Spectral multipliersfor the Ornstein-Uhlenbeck semigroup

J. Garcıa-Cuerva, G. Mauceri, P. Sjogren and J. L. Torrea ∗

May 7, 2004

Abstract

The setting of this paper is Euclidean space with the Gaussian measure. We let Lbe the associated Laplacian, by means of which the Ornstein-Uhlenbeck semigroup isdefined. The main result is a multiplier theorem, saying that a function of L whichis of Laplace transform type defines an operator of weak type (1,1) for the Gaussianmeasure. The (distribution) kernel of this operator is determined, in terms of anintegral involving the kernel of the Ornstein-Uhlenbeck semigroup. This applies inparticular to the imaginary powers of L . It is also verified that the weak type constantof these powers increases exponentially with the absolute value of the exponent.

1991 Mathematics Subject Classification. 42B15, 42B20, 42C10.

Keywords. Ornstein-Uhlenbeck semigroup, Mehler kernel, singular integrals, Fourier multi-pliers, imaginary powers of Laplacian.

∗The four authors have received support from the European Commission via the TMR network “HarmonicAnalysis”.

1

1 Introduction

We shall be working in a finite-dimensional Euclidean space Rd, where we shall consider theGaussian measure dγ(x) = e−|x|

2dx and also Lebesgue measure dx. The operator

L = −12∆ + x · ∇, defined on the space of test functions (i.e., the space C∞0 (Rd) of smooth

functions with compact support on Rd ), has a self-adjoint extension to L2(γ), also denotedL . The spectral properties of L are well known: L is positive semi-definite with discretespectrum {0, 1, ...}. For d = 1, the eigenfunctions are the Hermite polynomials, given byHn(x) = (−1)nex

2 dn

dxn e−x2

, n = 0, 1, ... . The eigenfunctions for arbitrary d are tensorproducts Hα = ⊗d

i=1Hαi, where α is a multiindex. For each nonnegative integer k, we

shall denote by Pk the orthogonal projection of L2(γ) onto the subspace generated by thehomogeneous Hermite polynomials of degree k. The operator L is the infinitesimal generatorof a “heat” semigroup, the so-called Hermite or Ornstein-Uhlenbeck semi-group (e−tL)t≥0,defined in the spectral sense as

e−tL =+∞∑k=0

e−tkPk.

In other words, e−tL is the bounded operator on L2(γ) which maps each Hα to e−t|α|Hα. Itcan be shown that the semi-group (e−tL)t≥0 is also given by the Mehler kernel

Mt(x, y) =1

πd/2(1− e−2t)d/2exp

(−|e

−tx− y|2

1− e−2t

),

in the sense that e−tLf(x) =∫Mt(x, y)f(y) dy for f ∈ L2(γ). See [Me, pp. 172-3] and also

the survey article [S3].During the last two decades, a considerable effort has been devoted to the study of

several operators naturally associated with the operator L and their boundedness propertiesbetween different spaces given by the Gaussian measure γ, mainly the spaces Lp(γ) andweak−L1(γ). At this time, a whole area of research, which is sometimes called GaussianHarmonic Analysis, is already well established. As a significant but incomplete sample ofthese investigations, we can cite the papers [S2], dealing with the maximal function associatedto the semigroup, [F-G-S], [F-S], [G-S-T], [Gz], [My], [Mu2], [Pe], [Pi], [U] and our recentarticle [GMST], dealing with Riesz transforms of different orders.

In [GMST] we prove the weak type (1, 1) boundedness for the Riesz transforms of order 2.Our method consists in decomposing the operator in a “local” part, given by a kernel livingclose to the diagonal, and the remaining or “global” part. The local part is essentially aCalderon-Zygmund singular integral, and the global part is delicately analysed by estimatingits size in different regions and using the “forbidden regions” method coming from [S2].

In the present paper we use the same philosophy to deal with multipliers of Laplacetransform type (in the terminology of [St1]) associated to L. Their boundedness in Lp(γ),for 1 < p <∞, is a result of the general theory given in [St1]. The main result of this paperis the weak type (1, 1) of these multipliers, with respect to γ. Two main aspects distinguishthe present work from [GMST]. First of all, we improve the treatment of the local part bymaking a smooth truncation and reducing the estimates to the general Calderon-Zygmund

2

theory. Then the global part can be immediately bounded by the maximal Mehler kernelused by P. Sjogren in [S2], so that the delicate estimates of [GMST] to deal with the globalpart are now simply replaced by an appeal to the bounds in [S2].

We also investigate how to define the multiplier operator in terms of its kernel, as a limitof truncated integrals. In particular, we see under what conditions the multiplier is given bya principal value integral.

Boundedness is also proved for the maximal multiplier operator, via a vector-valuedversion of the estimates.

Our result applies in particular to the imaginary powers of L. Here the growth of theoperator (quasi-)norm for large imaginary powers is of special interest, for reasons which weshall briefly describe.

The assumption that the function m is of Laplace transform type implies that m extendsto a holomorphic function on the right half plane {z : <z > 0} , which is bounded on everysector Sθ = {z : | arg z| < θ} , 0 < θ < π/2. Since the spectrum of the operator L on L1(γ)is the closed right half plane [D, pp. 115–116], it is natural to impose a holomorphy conditionon the multiplier m if we want the operator m(L) to be defined on L1(γ). On the otherhand, since the spectrum of L on Lp(γ), 1 < p < ∞ , is the set N of nonnegative integers,it seems too stringent to require holomorphy of the multiplier m to obtain boundedness ofm(L) on Lp(γ).

In [M], S. Meda gave a sufficient condition for the existence of a nonholomorphic func-tional calculus for the generator A of a symmetric contraction semigroup on Lp(M), 1 ≤p ≤ ∞ , where M is a σ -finite measure space. He proved that, if the norms of the imagi-nary powers Aiα of the generator, as an operator on Lp , grow polynomially as |α| tends toinfinity, then the multiplier operator m(A) is bounded on Lp, provided that the function msatisfies a Hormander condition of sufficiently high order on (0,+∞) [M, Theorem 4]. Inparticular there exist nonholomorphic functions m on (0,+∞) such that m(A) is boundedon Lp. The polynomial growth condition for the imaginary powers is satisfied for instanceby an invariant Laplacian or sublaplacian on groups of polynomial growth.

A standard method to obtain estimates of the norms of Aiα on Lp is via complex in-terpolation between a weak type (1,1) estimate and the L2 estimate. The L2 estimate istrivial, since by the spectral theorem the operators Aiα are unitary on L2 .

In view of these remarks it is important to obtain sharp estimates of the weak type(1,1) quasinorm of the imaginary powers of the operator L. Since L has a nontrivial kernel,to define imaginary powers we must first restrict L to the orthogonal complement of thekernel. This amounts to considering LiαΠ0, where Π0 = I−P0. This is a particular case of amultiplier operator m(L) of Laplace transform type, to which we can apply our main result.From its proof, one can see that the weak type (1, 1) constant of LiαΠ0 increases at mostexponentially as |α| → ∞. We prove that this estimate cannot be improved to polynomialgrowth. We also show that for negative powers of L there is not even a weak type (1, 1)estimate.

