Christ M., Kenig C.E., Sadosky C. Harmonic Analysis and Partial Differential Equations: Essays in Honor of Alberto P. Calderon

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154

Chapter 9: Riesz Tlansforms, Commutators, and Stochastic Integrals

The variable yj is a Hermite polynomial of degree 1.

Using the Hermite

expansion of a function in I?, Pisier proved that

11(lZf)(x)I1Rn < cp j IJ(x,y)Ipu(dy),

where J(x, y) =

J *

f (x cos 9 + y sin 9)0(9) do.

We have 0(9) = cot (1) +?P(9), where O E C°°; using the boundedness of the

Hilbert transform on LP of the circle we get the theorem.

In the spirit of Benedeck-Calderbn-Panzone [2], the same theorem remains

true for B-vector-valued functionals, where B is an UMD Banach space [17].

The previous proof of Pisier depends upon beautiful identities whose exten-

sion to a general situation is not reasonable to expect.

9.3

Littlewood-Paley-Calderon-Stein Area

Integral for Harmonic Extension

We shall replace identities of the previous section by computation of com-

mutators; again on the Gaussian space commutators have a very remarkable

expression. In the general case exact expressions will be replaced by ma-

jorizations.

Lemma 9.2. We have

VPtf = e-tPtV.

(9.3)

Denote C = v/ and C' = -G + 1 = A-'; then

Ve =e''V.

(9.4)

Proof. We can differentiate the Mehler formula relatively to x and get (9.3).

The semigroup a-'' can be expressed by the symbolic calculus as follows:

e =

f cc

Pagn(s) ds, where 4,,(s) =

7 s-1

exp (-;) .

(9.5)

Differentiating the right-hand side and using the commutator relation (9.3)

we get the same integral multiplied by exp(-s) which gives exp(-iC').

Given g E LP(X) we consider its harmonic extension h9 to the half-space

Z = X x R+ constructed by

h9 = 0.

h9(x, rj) = (exp(-jC)g)(x); then

{ ads + _C]

Cruzeiro and Malliavin

155

The gradient on Z has its horizontal component Vx and its vertical compo-

nent

. Then, for p E (1, oo), the area integral equivalence ([4],[20]) gives

[f00

1(dx) j Ig(x)I"p(dx),

(9.6)

for all g such that f g dp = 0.

For the vertical gradient the analogous equivalence holds true; we shall

state it for the harmonic extension h' = e-fi'g':

1P 1/

fx [Joy

\aa7h'\ 2

drl]

p(dx) '=' IIg'II,.

(9.7)

Theorem 9.3. (Meyer [18]). We have the following:

I Iof I IP dp =

IC f IP dp.

Proof. By differentiating (9.3) relative to 17 we get

V e-ncC f =

d-e 'vf. (9.8)

Take g = C f

g' = V f

. The identity (9.8) implies that the left-hand side of

(9.6) and (9.7) are equal; therefore their right-hand sides are equivalent.

9.4

Calderon Factorization through the

Cauchy Operator and Stochastic Integrals

We shall not discuss here the general theory of Gross-Sobolev-Stroock spaces,

theory where the Riesz transform plays a paramount role [21]. We limit

ourselves to a basic existence result for stochastic integrals.

In his paper on the uniqueness of the Cauchy problem, Calderdn [5] writes

any differential operator as the product of a power of the Cauchy operator

by a pseuodifferential operator of degree zero. This idea leads us 25 years

later to derive results in stochastic analysis.

We-have a stochastic process 0(7-) defined on the probability space of the

Brownian motion X; we consider partitions e of [0,1] by a finite number of

points; given 9 E e we denote by 9} E e the point of the partition which

is closest on the right to 9. With Nualart-Pardoux we form the following

renormalized Riemann sum:

Je(O) = 2[x(9+) - x(9)]0+1 eE6e f

qS(r) dT

ae-e

156

Chapter 9: Riesz Transforms, Commutators, and Stochastic Integrals

where Eae is the conditional expectation obtained by averaging on the a-

field generated by x(T) - x(8), r E [0, 0+]. We say that ¢ is stochastically

integrable if the limit of Je exists when the mesh of the partition e tends to

zero.

Theorem 9.4 (Watanabe [22]). Assume that

1 dT < oo.

f IIC'O(T)IILZ(X)

Then 0 is stochastically integrable and its stochastic integral satisfies the

following L2 majoration:

{(f'(T)dx(T))2]

0IIIIII.

