FRACTIONAL DIFFERENTIATION FOR THE GAUSSIANMEASURE AND APPLICATIONS.1

IRIS A. LOPEZ P. AND WILFREDO O. URBINA R.

Abstract. We define the Fractional Derivate of order 0 < α < 1, in-duced by the Ornstein Uhlenbeck operator L, associated with respectto the Gaussian measure. We obtain a characterization of the GaussianPotential Lp

α(γd) spaces, for 1 < p <∞ in terms of this operator and westudy the relation between Potential Lp

k(γ1) spaces and Sobolev W pk (γ1)

spaces for 1 < p <∞ and k ∈ N.Also, we obtain a version of Calderon’s reproduction formula for theGaussian measure in terms of this Fractional Derivate.

Dans cet article nous presentons la Derivation Fractionnelle de degree0 < α < 1, induite par l’operateur d’Ornstein–Uhlenbeck L, associe a lamesure gaussienne.Nous obtenons alors une caracterisation des espaces gaussiens Lp

α(γd)de potentiels, pour chaque 1 < p < ∞ en termes de cette derivation etnous donnons la relation entre les espaces gaussiens Lp

k(γ1) de Potencielset les espaces gaussiens W p

k (γ1) de Sobolev pour chaque 1 < p < ∞ etk ∈ N.De plus, nous obtenons une version de la formule de reproduction deCalderon associee a la mesure gaussienne en termes de cette derivation.

1. Introduction

Let us consider the Gaussian measure γd(x) = e−|x|2

πd/2 with x ∈ Rd and theOrnstein-Uhlenbeck differential operator

(1.1) L =124x − 〈x,∇x〉 .

Let β = (β1, ..., βd) ∈ Nd be a multi-index, let β! =∏di=1 βi!, |β| =∑d

i=1 βi, ∂i = ∂∂xi, for each 1 ≤ i ≤ d and ∂β = ∂β1

1 ...∂βdd .

2000 Mathematics Subject Clasification Primary 42C10; Secondary 26A33.Key words and phrases: Hermite expansions, Fractional Integral, Fractional Derivate,

Potencials Spaces, Sobolev Spaces, Meyer’s Multiplier Theorem.(1)Partially supported by Grant FONACIT #G-97000668 and by ECOS

Nord/FONACIT Action V00M02/PI-200000000860.

1

2 IRIS A. LOPEZ P. AND WILFREDO O. URBINA R.

Let us consider the normalized Hermite polynomials of order β, in dvariables

(1.2) hβ(x) = 1

(2|β|β!)1/2

d∏i=1

(−1)βiex2i∂βi

∂xβii

(e−x2i ),

then, it is well known, that the Hermite polynomials are eigenfunctions ofL,

(1.3) Lhβ(x) = − |β|hβ(x).

Given a function f ∈ L1(γd) its β-Fourier-Hermite coefficient is defined by

cfβ =< f, hβ >γd=∫

Rd

f(x)hβ(x)γd(dx).

Let Cn be the closed subspace of L2(γd) generate by the linear combinationsof {hβ : |β| = n}. By the orthogonality of the Hermite polynomials withrespect to γd it is easy to see that {Cn} is a orthogonal decomposition ofL2(γd),

L2(γd) =∞⊕n=0

Cn

which is called the Wiener chaos.Let Jn be orthogonal projection of L2(γd) onto Cn. If f is a polynomial,

Jnf =∑|β|=n

cfβhβ .

Let us define the Ornstein-Uhlenbeck semigroup {Tt}t≥0 as

Ttf(x) =1

(1− e−2t)d/2

∫Rd

e− e−2t(|x|2+|y|2)−2e−t〈x,y〉

1−e−2t f(y)γd(dy)(1.4)

=1

πd/2(1− e−2t)d/2

∫Rd

e− |y−e−tx|2

1−e−2t f(y)dy(1.5)

{Tt}t≥0 is a strongly continuous semigroup on Lp(γd), with infinitesimalgenerator L. Also, by a change of variable we can write,

(1.6) Ttf(x) =∫

Rd

f(√

1− e−2tu+ e−tx)γd(du).

Now, by Bochner subordination formula, we define the Poisson-Hermitesemigroup {Pt}t≥0 as

(1.7) Ptf(x) =1√π

∫ ∞

0

e−u√uTt2/4uf(x)du.

{Pt}t≥0 is also a strongly continuous semigroup on Lp(γd), with infinitesimalgenerator (−L)1/2. ¿From (1.4) we obtain, after the change of variable

FRACTIONAL DIFFERENTATION FOR γd 3

r = e−t2/4u,

(1.8)

Ptf(x) =1

2π(d+1)/2

∫Rd

∫ 1

0texp

(t2/4 log r

)(− log r)3/2

exp(−|y−rx|2

1−r2

)(1− r2)d/2

dr

rf(y)dy.

Then by (1.3)

Tthβ(x) = e−t|β|hβ(x),

Pthβ(x) = e−t√|β|hβ(x).

For α > 0, the Fractional Integral or Riesz potential of order α, Iγα, withrespect to the Gaussian measure is defined, as in the classical case, by

(1.9) Iγα = (−L)−α/2.

Now, if f ∈ L1(γd) with∫

Rd f(y)γd(dy) = 0, it can be proved that (see [8])

(1.10) Iγαf =1

Γ(α)

∫ ∞

0tα−1Ptfdt.

Observe that if f(x) = hβ(x), |β| > 0,

(1.11) Iγαhβ(x) =1

|β|α/2hβ(x).

For α ∈ Nd, let us consider the Riesz Transform of order |α| associated toL defined as, (see [12])

(1.12) Rα|α| = ∂αIγ|α|.