This paper is organized as follows. Section 2 contains the setup, the derivation of thekernel of the multiplier operator outside the diagonal and some estimates for it. In Section3 the weak type (1, 1) estimate is proved. Section 4 is devoted to the expression of the

3

operator in terms of the kernel. The maximal multiplier operator is estimated in Section 5.Finally, Section 6 contains the discussion of the imaginary and negative powers of L.

4

2 The multiplier operator and its kernel

For every complex-valued function m defined on the set N of the nonnegative integers, themultiplier operator

m(L) =+∞∑k=0

m(k)Pk

is a densely defined operator on L2(γ) with domain

Dm =

{f ∈ L2(γ) :

+∞∑k=0

|m(k)|2 ||Pkf ||2 <∞}.

The operator m(L) is bounded on L2(γ) if and only if the function m is bounded and thenorm of m(L) qua operator on L2(γ) is the supremum of |m| . The projection P0 onto thekernel of L is a bounded operator on each Lp(γ), 1 ≤ p < ∞. By subtracting m(0)P0 ifnecessary, we can thus assume that m(0) = 0. Following [St1], we say that the function mis of Laplace transform type if

m(k) = k∫ +∞

0φ(t)e−kt dt, k > 0,

where φ is a bounded measurable function on (0,+∞). Performing the change of variablese−t = r , we see that a function m is of Laplace transform type if and only if

m(k) = k∫ 1

0ψ(r)rk dr/r, k > 0,(2.1)

where ψ(r) = φ(− log r). It follows from the general Littlewood-Paley theory for semigroupsthat, if m is of Laplace transform type, then m(L) is bounded on all the spaces Lp(γ),1 < p < ∞ , [St1, Chapter IV, §6]. Our aim in this paper is to show that m(L) is alsoof weak type (1, 1) with respect to the Gaussian measure. If m is of Laplace transformtype, then m(L) is a continuous operator from the space of test functions to the space ofdistributions on Rd , and so it has a distributional kernel. In this section, we shall provethat, off the diagonal, this kernel has a density Kψ with respect to the measure dγ(x)⊗ dy,which satisfies the standard Calderon-Zygmund estimates in a suitable neighbourhood of thediagonal. See Lemma 2.1 and Theorem 2.2 below.

Our starting point is the family of operators rL, 0 ≤ r < 1, whose integral kernels

Mr(x, y) = π−d/2(1− r2)−d/2 exp

(−|rx− y|2

1− r2

)

may be obtained from the Mehler kernel by the change of variables t = − log r. Thus

rLf(x) =∫Mr(x, y) f(y) dy

5

for all test functions f . Since the Mehler kernel satisfies the heat equation ∂tMt = −LxMt,the kernel Mr satisfies the transformed equation r∂rMr = LxMr. If ψ ∈ L∞(R+), define

Kψ(x, y) =∫ 1

0ψ(r) ∂rMr(x, y) dr.

For t > 0 the local region Nt is the neighbourhood of the diagonal in Rd × Rd defined by

Nt =

{(x, y) : |x− y| < t

1 + |x|+ |y|

}.

Lemma 2.1 If x 6= y the integral defining Kψ is absolutely convergent. Moreover, for t > 0and each pair of multiindices α and β in Nd of lengths a and b, respectively, there exists aconstant C such that

|∂αx∂βyKψ(x, y)| ≤ C||ψ||∞

|x− y|d+a+b

for all (x, y) ∈ Nt , x 6= y .

Proof The formula

∂αx∂βyMr(x, y) = π−d/2(1− r2)−(d+a+b)/2(−r)aHα+β

(rx− y√1− r2

)exp

(−|rx− y|2

1− r2

)(2.2)

is a consequence of the definition of the Hermite polynomials. Thus an elementary compu-tation shows that the function r 7→ ∂r∂

αx∂

βyMr(x, y) is the product of the positive function

π−d/2(1−r2)−(d/2)−a−b−2 exp(−|rx−y|2

1−r2)

and of a polynomial in r of degree at most 2a+b+3,whose coefficients depend on x and y . Hence, as a function of r , it changes sign a finitenumber of times and there exists a constant C such that∫ 1

0|ψ(r)| |∂r∂αx∂βyMr(x, y)| dr ≤ C ||ψ||∞ max

0≤r<1

∣∣∣∂αx∂βyMr(x, y)∣∣∣ .(2.3)

for all x and y in Rd. By (2.2) we have that

∣∣∣∂αx∂βyMr(x, y)∣∣∣ ≤ C(1− r2)−(d+a+b)/2 exp

(−c0

|rx− y|2

1− r2

)(2.4)

for some positive constant c0 . Since, in the local region Nt , one has

|rx− y|2 ≥ |x− y|2 − 2(1− r)|x||x− y| ≥ |x− y|2 − 2(1− r)t,

the right-hand side in (2.4) can be estimated by

C(t)(1− r2)−(d+a+b)/2 exp

(−c0

|x− y|2

1− r2

)≤ C|x− y|−(d+a+b)

for all (x, y) in Nt . This proves the lemma.ut

6

Theorem 2.2 If the function m is of Laplace transform type, given by formula (2.1), then

m(L) =∫ 1

0ψ(r)L rL dr/r,(2.5)

where the integral converges in the weak operator topology of L2(γ). Moreover if f is a testfunction,

m(L)f(x) =∫Kψ(x, y)f(y) dy

for all x outside the support of f .

Proof For every pair of functions f , g in L2(γ), we shall denote by 〈f, g〉γ their innerproduct in L2(γ). Thus

〈m(L)f, g〉γ =∞∑k=1

m(k)〈Pkf, g〉γ

=∞∑k=1

k∫ 1

0ψ(r) rk dr/r 〈Pkf, g〉γ

Since∑∞k=0 |〈Pkf, g〉γ| ≤ ||f || ||g|| , we may interchange the order of summation and integra-

tion to get

〈m(L)f, g〉γ =∫ 1

0ψ(r)

∞∑k=1

krk〈Pkf, g〉γ dr/r

=∫ 1

0ψ(r)〈L rLf, g〉γ dr/r(2.6)

This proves (2.5). To compute the kernel of the operator m(L), assume that f and g aretest functions on Rd . Then

〈L rLf, g〉γ = 〈rLf, Lg〉γ=

∫∫Mr(x, y)f(y) dy Lg(x) dγ(x)

= 〈Mr dγ(x)⊗ dy, Lx(g ⊗ f)〉

Here 〈·, ·〉 denotes the pairing between distributions and test functions on Rd × Rd andMr dγ(x)⊗ dy is the distribution whose density with respect to the measure dγ(x)⊗ dy isMr . Since the operator L is symmetric with respect to the Gaussian measure,

〈L rLf, g〉γ = 〈(LxMr) dγ(x)⊗ dy, g ⊗ f〉

=∫∫

r∂rMr(x, y) g(x) f(y) dy dγ(x).

