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144

Chapter 8: On a Fourth-Order PDE in Conformal Geometry

the Neumann operator as is related to the geodesic curvature k via the

differential equation

an + k,,e" = k on M. (8.40)

Equations (8.6) and (8.40) suggest that we search for the right pair of cur-

vature functions and their corresponding differential operators through the

Gauss-Bonnet formula. Recall that on 4-manifolds, Paneitz operator P4 and

the fourth-order curvature operator Q are related via equation (8.7). It turns

out that on 4-manifolds there also exists a boundary local invariant of order 3

and a conformal covariant operator P3 of bidegree (0, 3), the relation of (Q, T)

to (P4, P3) on 4-manifolds is parallel to that of (K, k) to (A, A) on compact

surfaces. The expressions of P3 and T on general compact 4-manifolds, like

that of P4i are quite complicated but can be explicitly written down in terms

of geometric intrinsic quantities as in [16], [17]. In particular, via the confor-

mal change of metrics g,,, = e2wg, P3 and T satisfy the equation:

-P3w + T,,e3' = T on 8M,

(8.41)

and

(P3),, = e-3uP3 on OM.

(8.42)

Perhaps the best way to understand how T and P3 were discovered in [17]

is via the Chern-Gauss-Bonnet formula for 4-manifolds with boundary:

X(M) = (327r2)-1 fM(d2 + 4Q)dx + (47r2)-' OM (T - G4 - G5)dy,

(8.43)

where G4 and C5 are boundary invariant of order 3 that are invariant under

conformal change of metrics. Hence for a fixed conformal class of metrics,

Qdv + j Tds

M 8M

is a fixed constant. We would like to remark that in the original Chern-

Gauss-Bonnet formula T is not exactly the term as we have defined in [17];

actually it differs from T by

3AH,

which does not affect the integration

formula (8.43).

Thus on 4-manifolds with boundary it is natural to study the energy

functional

E[w] = 4

iP4w + 2

wQ + 2 ' wP3w + j wT.

(8.44)

Chang

145

In view of the complicated expressions of the operators P4, P3, Q, and T,

at this moment it is difficulty to study the functional E[w] defined as above

on general compact manifolds. But in the special case of (B4, S3) with the

standard metrics, we have

= (0)2, P3 = 21VQ, + % N + z and Q = 0,

and T=3,

(8.45)

where denotes the the Laplacian operator 0 on (S3, g). Thus the expres-

sion in E[w] becomes relatively simple. In this special case, we are able to

study the functional E[w). The main analytic tool is the following sharp

inequality of Lebedev-Milin type on (B4, S3).

Theorem 8.11. Suppose w E C°°(. ). Then

log {

§S3 e3(`'_ )dy}

(8.46)

fB4wA2w+jS32wP3W-4a+4z

-anT

under the boundary assumptions fs3 r[w]ds[w] = 0 where r is the scalar cur-

vature of S3. Moreover, the equality holds if and only if e2i''g on B4 is iso-

metric to the canonical metric g.

The key step in the proof of the theorem above is the following analytic

lemma:

Lemma 8.12. Suppose w solves

I Ozw=0

W IS3= U

k Is3- 0.

in R4

Then

Ow 2 Au+2{(-;K +1)a +1

(8.47)

6B4

- &w

2P3u + 20u - 2z 4

(8.48)

5B4

where P3 = (-D + 1) 12(-0) is the same as the P3 operator defined on S3 as

in §8.3.

We would also like to remark that the second term in (8.47) above, i.e.,

the term { (-,& + 1) s + 1 } is the Dirichlet to Neumann operator on (B4, S3).

We can get the following result as a direct corollary of Lemma 8.12.

146

Chapter 8: On a Fourth-Order PDE in Conformal Geometry

Corollary 8.13. On B4,

P3(w) = P3(w) on 0B4,

provided that &w = 0 in B4.

That is, on (B4, S3),

p3 is

an extension of the Dirichlet-Neumann with

respect to biharmonic functions. Thus the study of the

operator can be

viewed as an extension of the study of Dirichlet-Neumann operator.

Further developments in this field have been made since the paper was

written. We refer the reader to the survey article by S.-Y. A. Chang and

P. Yang, On a fourth-order curvature invariant, which will soon appear in

Contemporary Mathematics.