Proof: We shall first proceed to a transfer by duality.

Lemma 9.5 (of integration by parts (Gaveau-Trauber [13])). Assume

that 0 is stochastically integrable. Then, for every smooth functional f, the

directional derivative of f along 0 denoted by DO(f) satisfies the duality re-

lation

E(D4,f)=E(fJ

c5dx).

Remark. This lemma identifies the stochastic integral of 0 with "the diver-

gence of the vector field 0."

Proof. (of the theorem) Using the Riesz transform R' = C-'V f we have

V f = CR' f ; therefore, we write (9.9) as

((CR'fI0) = (R'fIC0) = (fI[R']*CO),

where the fact that C is a symmetric unbounded operator in L2(X) implies

the first equality; the second one is due to the hypothesis CO E L2(X; L2([0,11))

and, since R' is a bounded operator from L2(X) to L2(X; L2([0,1])), its

transpose is well defined. We get finally the following expression for the

stochastic integral, which, quite unexpectedly, establishes a beautiful link

between probability theory and singular integrals:

¢dx = [R']`Cf.

The Riesz transform has a norm as a bounded operator in Lz which is equal

to 1. This implies the announced inequality.

Cruzeiro and Malliavin

157

9.5 Commutators in Renormalized

Riemannian Geometry on Path Spaces

9.5.1

Ito Global Chart of Path Space

The Brownian motion sits in the Euclidean space R or R' and has therefore

a linear structure reflected for instance in its representation by the Wiener

series. We want to proceed in the nonlinear setting of the Brownian on a

compact Riemannian manifold M; we denote by 2L its Laplace-Beltrami

operator and by 7rt(mo, m) the fundamental solution of the heat operator

associated to 0, with pole at mo. Then the Brownian motion on M is char-

acterized by the property that it is a stochastic process p(r) having for law

of its "increments" ire+_e(p(O), *) dm for any finite subdivision 0 of [0, 11,

with the Markovian independence of those increments. Up to operations of

finite-dimensional geometry, we can limit ourselves to the case where the

Brownian motion starts at T = 0 from a fixed point. Then the probabil-

ity space of the Brownian motion will be the path space Pmo(M) with the

Brownian probability measure v.

The first question is how to construct a canonic global chart of P,,,o(M).

This ambition is feasible according to the fact that P,,,, is a contractible

space. We recall that Levi-Civita parallel transport t;,o along a smooth

curve r -4 q(T) is defined by solving the matrix differential equation

drtr-o = r(. dr)tQ -o,

(9.10)

where I'k,ti denotes the Christofel symbols of the metric ds2 = g;jdm'dmi

computed in local coordinates. By some smoothing procedure we can con-

struct a 1-parameter family pE of smooth curves such that lim fro pE = p.

Then

Theorem 9.6 (K. Ito [141). The following limit exists:

l o t"«-o

t;+-o.

Theorem 9.7 (Malliavin [15], Eells-Elworthy). We define a map J-':

P., (M) -+ Po (Rd) by

T o,(dp(o)),

x(T) = f

where the above integral is an Ito stochastic integral; then this map is an iso-

morphism of probability spaces when we take on P0(Rd) the Wiener measure.

We call 3 the Ito map.

Remark. The Ito map provides the desired global chart.

This chart pre-

serves the probability measures, but it does not preserve many other canonic

geometric objects which must be directly constructed on P, (M).

158

Chapter 9: Riesz Transforms, Commutators, and Stochastic Integrals

9.5.2

Structure Equation of the Path Space

Tangent process: Any infinitesimal variation Z of a Brownian path p on M

will be canonically parameterized by the vector-valued function tLT(ZT). We

call a tangent process along the Wiener space the data of a semimartingale

S which has its martingale part constructed by multiplying the Brownian

differential by an antisymmetric matrix: d(° = as dx'5 + c° dr. A tangent

process along Pm,,(M) is defined as to,T(((r)), where ( is a tangent process

on X.

Theorem 9.8 (Cruzeiro-Malliavin [9], Fang-Malliavin [121). The Ito

map can be differentiated; its differential .7' realizes an isomorphism between

the spaces of tangent process.

This theorem is false if we are dealing with restricted tangent processes,

that is, processes with a vanishing martingale part.