Thus, by (2.6),

〈m(L)f, g〉γ =∫ 1

0ψ(r)

∫∫∂rMr(x, y) g(x) f(y) dγ(x) dy dr.(2.7)

7

If f and g have disjoint supports, the triple integral in (2.7) is absolutely convergent in viewof Lemma 2.1. Thus, by Fubini’s theorem

〈m(L)f, g〉γ =∫∫

Kψ(x, y) f(y) dy g(x) dγ(x).

This proves that Kψ is the restriction to the complement of the diagonal of the kernel ofm(L).

ut

3 The weak type estimate

Let µ denote either the Lebesgue or the Gauss measure on Rd . In this section we shallconsider a linear operator T mapping the space of test functions into the space of measurablefunctions on Rd, satisfying the following assumptions:

(a) T either extends to a bounded operator on Lq(µ) for some q, 1 < q < ∞, or is ofweak type (1, 1) with respect to µ;

(b) there exists a measurable function K, defined in the complement of the diagonal inRd × Rd , such that for every test function f

Tf(x) =∫K(x, y) f(y) dy,

for all x outside the support of f ;

(c) the function K satisfies the estimates

|K(x, y)| ≤ C

|x− y|d, |∂xK(x, y)|+ |∂yK(x, y)| ≤ C

|x− y|d+1,

for all (x, y) in the local region N2, x 6= y .

Let ϕ be a smooth function on Rd × Rd such that ϕ(x, y) = 1 if (x, y) ∈ N1, ϕ(x, y) = 0 if(x, y) /∈ N2 and

|∂xϕ(x, y)|+ |∂yϕ(x, y)| ≤ C |x− y|−1 if x 6= y.(3.1)

We define the global and local parts of the operator T by

Tglobf(x) =∫K(x, y)(1− ϕ(x, y))f(y) dy,

Tlocf(x) = Tf(x)− Tglobf(x).

We shall prove that, if the operator T satisfies assumptions (a), (b) and (c), then its localpart is bounded on Lp(γ) and on Lp( dx), for 1 < p < ∞. Moreover Tloc is of weak type(1,1), both with respect to the Lebesgue and the Gauss measure (see Theorem 3.7 below).The boundedness of the local part of the multiplier operator m(L) defined in the previous

8

section will follow at once. We shall need the following covering lemma whose proof can befound in [GMST]; item 3. below is modified in a simple way. If B is a ball in Rd, we shalldenote by δB the ball that has the same centre as B and radius δ times that of B. Weshall denote by |E| the Lebesgue measure of a set E in Rd.

Lemma 3.1 There exists a collection of balls

Bj = B(xj,κ

1 + |xj|),

where κ = 1/20, such that

1. the collection {Bj : j ∈ N} covers Rd ;

2. the balls {14Bj : j ∈ N} are pairwise disjoint;

3. for any A > 0, the collection {ABj : j ∈ N} has bounded overlap, i.e.,sup

∑j χABj

<∞;

4. Bj × 4Bj ⊂ N1 for all j ∈ N;

5. x ∈ Bj ⇒ B(x, κ1+|x|) ⊂ 4Bj.

Lemma 3.1 will be basic in passing from estimates with respect to the Lebesgue measureto estimates with respect to the Gaussian measure and viceversa. Indeed the Lebesgue andthe Gaussian measures are equivalent on each ball of the collection {4Bj : j ∈ N}. Moreprecisely, there exist two positive constants c and C such that for all j ∈ N

c e−|xj |2|E| ≤ γ(E) ≤ C e−|xj |2|E|(3.2)

for every subset E of 4Bj.We shall also need the remark that

N1/7 ⊂⋃j

(Bj × 4Bj) ⊂ N1.(3.3)

This follows from 4. of Lemma 3.1 and the simple observation that, for x in Bj,

1 + |xj| ≤21

20(1 + |x|).

Similarly, one verifies that

(x, y) ∈ N2 and x ∈ Bj ⇒ y ∈ ABj(3.4)

for some constant A > 0.

9

Lemma 3.2 Let µ be a nonnegative Borel measure on Rd. Given a sequence of nonnegativemeasurable functions (fj), let f =

∑j χBj

fj, where {Bj : j ∈ N} is the collection of balls inLemma 3.1. Then

µ{x : f(x) > λ} ≤∑j

µ{x ∈ Bj : fj(x) > λ/M}(3.5)

for all λ > 0, and

||f ||q ≤M

∑j

∫Bj

|fj|q dµ

1/q

,(3.6)

for 1 ≤ q <∞.

Proof Since every point belongs to at most M balls Bj, the level set {x : f(x) > λ} iscontained in

⋃j{x ∈ Bj : fj(x) > λ/M}. This proves (3.5). To prove (3.6), we simply observe

that ∫Rd|f |q dµ ≤

∫Rd

(M max

j|fj|χBj

)qdµ ≤M q

∫Rd

∑j

|fj|qχBjdµ.

ut

Lemma 3.3 Let µ denote either the Lebesgue or the Gauss measure on Rd. Let T be alinear operator mapping the space of L∞ functions with compact support into the space ofmeasurable functions on Rd. Given the covering {Bj} in Lemma 3.1, we define the operator

T 1f(x) =∑j

χBj(x)T (χ4Bj

f)(x),

for measurable and locally bounded functions f. Then(i) if T is of weak type (1, 1) with respect to the measure µ, it follows that T 1 is of weaktype (1, 1) with respect to the Lebesgue and the Gaussian measure; and(ii) if T extends to a bounded operator on Lq(µ) for some q, 1 < q <∞, it follows that T 1

is bounded on Lq(dx) and on Lq(γ).

Proof To prove (i), we observe that

µ({x ∈ Bj : |T (χ4Bjf)(x)| > λ/M}) ≤ C

λ

∫4Bj

|f | dµ.

uniformly in j. Because of (3.2), this holds for both the Lebesgue and Gauss measures. Itis now enough to apply (3.5) and use the bounded overlap of the collection {4Bj}.

The proof of (ii) is analogous.ut

Remark Lemma 3.3 is also true in the vector-valued setting in the following sense. Given twoBanach spaces E and F and a linear operator T bounded from L1

E( dµ) to weak-L1F ( dµ),

then defining the operator T 1 as in Lemma 3.3, we have that T 1 is bounded from L1E( dx)

into weak-L1F ( dx) and from L1

E( dγ) into weak-L1F ( dγ). Analogously, if T is bounded from

LqE( dµ) into LqF ( dµ), for some q , 1 < q < ∞ , then T 1 is bounded from LqE( dx) intoLqF ( dx) and from LqE( dγ) into LqF ( dγ).

10

Proposition 3.4 Under the assumptions (a), (b) and (c) made on T at the beginning of thissection, the operator Tloc inherits from T either the Lq−boundedness or the weak type (1, 1)as the case might be. Besides, the corresponding boundedness holds for both the Lebesgue andthe Gauss measures.

