)).

j = !

This observation leads immediately to the following.

139

LEMMA 3.1. (a) :t:

B((x,

s),

6)--~B(x, 6) and this mapping is onto.

(b) ~r (exp (E

aj

3) (x, s)) = exp (E

aj Yj) (x).

(c)

I f xl, x2 E s and sl, s2 E V then O(xl,

x2)~<0((xl, Sl), (x2, s2)).

Now fix ~ E C o ( 1 0 . As in Rothschild-Stein [RS], we can define a restriction mapping from functions on f l x V to functions on fl by the formula

RF(y) fvF(Y, s) ~(s) ds

for y E Q, Fly, s) a function on f~x V. In order to estimate these restrictions for suitable F we need

LEMMA 3.2.

Let Ec:x=:~ be a compact set. There is a constant C so that if x E E and if yEB(x, 6) then

fvXa((x,o),~)(y, s) dp(s) ds <~ C

I/~((x' 0), 6) I

18fx, 6)1

Proof.

By Theorem 7, we can choose an N-tuple of vector fields I",. ... Y;N with the following properties: Let Ui= Y;~ for

I<<.j<<.N

and let

V 1 ... Vq_N

denote the remain- ing vector fields. For u E R N, v E R q-N, define

*o(u)

= exp (u' U + o . V) (x).

140 ALEXANDER NAGEL, ELIAS M. STEIN AND STEPHEN WAINGER

< ~d(v)

Then if vj r12o , l<~j<~q-N the mapping

u ~ cI,~(u)

< -e(t~)

is globally one to one for luj[ rl~ o , and the image o f this map contains the ball B(x, ri: 6).

We also consider the map 4): Rq--->R q given by

4)(u, v) = exp (u" U + v. V) ((x, 0)).

N o t e that

sto 4)(u, v) = r Let y E B(x, O) and let

q ~d(~J

Zy = {(U, V) E R [[ujl < r/t o , Iv~l < r]z 6 d(vj), and (I)o(u) = y}.

Since the mapping cI, o is globally one to one, for each v there is a unique O(v)E R N so that

(O(v), v) = Zy i.e. dPo(O(v))=y.

If we differentiate this last equation with respect to vj and use the fact that the Jacobian of r is b o u n d e d b e t w e e n two positive multiples o f ;tAx), it follows that

IJO(u)l ~ clZ~(x)l -~.

Next consider the map Z:

Rq-N'-">Rq-N

given b y X(v) = zl2 4)(O(v), v).

Since 4) is a diffeomorphism

[Jz[ <

C'lJOl <

c"{At(x)l-'.

N o w in the integral

f vX ~C (x, o), ~) (Y , s) r ( s ) d s

BALLS AND METRICS DEFINED BY VECTOR FIELDS I: BASIC PROPERTIES 141 we are just integrating 9 over the image of the map X. The change of variables formula for multiple integrals now gives

fV dE d(Uj)+d(Vj)

z <- <- c

I21(x)l ~d(V)

-< c 0),

IB(x,

We turn now to the proof of Theorem 5.

Choose e so small that L e m m a 3.2 holds. We then write

where

Thus

k(x,

O;y, s) = E

k(x,

0;y,

s)~j(x,

0;y, s) J

)~J(X' O; y' s) = { lo

otherwise.ife2-J<'O(x'0;y's)<2e'2-J

[k(x,

0;y, s) I ~< C E (2e2-J)~

2_j>~Q(x,y)/2e

Vol B(X, 0, e2 -i)

Z B(x, o, 2,.2-~) (y' s)

where ZB is the characteristic function of/~. Now we apply Lemma 3.2 to obtain

9 Vol

(B(x,

2-J)) "

2 -J>~Q(x, y)/2e

We then obtain the estimate for

R k

by comparing the sum to an integral. The estimation of

Xil ... X b g k

is similar. (See the discussion

of X i g k - R X j k

in [RS] p. 302.)