The expression of the derivative of the Ito map incorporates a holon-

omy factor which is a stochastic integral of the curvature of the underlying

Riemannian manifold. Then the derivative of a smooth functional f on the

path space and along a tangent process tLo(((r)), that we denote by Ds,

is computed via the transfer to the Wiener space through the Ito map. It

corresponds to the derivative D- (f o ,7), where

d(+

Sl(o',

odx(o)1 odx(r)

[fT

with S2 the curvature tensor of M and where the stochastic integral is of

Stratonochich type.

Theorem 9.9 (Bismut [3], Driver [11]). Tangent processes have a for-

mula of integration by parts:

E.,(D(f) = E (f

[a,

12RM

(fTc)]

dx(r))

where R`N = to+-To Ricci o to1-T

Theorem 9.10 (Structure theorem, Cruzeiro-Malliavin [9]). The Lie

bracket of two restricted tangent processes z1, z2 has in the parallelism the

following expression:

[zi, z2] = Qz,z2 - QZ,zI where Q., (7) = f lp(,,)(z.,,odp(a)),

where Q,, denotes the curvature tensor read in the parallelism to,,,.

Cruzeiro and Malliavin 159

We encounter therefore a new hypoelliptic phenomenon, which leads to

a necessary renormalization and is present in all subsequent developments

of the analysis on the path space. Nevertheless, we still have a natural Lie

algebra structure:

Theorem 9.11 (Cruzeiro-Malliavin [9], Driver [11]). The Lie bracket

of two tangent processes is a tangent process.

9.5.3 Energy Estimates for Stochastic Integrals

The structure theorem provides tools that are needed to compute commuta-

tors, which, in the spirit of Calderon [6], are the basic gap to reach elliptic

estimates. The commutator effect will bring a small loss in the exponent of

the integrability.

The Levi-Civita covariant derivative VI on M induces a Markovian co-

variant V" derivative on the path space.

Theorem 9.12 (Cruzeiro-Fang [8]). Given a process Z,. such that for some

p>2

< oo,

E [(f

IOPZQ12 drdo

the divergence of Z, b(Z), exists and satisfies E(6 (Z)2) < oo.

By Bismut's integration by parts formula we can identify the stochastic

integral fo ZT dp(r) with the divergence, modulo the Ricci correction term.

9.6

Elliptic Estimates and Riesz Transforms

on Pmn(M)

In its essence the Littlewood-Paley-Calderon-Stein area integral approach

outlined in §9.3 can be worked out in the nonlinear setting [10]. Technically,

though, some major gaps have to be filled.

We do not have a representation of the type of Mehler formula for the

semigroup and the simple commutators of the Gaussian case are no longer

available. The Ornstein-Uhlenbeck operator L f = -bD f has an explicit

expression which has recently been computed by Kazumi.

We take the point of view of Airault-Malliavin [1] that consists in evalu-

ation the intertwining of the differential with the Laplace-Beltrami operator

operating on functions, making in this way the Laplace-Beltrami operator on

forms, that is, the de Rham-Hodge operator, appear. We denote by A the

160

Chapter 9: Riesz Transforms, Commutators, and Stochastic Integrals

de Rham-Hodge operator on differential forms wz of degree one and, given

the underlying Hilbert structure of the restricted tangent space, we identify

the form with the restricted tangent process z. Then we write

A=6d+db

with (df, z) = Da f .

The semigroup a-IL on the path space satisfies

(de-tU f) = dbde-tL f =

A(de-tL f)

since ddu = 0. The problem of estimating the commutator between De-tL

and

a-tLD

reduces therefore to estimating the difference between the op-

erators L and A on differential forms, for which one has to obtain explicit

expressions. A Weitzenbock formula for 1-differential forms is then derived

in the spirit of the results of [9], where such type of theorem was shown with

respect to a particular Markovian covariant derivative on the path space.

Theorem 9.13 (Cruzeiro-Malliavin [101). The difference between the Orn-

stein-Uhlenbeck and the de Rham-Hodge operator defined for 1-differential

forms on the path space is of type

(wZ) - L(w2) = A(z) + B(Dz),

where A and B are operators with kernel representations expressed in terms

of stochastic integrals.

The stochastic integrals appearing in this Weitzenbock formula come from

the structure equations on one side and, on the other, from the integration

by parts formula. Then a commutation formula allows us to derive stochas-

tic integrals. Since one integrates geometric bounded quantities, the final

estimates follow from Ito's stochastic calculus.

References

[1] H. Airault and P. Malliavin, Semi-martingales with values in a Euclidean

vector bundle and Ocone's formula on a Riemannian manifold, Proc. Cor-

nell Conf. 1993, Amer. Math. Soc., Providence, 1995.