Proof Assume that x is in the ball Bj from the covering in Lemma 3.1. Then

Tlocf(x) = Tf(x)− Tglobf(x)

= T (fχ4Bj)(x) + T (f(1− χ4Bj

))(x)−∫

(1− ϕ)K(x, y) f(y) dy

= T (fχ4Bj)(x)−

∫(ϕ(x, y)− χ4Bj

(y))K(x, y) f(y) dy.

By multiplying by χBjand adding over j, we get

|Tlocf(x)| ≤

∣∣∣∣∣∣∑j

χBj(x)T (fχ4Bj

)(x)

∣∣∣∣∣∣+∫ ∑

j

χBj(x)|ϕ(x, y)− χ4Bj

(y)||K(x, y)| |f(y)| dy

= |T 1(f)(x)|+ T 2(|f |)(x),

with T 1 as before and T 2 defined here. By assumption (a) on the operator and Lemma 3.3,we know that T 1 is either bounded on Lq for both measures or of weak type (1, 1) for bothmeasures.

Let us now consider T 2. This operator is defined by the kernel

H(x, y) =∑j

χBj(x)|ϕ(x, y)− χ4Bj

(y)||K(x, y)|,

which is supported in the region N2 \ N 17

because of (3.3). Thus, by assumption (c),

|H(x, y)| ≤ C(1 + |y|)dχN2(x, y). Let µ be either the Lebesgue or the Gauss measure. Thestrong type (1, 1) property of T 2 with respect to µ follows by Fubini’s theorem and the factthat

µ

(B(y,

2

1 + |y|)

)≤ C(d)(1 + |y|)−d dµ

dy(y).

The boundedness of T 2 on L∞(µ) follows from the fact that T 2(|f |)(x) is dominated by the

mean value of |f | on the ball B(x, 2

1+|x|

). By interpolation, it is bounded on Lp, 1 ≤ p ≤ ∞,

for both measures.ut

Let us record as a separate lemma the fact that, for an operator whose kernel lives nearthe diagonal, boundedness with respect to Lebesgue and Gauss measure are equivalent.

Definition 3.5 We shall say that an operator S : C∞0 −→ (C∞0 )′ is local if its kernel issupported in N2.

Lemma 3.6 If S is a local operator, then strong type (p, p) for Lebesgue and Gauss mea-sures are equivalent. The same holds for weak type (p, p), 1 ≤ p ≤ ∞.

11

Proof From (3.4) we get S(f) =∑j χBj

S(χABjf). The conclusion follows by Lemma 3.2

and the bounded overlap property, i.e. item 3. of Lemma 3.1. ut

Theorem 3.7 Under the assumptions (a), (b) and (c) made on T at the beginning of thissection, the operator Tloc is bounded on Lp(γ) and on Lp( dx) for 1 < p < ∞. Moreover,Tloc is of weak type (1, 1) both with respect to the Lebesgue and the Gauss measure.

Proof By Proposition 3.4, Tloc is bounded in Lq for both measures or of weak type (1, 1)for both measures. Next we remark that Tloc is a Calderon-Zygmund operator. Indeed letKloc = Kϕ . Then for every test function f on Rd

Tlocf(x) =∫Kloc(x, y) f(y) dy

for all x outside the support of f . Moreover, by assumption (c) and (3.1), the kernel Kloc

satisfies the estimates

|Kloc(x, y)| ≤C

|x− y|d, |∂xKloc(x, y)|+ |∂yKloc(x, y)| ≤

C

|x− y|d+1,

for all (x, y) in Rd , x 6= y .Notice that in the case when Tloc is bounded in Lq for some q > 1, the weak type (1, 1)

follows by standard Calderon-Zygmund theory. Thus, we have the weak type (1, 1) propertyin all cases. From there, the boundedness of Tloc on Lp for all p, 1 < p <∞ can be deducedvia a good-lambda inequality as in [C-F, proof of Theorem III]. Since Tloc is a local operator,the conclusion follows from Lemma 3.6.

utWe can now state our main result

Theorem 3.8 If the function m is of Laplace transform type, then the multiplier operatorm(L) is of weak type (1, 1) with respect to the Gaussian measure.

Proof By Lemma 2.1 the operator m(L) satisfies assumptions (a), (b) and (c) from thebeginning of the section with q = 2 and µ = γ. Therefore, by Theorem 3.7, its local part isof weak type (1, 1) both with respect to the Lebesgue and the Gaussian measure. We onlyhave to prove that the global part of the operator is of weak type (1, 1) with respect to theGaussian measure. But, by using (2.3), we have

|m(L)globf(x)| ≤ C‖ψ‖∞∫

max0≤r<1

Mr(x, y)(1− ϕ(x, y))|f(y)| dy

≤ C‖ψ‖∞∫

max0≤r<1

Mr(x, y)(1− χN1(x, y))|f(y)| dy

and the latter operator is of weak type (1, 1) (see [S2]). ut

12

4 Pointwise expression for the operator in terms of its

kernel

The operator m(L) is in general not a principal value singular integral. In this section,we investigate how it can be expressed as a limit of the sum of a truncated integral and avariable multiple of the identity.

Theorem 4.1 Let m be of Laplace transform type. There exists a bounded function αdefined on (0,+∞) such that

m(L)f(x) = limε→0+

(α(ε)f(x) +

∫|x−y|>ε

Kψ(x, y)f(y) dy

)

for a.e. x, whenever f is a test function. If (εj) is a sequence tending to 0, such thatlimj→∞ α(εj) = α0 exists, then

m(L)f(x) = α0f(x) + limj→∞

∫|x−y|>εj

Kψ(x, y)f(y) dy.

If the function ψ is continuous at 1, then limε→0 α(ε) = ψ(1) and m(L) is the sum of ψ(1)Iand a principal value integral.

We begin by proving a technical lemma.

Lemma 4.2 For any compact subset K of Rd and any multiindex β of length b, there existsa constant C = C(K, b) such that for x and y in K∫ 1/2

0|∂βy (Mr(x, y)−M0(x, y)) |

dr

r≤ C(4.1)

and ∫ 1

1/2|∂βy (Mr(x, y)−M0(x, y)) |

dr

r≤ C(1 + hb(|x− y|))(4.2)

where

hb(ρ) =

ρ2−d−b if d+ b > 2;log(ρ) if d+ b = 2;1 if d+ b = 1.

Proof Since the function (r, x, y) 7→ ∂βyMr(x, y) is smooth in [0, 1/2]×Rd×Rd , there existsa constant C(K, b) such that for 0 ≤ r ≤ 1/2 and x , y in K

|∂βy (Mr(x, y)−M0(x, y)) | ≤ C(K, b) r.