Appendix: Exponential mappings and the Campbell-Hausdorff formula

In this appendix we briefly recall the basic properties of exponential mappings induced by vector fields. Thus suppose that f ~ c R N is an open set, and that Y is a C ~ real vector field defined on f~. Y then induces a local one parameter group of transforma- tions on f~ as follows: for each x E l i let

Er(t, x)=E(t, x)

be the unique solution to the initial value problem

OE (t, x) = Y(E(t, x)), E(O, x) = x

(1)

142 ALEXANDER NAGEL, ELIAS M. STEIN AND STEPHEN WAINGER

This unique solution always exists for Itl sufficiently small. In fact, if Kc:c:fl is compact, there is a number 60>0 so that E:(-6o,6o)XK-->f2 is a Coo mapping.

Moreover, if the vector field Y= Y(u~ ... up) depends in a C ~ manner on a parameter u=(ul ... up) E U c R p where U is open, then if K c c f ~ and L c c U are compact, there is a constant 6o>0 so that the solution

g ( u I . . . Up, t, X) = E: L x (-60, 6o) x K--> f/

is a C ~ mapping. (A discussion of existence and uniqueness of solutions of (1), and of smoothness of dependence on parameters can be found in Dieudonn6 [D], Chapter X.)

The uniqueness of the solution to (1) shows that

and

E ( s , E ( t , x ) ) = E ( s + t , x ) if x E K ,

Isl+ltl<6o

⁽²⁾

Er(2t, x)=E~r(t,x) if x E K , 12t1<6o. (3)

Equation (2) shows that the mapping

x~--~ E(t,x)

is a diffeomorphism on compact subsets of f~ if t is sufficiently small. Equation (3) shows that Ety(1,x) is well defined on compact subsets for all t sufficiently small. We define

exp (Y) (x) =Ey(1,x)

whenever the right hand side is defined. In particular, exp (tie)(x) is always defined for small

Itl.

If f is a C ~ function defined near a point x0 E Q the function F(t) = f ( e x p (t Y) (Xo))

is defined and C ~ near t=0, and the differential equation (1) shows that F'(t) = (Yf) (exp (tY) (x0)).

This shows that the Taylor series expansion o f F at t=0 is given by

F(t) ~ ~ 1_~ (y~f) (Xo) t" = etrf(Xo), (4)

n~O n!

where the last expression is thought of as a formal power series in (tY).

BALLS AND METRICS DEFINED BY VECTOR FIELDS I: BASIC PROPERTIES 143 Now suppose that Y~ . . . Ye are real C | vector fields on ~ . I f u = ( u l . . . Up) E R P, then X~= 1 uj Yj is again a real C | vector field on Q, and we may consider exp (Xje=l uj Yj.) (x) for x E f~, u E R e. It is clear that this is well defined for

sufficiently small. In fact if K c c f ~ is compact, there exists 6o>0 so that exp (E uj Yj)(x) exists for x E K and lu]<do. Moreover the mapping

(u, ... u,;x)~-~exp ( E uj Yy) (x )

is infinitely differentiable. F o r x E f~, we may define

Then expx is a C ~~ mapping o f a neighborhood o f 0 in R p to f~ with expx (0)=x. This induces the differential mapping d(expx): ToRP---~TxQ between tangent spaces. It follows easily from the differential equation (1) that

In particular, if P = N and Y1 ... YN is a basis for the tangent space Tx g2, this shows that d(expx) is a bijection. The inverse function theorem, together with the smoothness of the mapping (ul ... UN; X)~--~exp (E Uj Yj) (x), proves:

PROPOSITION 4.1. Let Yl ... YN be C ~ real vector fields on an open set ~ c R N.

Suppose Y1 ... YN are linearly independent at each point o f if2. let K c c t ) be com- pact. Then there are finite positive constants 61, 62, and C so that:

(a) For x EK, expx is a diffeomorphism o f {u E RNI

lul<61}

onto a neighborhood Vx o f x in f2.

(b) For x E K, {y E RNI Ix--y] <62) t"- Vx"

]eXpx(uO-exp~(u2)] <<. Cluru21.

COROLLARY 4.1. Let XoEQ. Then there is an open neighborhood U o f xo and constants 60>0, C<oo so that

144 ALEXANDER NAGEL, ELIAS M. STEIN AND STEPHEN WAINGER

(a) I f x E U, exp,~ is a diffeomorphism o f (u E RN[ [ul<6o} onto an open set in if2 containing U.

(b) I f x, y E U and y = e x p x ( u ) with

lul<Oo

then expx(tU) E if2 f o r 0~<t~<l.