[2] A. Benedeck, A. Calder5n, and R. Panzone, Banach-valued convolution

operators, Prop. Nat. Acad. 48 (1962), 356-365.

[3] J. M. Bismut, Large deviations and Malliavin calculus, Birkhauser, Boston,

1984, 216 pages.

Cruzeiro and Malliavin 161

[4] A. Calderon, A theorem of Marcinkiewicz and Zygmund, Trans. Amer.

Math. Soc. 68 (1950), 54-61.

[5] , Uniqueness in the Cauchy problem, Amer. J. Mathematics 80

(1958), 16-36.

[6] ,

Commutators, singular integrals on Lipschitz curves, applica-

tions, ICM Helsinki. vol. 1, Helsinki (1978), 84-96.

[7] A. Calderon and A. Zygmund, On singular integrals, Acta Mathematica

88 (1952), 85-139.

[8] A. B. Cruzeiro and S. Fang, Une inegalite L2 pour les integrales stochas-

tiques anticipatives sur une variete riemannienne. C. R. Acad. Sci. Paris

321 (1995), 1245-1250.

[9] A. B. Cruzeiro and P. Malliavin, Renormalized differential geometry on

path space: Structural equation, curvature, J. Fint. Analysis 139 (1996),

119=181.

[10] , Commutators on

path spaces and parabolic, elliptic estimates,

in preparation.

[11] B. Driver, A Cameron-Martin type quasi-invariance theorem for the

Brownian motion on a compact manifold, J. Funct. Analysis 109 (1962),

272-376.

[12] S. Fang and P. Malliavin, Stochastic analysis on the path space of a

Riemannian manifold, J. Funct. Analysis 118 (1993), 249-274.

[13] B. Gaveau and P. 'I'Yauber, L'integrale stochastique comme operateur

divergence, J. Funct. Analysis 38 (192), 230-238.

[14] K. Ito, The Brownian motion and tensor fields on Riemannian manifolds,

Proc. Int. Cong. Math. Stockholm, Stockholm (1962), 536-539.

[15] P. Malliavin, Formule de la moyenne, calcul de perturbations et theoreme

d'annulation pour les formes harmoniques, J. Funct. Anal.

17 (1974),

274-291.

[16]

Stochastic Analysis, Springer, Berlin, 1996, 370 pages.

[17] P. Malliavin and I). Nualart, Quasi-sure analysis of stochastic flows and

Banach space valued smooth functionals, J. Funct. Analysis 112 (1993),

429-457.

162 Chapter 9: Riesz Transforms, Commutators, and Stochastic Integrals

[18] P. A. Meyer, Transformations de Riesz pour les lois gaussiennes, Sem.

Proba. XVII, Springer Lecture Notes 1059 (1984), 179-193.

[19] G. Pisier, Riesz transform: A simpler proof of P. A. Meyer inequalities,

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[20] E. Stein, Abstract Littlewood-Paley theory and semi-group, Princeton

Univ. Press, Princeton, 1970.

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Chapter 10

An Application of a Formula of

Alberto Calderon to Speaker

Identification

Ingrid Daubechies and Stephane Maes

It was an honor as well as a great pleasure for me to be asked to contribute

to Alberto Calderon's seventy-fifth birthday conference. More than at any

other similar conference, one could feel in the air the friendship, the sense of

community among the participants representing the many different branches

of analysis with whom Calderdn himself is associated. It was wonderful to

have been invited to be a part of this celebration.

As the subject of my presentation at a meeting dedicated to Alberto

Calderon's wide ranging interests, I chose a topic that may have reminded

him of the interest in engineering of his youth. Although most of my work

is in mathematics, I make occasional excursions into engineering, assisted by

students or other collaborators who are the real engineers in the project. The

paper below reports on such an excursion. There was an additional reason to

dedicate this presentation to Alberto Calderon: the whole project had been

directly motivated by a different reading of a classical integral formula due

to him (see formulas (10.2, 10.3) in the paper).

- Ingrid Daubechies

Research by Ingrid Daubechies partially supported by NSF grant DMS-9401785. The

bulk of this paper appeared under the title A Nonlinear Squeezing of the Continuous

Wavelet T}nnsform Based on Auditory Nerve Models, 527-546, in Wavelets in Medicine

and Biology (A. Aldroubi and M. Unser, editors), CRC Press, 1996. This material was

reprinted by permission from CRC Press, Boca Raton, Florida.

163