This proves (4.1). Next we observe that by (2.4)

∫ 1

1/2|∂βyMr(x, y)|

dr

r≤ C

∫ 1

1/2(1− r2)−(d+b)/2 exp

(−c0

|rx− y|2

1− r2

)dr

13

for some positive constant c0. Since |rx− y|2 ≥ |x− y|2 − C(K)(1− r) for x and y in K ,the right hand side is bounded by

C∫ 1

1/2(1− r2)−(d+b)/2 exp

(−c0

|x− y|2

1− r2

)dr ≤ Chb(|x− y|).

Moreover for any y in K ∫ 1

1/2|∂βyM0(x, y)|

dr

r≤ C.

This proves (4.2). ut

Proof of Theorem 4.1 By Theorem 2.2 , if f and g are test functions on Rd ,

〈m(L)f, g〉γ =∫ 1

0ψ(r)〈rLLf, g〉γ

dr

r

=∫ 1

0ψ(r)

∫∫Mr(x, y)Lf(y) dy g(x) dγ(x)

dr

r

=∫ 1

0ψ(r)

∫∫(Mr(x, y)−M0(x, y))Lf(y) dy g(x) dγ(x)

dr

r,

since∫M0(x, y)Lf(y) dy = 〈Lf, 1〉γ = 0. By Lemma 4.2, the above integral is absolutely

convergent. Thus, by Fubini’s theorem

〈m(L)f, g〉γ =∫g(x)

∫ 1

0ψ(r)

∫(Mr(x, y)−M0(x, y))Lf(y) dy

dr

rdγ(x).

This proves that for a.e. x in Rd

m(L)f(x) =∫ 1

0ψ(r)

∫Rd

(Mr(x, y)−M0(x, y)) Lf(y) dydr

r.(4.3)

For x fixed, let Nr(y) = (Mr(x, y)−M0(x, y)) γ−1(y). Thus

m(L)f(x) = limε→0

∫ 1

0ψ(r)

∫|x−y|>ε

Nr(y)Lf(y) dγ(y)dr

r.

From Green’s formula one can deduce that∫|x−y|>ε

Nr(y)Lf(y) dγ(y) =∫|x−y|>ε

LNr(y) f(y) dγ(y)

+1

2

∫Sε

∂nf(y)Nr(y)γ(y) dσ(y)− 1

2

∫Sε

f(y) ∂nNr(y)γ(y) dσ(y)

= I1(ε, r) + I2(ε, r) + I3(ε, r),

where Sε is the surface of the ball of radius ε centred at x , ∂n the interior normal derivativeand dσ the surface measure on Sε . Since LNr(y) = r∂rMr(x, y)γ

−1(y), we get∫ 1

0ψ(r) I1(ε, r)

dr

r=

∫ 1

0ψ(r)

∫|x−y|>ε

r∂rMr(x, y) f(y) dydr

r

=∫|x−y|>ε

f(y)∫ 1

0ψ(r)∂rMr(x, y) dr dy

=∫|x−y|>ε

Kψ(x, y) f(y) dy.(4.4)

14

To prove the first part of the theorem we only need to show that, as ε tends to 0,∫ 1

0ψ(r) I2(ε, r)

dr

r= O(ρd(ε))(4.5) ∫ 1/2

0ψ(r) I3(ε, r)

dr

r= O(ρd(ε))(4.6) ∫ 1

1/2ψ(r) I3(ε, r)

dr

r= α(ε)f(x) +O(ρd(ε)),(4.7)

where ρd(ε) = ε if d ≥ 3, ρ2(ε) = ρ1(ε) = ε(1 + | log ε|) and α is a bounded function. Inthis proof, O symbols and constants C may depend on x. Indeed, by Lemma 4.2,

|∫ 1

0ψ(r) I2(ε, r)

dr

r| ≤ C‖ψ‖∞ max |∂f |(1 + h0(ε))ε

d−1,

which is O(ρd(ε)) provided that d ≥ 2. If d = 1 we get∫ 1

0ψ(r) I2(ε, r)

dr

r(4.8)

=1

2f ′(x− ε)γ(x− ε)

∫ 1

0ψ(r)Nr(x− ε)

dr

r− 1

2f ′(x+ ε)γ(x+ ε)

∫ 1

0ψ(r)Nr(x+ ε)

dr

r.

Arguing as in the proof of Lemma 4.2, we see that, for y in a compact set∫ 1

0|∂yNr(y)|

dr

r≤ C(1 + | log |x− y| |).

Thus ∣∣∣∣∣∫ 1

0ψ(r) (Nr(x− ε)−Nr(x+ ε))

dr

r

∣∣∣∣∣ ≤ ‖ψ‖∞∫ ε

−ε

∫ 1

0|∂yNr(y)|

dr

rdy

≤ C‖ψ‖∞∫ ε

−ε(1 + | log |t|) dt

= O(ε(1 + | log ε|)).

Since the function f and the density γ are smooth, this suffices to prove that (4.8) isO(ε(1 + | log ε|)).

This proves (4.5). A similar argument proves (4.6). To prove (4.7) we observe that

γ(y)∂nNr(y) = ∂nMr(x, y) + 2〈y, n〉Mr(x, y).

Arguing as in the proof of (4.5), one can easily see that the contribution of the secondsummand is O(ρd(ε)). To analyse the contribution of the first summand, we shall compareit with a convolution kernel . Let

φ(s) = 2π−d/2(1− r2)−d/2−1〈sx− y, n〉 exp

(−|sx− y|2

1− r2

).

15

Then

∂nMr(x, y) = φ(1)−∫ 1

rφ′(s) ds

= 2π−d/2(1− r2)−d/2−1|x− y| exp

(−|x− y|2

1− r2

)− Er(x, y).

Thus∫ 1

1/2ψ(r) I3(ε, r)

dr

r= −π−d/2ε

∫ 1

1/2ψ(r) (1− r2)−d/2−1 exp

(− ε2

1− r2

)dr

r

∫Sε

f(y) dy

+1

2

∫ 1

1/2ψ(r)

∫Sε

Er(x, y)f(y) dydr

r.(4.9)

Observe that∫Sεf(y) dσ(y) = ωdε

d−1 (f(x) +O(ε)) , where ωd is the measure of the unitsphere in Rd. If we perform the change of variables ε2 = (1− r2)t, we conclude that the firstsummand in the right hand side of (4.9) equals α(ε) (f(x) +O(ε)) , where

α(ε) = π−d/2ωd

∫ ∞

4ε2/3ψ(√

1− ε2/t)(1− ε2/t)−1td/2−1e−t dt.(4.10)

To estimate the integral of the error term Er(x, y) we put A = sx−y√1−r2 and we observe that

|φ′(s)| = 2π−d/2(1− r2)−d/2−1|〈x, n〉 − 2〈A, n〉〈A, x〉| exp(−|A|2

)≤ C|x|(1− r2)−d/2−1 exp(−c0|A|2),

where c0 is some small positive constant. Thus

|Er(x, y)| ≤ C|x|(1− r2)−d/2 supr≤s<1

exp

(−c0

|sx− y|2

1− r2

)

≤ C|x|e2c0|x|ε(1− r2)−d/2 exp

(−c0

ε2

1− r2

),

because |sx− y|2 ≥ |x− y|2 − 2(1− r)|x|ε , for y ∈ Sε and r ≤ s < 1.Therefore, if d ≥ 2,∫ 1

1/2ψ(r)

∫Sε

f(y)Er(x, y) dσ(y)dr

r= O(ρd(ε)).