We shall need a generalization of (a) which is proved in the same way:

COROLLARY 4.2. Suppose VI . . . Vr are some additional vector fields. Then i f

U = ( V l . . .

Or)

with

Ivjl<ao

expx (ul

YI+...+UN

YN+vl

Vl+,..+UrVr)

is a diffeomorphism o f ]ul<6o onto an open subset o f a neighborhood o f expx (vl V1 + . . . + v, V,).

We shall call U a normal neighborhood of Xo. If U is a normal neighborhood o f Xo then for each x E U we can introduce coordinates in U via the mapping (expx)-~:

U---~R N. We call these the canonical coordinates at x, relative to the linearly independent vector fields Y1 . . . YN.

Next, we return to the more general situation of P (not necessarily linearly independent) vector fields Y i , . . . , Ye on g2. Ifxo E g2 a n d f i s a C ~~ function defined near Xo the function

is defined and C ~ near 0 E R N. As a generalization of equation (4) we have PROPOSITION 4.2. The f o r m a l Taylor series o f F at 0 E R e is given by

1 u

F(u, ... ue)-- ~ ~ jYs f(xo)=-eE'rSf(Xo).

n=O

Proof. Define

G(t, ul . . . ue) = F(tui . . . tUB).

Then G(t, Ul . . . u e ) = f ( e x p t(Z uj Yj)(Xo)), and from equation (4) we have O"___GG (0, u I . . . Up) = uj f(Xo).

at n

BALLS AND METRICS DEFINED BY VECTOR FIELDS I: BASIC PROPERTIES 145 On the other hand

n! a aaF,.,, a"G (o,.,, u,,) = 7

- - ..., U -~ua (U)

a f lal=n

.jr f(x0).

ral=n aU a

Q.E.D.

We have seen that if K = c Q is compact, there exists 6o>0 so that the mapping K---~f2 given by

is a diffeomorphism if ]ul<60. We shall often have need to compose two such diffeo- morphisms. Let Xo E fl and let f be a C ~ function defined near x0. Then the function

F(s 1 ... se, t l , . . . , t e ) = f ( e x p ( E s j Y j ) o e x p ( E t j

Yj) (x0))

is defined and C = near 0 E R 2P. According to Proposition 2, for Itl sufficiently small

lalfn a! ~ (tl, t) = ~

Sj Yj

exp tj Yj (Xo) .

Applying the proposition again we have

a~n satfl alal+Ls[ F 1 1 ( E ) m ( E )n

I = av-~! asaOt---~

( 0 , 0 ) -

m'-. nt tjYj sjYj f(Xo).

L81 =m

Thus the formal power series of F at (0, 0) is given by

F(sl ... Sp, t I ... te)~ ~ m ~ n ~ 1 ( E tj y j . ) m ( ~ 5 yj)n f(Xo)"

t r / , n = 0 " "

But the right hand side, viewed as a formal series in t and s is exactly e E5 rj

e~Sj rJf(xo)"

ESrj e~Sj~

According to the Campbell-Hausdorff formula, the formal series e can be written as e z where

146 ALEXANDER NAGEL, ELIAS M. STEIN AND STEPHEN WAINGER

Z=~__l(-1)n+ln-l ~ . o ( a ' f l ' ( c t + f l ) ) - ' ( a d ( ~ t j Y j ) ) a Z ( a d ( ~ s j Y j ) ) # ~

\~.-1 s)

⁽⁵⁾

(see H 6 r m a n d e r [H], p. 160, o r H o c h s c h i l d [Ho], C h a p t e r X). I f M is a positive integer, let ZM be thefinite partial sum o f the formal series for Z so that

Z-Z~=O(lsIM+ltlM).

We now have

Since f is arbitary, we also have:

PROPOSITION 4.3. Let Kc:z:t2 be compact, and let M be a positive integer. There exists a constant C so that f o r x E K

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[FP] FEFFERMAN, C. & PHONG, D. H., Subelliptic eigenvalue problems, in Conference on Harmonic Analysis in honor of Antoni Zygmund, vol. 2, pp. 590--606. Wadsworth, 1983.

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[Sa] SANCHEZ, A . , Estimates for kernels associated to some subelliptic operators. Thesis, Princeton University, 1983.

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Received November 11, 1983

Dans le document Introduction Contents Balls and metrics defined by vector fields I" Basic properties (Page 36-45)