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Chapter 8: On a Fourth-Order PDE in Conformal Geometry

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Chapter 9

Riesz Transforms,

Commutators, and Stochastic

Integrals

A. B. Cruzeiro and P. Malliavin

9.1

Renormalization in Probability Theory

and Elliptic Estimates in Infinite

Dimension

The state at time n of the gambler's fortune F,, in a game of heads or tails

is almost surely bounded by n2+`; denoting by gk the gain at step k we have

F. = Eo<k<n+i A. Therefore F,, appears as a sum which is small because

of a large number of cancellations. This elementary example is quite typical

of probabilistic situations associated to the Laplace central limit theorem

where semiconvergent series always appear. We shall refer to this fact as an

example of the more general concept of renormalization.

In some branches of classical analysis, as for instance in singular integrals,

cancellations play a fundamental role; a methodology for proving convergence

consists in obtaining a priori estimates for a sequence of smooth approxima-

tions. In our probabilistic context the same approach can work if a priori

elliptic estimates in infinite dimension are available. Indeed, limit theorems

concern games where the number of trials is tending to infinity and the cor-

responding probability space must be of infinite dimension.

The appearance of elliptic operators is a more hidden fact:

it is linked

to the concept of some kind of "universal chaos," that is, some probability

151

152

Chapter 9: Riesz Transforms, Commutators, and Stochastic Integrals

space containing in itself all the random fluctuations which could appear in

the universe. The Ito theory of stochastic differential equations gives to the

probability space X of the Brownian motion this status of "universal chaos;"

it is impossible to justify in few words the paramount importance of this

space.

The Brownian motion can be characterized as a Gaussian process which

has independent increments on disjoint intervals of time; for the sake of

brevity we shall grasp the Brownian motion through its Wiener series:

00 sin k7rr

x(r)=rSo+vL

0<r<1,

k=1

where the ek are independent Gaussian normal variables. We denote by X

the probability space generated by those Gaussian random variables: it is

isomorphic to Rx with the Wiener measure p being defined as the infinite

product of the Gaussian measure pj on R. µl = sR exp (- 2) <. The data

of the l;k constitute some kind of "heat bath." The next step is to say that

this heat bath takes its stochastic regularity from the fact that it represents

some kind of "stochastic equilibrium."

On 1R, the space L2(1R, p1) has for its orthonormal basis the Hermite

polynomials H,,; these polynomials can be characterized as the eigenfunctions

of the following Sturm-Liouville operator:

G12exp 22}dal 2fd

which will be called the Ornstein-Uhlenbeck operator.

It is a symmetric

operator in L2(R, µ1); therefore, the associated parabolic semigroup 1Pt =

etG' preserves µ1.

This semigroup can be explicitly written through the

Mehler formula:

1I't(f)(M = f f(e-teo+(1 -e'ZC)zrl)111(d1l).

It is clear from this formula that 1Pt acts as a contraction on all the L.

This Mehler formula stays meaningful in infinite dimensions when we

replace u1 by the Wiener measure p; we get in this way on X a semigroup

Pt; its infinitesimal generator will be formally denoted by G. The elliptic

estimates that we shall discuss will be related to G.

Cruzeiro and Malliavin

153

9.2

Pisier Transplantation of the Calder on-

Zygmund Rotation Method

The inverse of the Cauchy operator, A, is defined by

1 °O

a-t

A = 1 r P

dt,

.ir

where the semigroup Pt is define by the Mehler formula.

It is possible to say that R' = limR"; therefore it is sufficient to prove

the Riesz transform inequalities for the finite-dimensional case, with constants

independent of the dimension. The gradient V of a smooth function defined

on R" is an R"-valued function. The Riesz transform is defined

R f = VA f ; -the r.h.s. is an R" - valued function.

Theorem 9.1. Given P E (1, oo), there is a constant cp, independent of the

dimension n, such that

IIRfIIRn d

< cpf

Iflpd,

where I I -

I IRn denotes the Euclidean norm and where An denotes the Gaussian

normal measure on W.

Proof. We shall present the Pisier approach. We denote X = Y = R" and

we define on X x Y the measure v = 4!. The rotation matrix on R2 acts

on X x Y; this action preserves v; the orbit of each point is a circle. Making

in the Mehler formula the change of variable t = logcos(9)I, we get

(A f) (x) =

ff (x cos B + y sin 0)1 log(cos 9) I - 2 sin B d9 dµ. (9.2)

The gradient satisfies the identity

so_ Vyf(xcos0+ysin0).

In 0

This identity makes possible to transfer the differentiation relative to x of

the left-hand side of (9.2) into a differentiation relative to y, on which an

integration by parts can be made; we get

(Af)

(x)

f (x cos

9 + y sin 9)0(9)yi dep(dy),

where 0(e) = signum(9) (cos 9) I log cos 9I- a .