If d = 1 we replace the integral over Sε by the sum of the values of the integrand at thepoints y = x± ε. Let A(t) = sx−(x+t)√

1−r2 . Then

Er(x, x+ ε) + Er(x, x− ε)

= 2π−1/2(1− r2)−3/2x∫ 1

r

((2A2(ε)− 1)e−A

2(ε) − (2A2(−ε)− 1)e−A2(−ε)

)ds.

16

An argument similar to the one used in the estimate of Er(x, y) shows that

|∂t((2A2(t)− 1)e−A

2(t))| ≤ C(x, ε)(1− r2)−1/2 exp

(−c0

t2

1− r2

),

for |t| ≤ ε , where the constant C(x, ε) stays bounded for x and ε bounded. Thus

|Er(x, x+ ε) + Er(x, x− ε)| ≤ C(x, ε)(1− r2)−1∫ ε

0exp

(−c0

t2

1− r2

)dt,

and ∣∣∣∣∣∫ 1

1/2ψ(r) (Er(x, x+ ε) + Er(x, x− ε))

dr

r

∣∣∣∣∣≤ C‖ψ‖∞

∫ 1

1/2(1− r2)−1

∫ ε

0e−c0 t2

1−r2 dtdr

r.

The right hand side here can be easily estimated by C‖ψ‖∞ε(1 + | log ε|). Since thefunction f is smooth, this suffices to prove (4.7) also for d = 1. The first claim of thetheorem follows, and the other two claims are easy consequences of the first and of theexpression for α in (4.10). ut

5 Maximal Operators

Let T be a linear operator satisfying the assumptions (a), (b) and (c) at the beginning ofSection 3. Let T ∗ be the operator defined by

T ∗f(x) = supε>0

∣∣∣∣∣∫|x−y|>ε

K(x, y)f(y) dy

∣∣∣∣∣ .Notice that the function T ∗f is measurable since the supremum over all ε > 0 coincideswith the supremum over all rational ε > 0, because the operator T satisfies assumption (c).We consider the vector-valued (in fact l∞−valued ) operator S given by

Sf(x) ={∫

K(x, y)χ{|x−y|>ε}f(y) dy}ε∈Q+

={∫

Kε(x, y)f(y) dy}ε∈Q+

.

It is clear that T ∗f(x) = ‖Sf(x)‖l∞ . In order to prove that T ∗ maps L1(γ) into weak−L1(γ),it is enough to prove that S maps L1(γ) into weak−L1

l∞(γ).To handle S, we define, in analogy with Section 3, the operators Sglob and Sloc, given by

Sglobf(x) ={∫

Kε(x, y)(1− ϕ(x, y))f(y) dy}ε

Slocf(x) = Sf(x)− Sglobf(x).

Theorem 5.1 Sloc maps L1(γ) into weak−L1l∞(γ).

17

Proof Observe that

Slocf(x) ={∫

K(x, y)ϕ(x, y)χ{|x−y|>ε}f(y) dy}ε.

In other words, Sloc is the l∞−valued version of the maximal operator associated to theCalderon-Zygmund operator Tloc.

It now follows that Sloc maps L1( dx) into weak−L1( dx)l∞ ; see Theorem 5.20(iv) ofChapter II of [GR], whose proof can be found in Section 4 of Chapter V. (This is writtenonly for convolution operators, but it can be extended to our case.) In a more generalcontext, this is also proved in [St2, Corollary 2 of Section 7, Chapter I, page 36].

In order to prove boundedness with respect to the Gaussian measure, we continue as inthe proof of Proposition 3.4. In particular we have

‖Slocf(x)‖l∞ ≤∑j

χBj(x)

∥∥∥Sloc(fχ4Bj)(x)

∥∥∥l∞

+∫ ∑

j

‖{Kε(x, y)}ε‖l∞ χBj(x)|ϕ(x, y)− χ4Bj

(y)| |f(y)| dy

= ‖S1locf(x)‖l∞ + S2(|f |)(x).

Now by the remark after Lemma 3.3, S1loc maps L1(γ) into weak-L1

l∞(γ). On the otherhand as ‖ {Kε(x, y)}ε ‖l∞ = |K(x, y)|, we have that S2 coincides with the operator T 2

introduced in the proof of Proposition 3.4. In particular S2 is bounded from L1(γ) intoL1(γ). Hence Sloc maps L1(γ) into weak−L1

l∞(γ). utWe assume that m is of Laplace transform type. Let α be the bounded function of

Theorem 4.1. The maximal operator associated with m(L) is defined by

m(L)∗f(x) = supε>0

∣∣∣∣∣α(ε)f(x) +∫|x−y|>ε

Kψ(x, y)f(y) dy

∣∣∣∣∣ .Theorem 5.2 If the function m is of Laplace transform type, then the maximal operatorm(L)∗ associated with the multiplier operator m(L) is of weak type (1, 1) with respect to theGaussian measure. Moreover, whenever f is a function in L1(γ),

m(L)f(x) = limε→0+

(α(ε)f(x) +

∫|x−y|>ε

Kψ(x, y)f(y) dy

), a.a. x.

Proof Because of Theorem 4.1 and a standard approximation argument, is of weak type(1, 1) with respect to the Gaussian measure. Since α is a bounded function, it is enough toshow that the operator

T ∗f(x) = supε>0

∣∣∣∣∣∫|x−y|>ε

Kψ(x, y)f(y) dy

∣∣∣∣∣is of weak type (1, 1) with respect to γ . But this is equivalent to proving that the operator

S(f)(x) ={∫

Kψ(x, y)χ{|x−y|>ε}f(y) dy}ε

18

maps L1(γ) into weak-L1l∞(γ). As the kernel of m(L) satisfies assumptions (a), (b) and (c),

we can apply Theorem 5.1 to conclude that Sloc maps L1(γ) into weak-L1l∞(γ). On the other

hand, it is clear from (2.3) that

‖Sglobf(x)‖`∞ ≤∫ ∥∥∥Kψ(x, y)χ{|x−y|>ε}(1− ϕ(x, y)

∥∥∥`∞|f(y)| dy

≤∫|Kψ(x, y)|(1− ϕ(x, y)|f(y)| dy

≤ C‖ψ‖∞∫

max0≤r<1

Mr(x, y)(1− χN1(x, y))|f(y)| dy.

The latter operator is of weak type (1, 1) (see [S2]). ut

6 Negative and imaginary powers of L

Writing, as in the Introduction, Π0 = I − P0 , we prove negative results for the imaginarypowers LiαΠ0 , α ∈ R , and the negative powers L−bΠ0 , b > 0.

If in Theorem 2.2 we let ψ(r) = Γ(1− iα)−1(− log r)−iα for some α ∈ R \ {0} andinterpret m(0) as 0, the resulting operator m(L) will be LiαΠ0 . It is thus in particularof weak type (1, 1). Considering the proof of Theorem 3.8, one can see that the constantinvolved increases at most exponentially as |α| → ∞ . This cannot be improved to polynomialgrowth, as our next result shows.

Theorem 6.1 For some c > 0, the weak type (1, 1) quasinorm of LiαΠ0 is bounded frombelow by ec|α| , when α ∈ R with |α| large.

Proof We denote by c > 0 and C <∞ various constants depending only on the dimension.In order to be able to integrate by parts in the expression for the kernel, we consider theoperator LiαΠ0 + P0, whose kernel is

K−iα(x, y) =1

Γ(1− iα)

∫ 1

0

(log

1

r

)−iα∂r (Mr(x, y)− (1− r)M0(x, y)) dr.

After an integration by parts, we get

K−iα(x, y) =π−d/2

Γ(−iα)

∫ 1

0

(log

1

r

)−iα−1(

(1− r2)−d2 e− |rx−y|2

1−r2 − (1− r)e−|y|2

)dr

r.

We apply this operator to L1 functions approximating a point mass at a point y withη = |y| large. Since the above kernel is to be integrated against L1(γ) functions andLebesgue measure, we need only integrate it against the measure f = eη

2δy and verify

that the weak−L1(γ) quasinorm of the resulting function (LiαΠ0 +P0)f(x) has exponentialincrease as |α| → ∞ .

19

Consider the value (LiαΠ0 +P0)f(x) = K−iα(x, y)eη2

at points x = ξy/η+ v with ξ ∈ Rand v ⊥ y , where η/3 < ξ < 2η/3 and |v| < 1. As in [GMST], one has

(LiαΠ0 + P0)f(x) =π−d/2

Γ(−iα)

∫ 1

0

(log

1

r

)−iα−1(eξ

2

(1− r2)−d2 e− |ξ−rη|2

1−r2 − r2|v|2

1−r2 − (1− r)

)dr

r.

We shall split the integral here into several parts. The main part will be

I1 =π−d/2

Γ(−iα)eξ

2∫|ξ−rη|<A

(log

1

r

)−iα−1

(1− r2)−d2 e− |ξ−rη|2

1−r2 − r2|v|2

1−r2dr

r,

where A = A(d) is a large constant to be determined. In I1 the variable of integration rruns over an interval of length 2A/η , and so arg(log 1/r)−iα = −α log log 1/r varies by atmost C|α|A/η << 1 if |α|/η is small enough. By the change of variables s = ξ − rη , wethen easily get

|I1| > ceξ

2

|Γ(−iα)|1

η

∫|s|<A

e−cs2

ds ≥ c0eξ

2

|Γ(−iα)|1

η,

for some constant c0 depending only on d . Let

I2 =π−d/2

Γ(−iα)eξ

2∫|ξ−rη|>A, 1

6<r< 5

6

(log

1

r

)−iα−1

(1− r2)−d2 e− |ξ−rη|2

1−r2 − r2|v|2

1−r2dr

r.

Changing variables as above, we obtain

|I2| ≤ Ceξ

2

|Γ(−iα)|1

η

∫|s|>A

e−cs2

ds.

With A large enough, we thus have

|I2| < |I1|/6.

We further set

I3 =π−d/2

Γ(−iα)eξ

2∫ 1

56

(log

1

r

)−iα−1

(1− r2)−d2 e− |ξ−rη|2

1−r2 − r2|v|2

1−r2dr

r.

For these r , one has |ξ − rη| > η/6, and so

|I3| ≤ Ceξ

2

|Γ(−iα)|

∫ 1

56

(1− r)−1−d/2e−cη2

1−r dr ≤ Ceξ

2

|Γ(−iα)|

∫ 1

56

e−cη2

dr ≤ |I1|/6

for large η . The next part of (LiαΠ0 + P0)f(x) we consider is

I4 =π−d/2

Γ(−iα)eξ

2∫ 1

6

1/η2

(log

1

r

)−iα−1

(1− r2)−d2 e− |ξ−rη|2

1−r2 − r2|v|2

1−r2dr

r.

20

Here again |ξ − rη| > η/6, and for large η

|I4| ≤ Ceξ

2

|Γ(−iα)|

∫ 16

1/η2e−cη

2 dr

r≤ |I1|/6.

Defining

I5 =π−d/2

Γ(−iα)

∫ 1

1/η2

(log

1

r

)−iα−1

(1− r)dr

r,

we see that

|I5| ≤ Clog log η

|Γ(−iα)|≤ |I1|/6

when η is large, since ξ > η/3. What remains to be estimated is

I6 =π−d/2

Γ(−iα)

∫ 1/η2

0

(log

1

r

)−iα−1(

(1− r2)−d2 e

−r2ξ2−r2η2−r2|v|2+2rξη

1−r2 − (1− r)

)dr

r.

Here the exponent of e is positive and bounded by Crη2 for large η , and therefore

|I6| ≤C

|Γ(−iα)|

∫ 1/η2

0

((1 + Cr2)eCrη

2 − 1 + r) dr

r

≤ C

|Γ(−iα)|

∫ 1/η2

0rη2 dr

r≤ C

|Γ(−iα)|≤ |I1|/6.

Summing up, we conclude that

|(LiαΠ0 + P0)f(x)| > I16≥ c

eξ2

|Γ(−iα)|1

η≥ c

e(η/3)2

|Γ(−iα)|1

η

in the set {x : η/3 < ξ < 2η/3, |v| < 1} . The γ measure of this set is at least cη−1e−(η/3)2 .Thus we get, for the weak−L1 quasinorm

‖ (LiαΠ0 + P0)f ‖L1,∞(γ)≥c

|Γ(−iα)|1

η2.

In the above argument, we must have η large and |α|/η smaller than some constant depend-ing only on d . This allows us to have η−2 ≥ c|α|−2 , so that

‖ (LiαΠ0 + P0)f ‖L1,∞(γ)≥c

|Γ(−iα)|1

α2> ec|α|

for some c and large |α| . The result follows, since P0 is strong type (1,1). ut

Proposition 6.2 For any b > 0, the operator L−bΠ0 is not of weak type (1, 1) with respectto γ .

21

Proof Because of Theorem 2.2, the operator L(L+ εI)−b−1, ε > 0, has kernel

1

Γ(b+ 1)

∫ 1

0rε(log

1

r

)b∂r(Mr(x, y)−M0(x, y)) dr.

Integrating by parts and letting ε→ 0, we see that the kernel of L−bΠ0 is

Kb(x, y) =π−d/2

Γ(b)

∫ 1

0

(log

1

r

)b−1(

(1− r2)−d2 e− |rx−y|2

1−r2 − e−|y|2

)dr

r.

We remark that this expression can also be obtained from Lemma 2.2 of [GMST].Choosing y with η = |y| large as in the preceding proof, we see that it is enough to verify

that the weak−L1(γ) quasinorm of Kb(., y)eη2

tends to ∞ as η →∞ . Notice that

πd/2Γ(b)Kb(x, y) =∫ 1

0

(log

1

r

)b−1(

(1− r2)−d2 e

−r2|x|2−r2η2+2rx·y1−r2 − 1

)dr

r.(6.1)

We shall consider x with |x| < 1 and x · y < 0. If 1/η < r < ε for some ε = ε(d) > 0, wehave

(1− r2)−d2 e

−r2|x|2−r2η2+2rx·y1−r2 ≤ eCr

2

e−1 < 1/2.

Thus ∫ ε

1/η

(log

1

r

)b−1(

(1− r2)−d2 e

−r2|x|2−r2η2+2rx·y1−r2 − 1

)dr

r

≤ −1

2

∫ ε

1/η

(log

1

r

)b−1 dr

r= − 1

2b

((log η)b − (log 1/ε)b

)≤ − 1

4b(log η)b

for large η . Further,∣∣∣∣∣∫ 1/η

0

(log

1

r

)b−1(

(1− r2)−d2 e

−r2|x|2−r2η2+2rx·y1−r2 − 1

)dr

r

∣∣∣∣∣≤ C

∫ 1/η

0

(log

1

r

)b−1

(r2 + η2r2 + ηr)dr

r≤ C(log η)b−1.

The remaining part of the integral in (6.1) is for large η bounded by

C∫ 1

ε(1− r)b−1

((1− r2)−

d2 e− 1

1−r2 + 1)

dr

r≤ C.

Altogether, this means that |L−bΠ0f(x)| ≥ c(log η)b in the set {|x| < 1, x · y < 0} , for largeη and some c = c(b, d). This implies that the weak−L1(γ) quasinorm of L−bΠ0f is large,and the result follows. ut

22

References

[C-F] R. Coifman and C. Fefferman, Weighted norm inequalities for Maximal Functionsand Singular Integrals, Studia Math. 51 (1974), 241–250.

[D] E. B. Davies, Heat kernels and spectral theory, Cambridge Tracts in Mathematics92, Cambridge University Press, Cambridge 1989.

[F-G-S] E. B. Fabes, C. E. Gutierrez and R. Scotto, Weak-type estimates for the Riesztransforms associated with the Gaussian measure, Revista Matematica Iberoamer-icana 10 (1994), 229–281.

[F-S] L. Forzani and R. Scotto, The higher order Riesz transform for Gaussian measureneed not be weak type (1, 1), to appear in Studia Math.

[GMST] J. Garcıa-Cuerva, G. Mauceri, P. Sjogren and J. L. Torrea, Higher-order Rieszoperators for the Ornstein-Uhlenbeck semigroup, Preprint 1996-34, Dept. of Math.,Chalmers Univ. of Techn. and Univ. of Goteborg, to appear in Potential Analysis.

[GR] J. Garcia-Cuerva, J. L. Rubio de Francia, Weighted Norm Inequalities and relatedTopics, North-Holland Mathematics Studies 116, North-Holland, 1985.

[Gu] R. F. Gundy, Sur les transformations de Riesz pour le semigroupe d’Ornstein-Uhlenbeck, C. R. Acad. Sci. 303 (1986), 967–970.

[Gz] C. Gutierrez, On the Riesz transforms for Gaussian measures, J. Funct. Anal.120 (1) (1994), 107–134.

[G-S-T] C. Gutierrez, C. Segovia and J. L. Torrea, On higher Riesz transforms for Gaus-sian measures, J. Fourier Anal. Appl. 2 (1996), 583–596.

[M] S. Meda, A general multiplier theorem, Proc. Amer. Math. Soc. 110 (3) (1990),639–647.

[Me] F. G. Mehler, Ueber die Entwicklung einer Funktion von beliebig vielen Variablennach Laplaceschen Funktionen hoherer Ordnung, J. reine angew. Math. 66 (1866),161–176.

[M-P-S] T. Menarguez, S. Perez and F. Soria, Pointwise and norm estimates for operatorsassociated with the Ornstein-Uhlenbeck semigroup, C. R. Acad. Sci. Paris Serie I326 (1998), 25–30.

[My] P. A. Meyer, Transformations de Riesz pour les lois Gaussiennes, Springer LectureNotes in Mathematics 1059 (1984), 179-193.

[Mu1] B. Muckenhoupt, Poisson integrals for Hermite and Laguerre expansions, Trans.Amer. Math. Soc. 139 (1969), 231-242.

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[Mu2] B. Muckenhoupt, Hermite conjugate expansions, Trans. Amer. Math. Soc. 139(1969), 243-260.

[Pe] S. Perez, Estimaciones puntuales y en la norma para operadores relacionadoscon el semigrupo de Ornstein-Uhlenbeck, Ph.D. thesis, Universidad Autonoma deMadrid, 1996.

[Pi] G. Pisier, Riesz transforms: a simpler analytic proof of P. A. Meyer’s inequality,Springer Lecture Notes in Mathematics 1321 (1988), 485–501.

[S1] P. Sjogren, Weak L1 characterizations of Poisson integrals, Green potentials, andHp spaces, Trans. Amer. Math. Soc. 233 (1977), 179–196.

[S2] P. Sjogren, On the maximal function for the Mehler kernel, in Harmonic Analysis,Cortona 1982, Springer Lecture Notes in Mathematics 992 (1983), 73–82.

[S3] P. Sjogren, Operators associated with the Hermite semigroup — a survey, J.Fourier Anal. and Appl. 3 Special issue (1997), 813–823 (Proceedings of the 5thInternational Conference on Harmonic Analysis and Partial Differential Equa-tions, El Escorial, Spain, June 1996)

[St1] E. M. Stein, Topics in Harmonic Analysis Related to the Littlewood-Paley Theory,Annals of Math. Study #63, Princeton Univ. Press, 1970.

[St2] E. M. Stein, Harmonic Analysis, Princeton Mathematical Series, 43, PrincetonUniv. Press, 1993.

[Th] S. Thangavelu, Lectures on Hermite and Laguerre expansions, MathematicalNotes 42, Princeton University Press, Princeton 1993.

[U] W. O. Urbina, On singular integrals with respect to the Gaussian measure, Ann.Scuola Norm. Sup. Pisa, Classe di Scienze, Serie IV, Vol. XVIII, 4 (1990), 531–567.

First and last author’s address: Departamento de Matematicas, C-XV, Uni-versidad Autonoma, 28049, Madrid, SPAIN

E-mail: jose.garcia-cuerva@uam.es joseluis.torrea@uam.esSecond author’s address: Dipartimento di Matematica, via Dodecaneso 35,16146 Genova, ITALY

E-mail: mauceri@dima.unige.itThird author’s address: Department of Mathematics, Chalmers Universityof Technology and Goteborg University, S-412 96 Goteborg, SWEDEN

E-mail: peters@math.chalmers.se

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