If f ∈ L1(γd) with∫

Rd f(y)γd(dy) = 0, then it can be proved that(1.13)

Rα|α|f(x) = Cd,α

∫Rd

∫ 1

0r|α|−1

(−logr1− r2

) |α|−22

hα

(y − rx√1− r2

)e− |y−rx|2

1−r2

(1− r2)d/2+1drf(y)dy

and if for i = 1, · · · , d βi ≥ αi, we have (see [13]))(1.14)

Rα|α|hβ(x) =

(2|α|

|β||α|

)1/2 [ n∏i=1

βi(βi − 1) . . . (βi − αi + 1)

]1/2

hβ−α(x).

In particular for the j-th Gaussian Riesz transforms of first order, Rj1 = ∂jIγ1 ,

j = 1, · · · , d, we have

(1.15) Rj1hβ =

√2βj|β|

hβ−ej.

Additionally, consider the j-th adjoint operator (Rj1)∗ of the Gaussian

Riesz transform Rj1, defined by

(1.16) 〈(Rj1)∗f, g〉γd

= 〈f,Rj1g〉γd.

4 IRIS A. LOPEZ P. AND WILFREDO O. URBINA R.

Observe that integrating by parts with respect xj and using the fact thatIγ1 = (−L)−1/2 is a self adjoint operator since L is, we have

〈f,Rj1g〉γd= 〈f, ∂iIγ1 g〉γd

= 〈δjf, Iγ1 g〉γd= 〈Iγ1 δjf, g〉γd

.

where δj is the adjoint operator of the partial derivate ∂j with respect γd.Explicity,

δj(.) = −∂j(.) + 2xj(.).Therefore for each j = 1, · · · , d we define

(Rj1)∗ = Iγ1 δj = (−L)−1/2δj .

In particular, by using properties of Hermite polynomials, we have that

(1.17) δjhβ =√

2(βj + 1)hβ+ej,

and then

(1.18) (Rj1)∗hβ =

√2(βj + 1)|β|+ 1

hβ+ej.

The j-th adjoint operator of the Gaussian Riesz transform (Rj1)∗, is also

Lp(γd)-continuous, 1 < p <∞, that is to say, there exist Cp > 0 such that

(1.19)∥∥∥(Rj1)∗f∥∥∥

p,γd

≤ Cp ‖f‖p,γd,

since, using the definition of the norm by duality, Holder’s inequality andthe Lp(γd) continuity of the Riesz transform, we get∥∥∥(Rj1)∗f∥∥∥

p,γd

= Sup‖g‖q≤1

∣∣∣ ∫Rd

(Rj1)∗f(x)g(x)γd(dx)

∣∣∣= Sup

‖g‖q≤1

∣∣∣ ∫Rd

f(x)Rj1g(x)γd(dx)∣∣∣

≤ Sup‖g‖q≤1

‖Rj1g‖q,γd‖f‖p,γd

≤ Cq‖f‖p,γd.

The j-th adjoint Gaussian Riesz operator (Rjk)∗ of higher order k, k ≥ 1,

is defined(Rjk)

∗ = Iγk δkj = (−L)−k/2δkj

for each j = 1, · · · , d.If α ∈ Nd, we can define the higher order adjoint operator of the Riesz

transform by(Rα|α|)

∗ = Iγ|α|δαdd ◦ ... ◦ δα1

1 .

Then, we get(1.20)

(Rα|α|)∗hβ(x) =

(2|α|

|β + α||α|

)1/2 [ n∏i=1

(βi + 1) . . . (βi + αi)

]1/2

hβ+α(x).

FRACTIONAL DIFFERENTATION FOR γd 5

and obviously if 1 < p <∞ and f ∈ Lp(γd), by the same before argument

‖(Rα|α|)∗f‖p,γd

≤ ‖f‖p,γd.

In the classical case of the Lebesgue measure, the Fractional Derivate forthe Laplacian operator is defined as, (see [9])

(−4)α/2 f(x) = aα limε→0

∫|y|≥ε

f(x+ y)− f(x)|y|d+α

dy

with 0 < α < 2, aα = 2αΓ(d+α/2)

πd/2Γ(−α/2) .For the case of doubling measures, and more recently for s-dimensional

non doubling measures, this has been generalized by A. E. Gatto and others,see [3]. Observe that

(1.21)∫

Rd

f(x+ y)− f(x)|y|d+α

dy = Cα,d

∫ ∞

0t−α−1 (Ptf(x)− f(x)) dt,

where Pt is the classical Poisson’s semigroup.Then, following the classical case, the Fractional Derivate of order α > 0

with respect to the Gaussian measure Dγα, is defined formally as

Dγα = (−L)α/2.

For the Hermite polynomials, we have

(1.22) Dγαhβ(x) = |β|α/2 hβ(x),

and therefore, by the density of the polynomials in Lp(γd), Dγα can then be

then extended to Lp(γd).In the case of 0 < α < 1 we can write, for f a polynomial,

(1.23) Dγαf =

1cα

∫ ∞

0t−α−1(Ptf − f)dt,

where

(1.24) cα =∫ ∞

0u−α−1(e−u − 1)du,

since for the Hermite polynomials, we have, by the change of variable u =√|β|t and the definition of cα,

Dγαhβ(x) = hβ(x)

{1cα

∫ ∞

0t−α−1

(e−t

√|β| − 1

)dt

}= |β|α/2 hβ(x)

{1cα

∫ ∞

0u−α−1

(e−u − 1

)du

}= |β|α/2 hβ(x).

The equality (1.23) will be very important in the development of a versionof Calderon’s reproduction formula.

Now, if f is a polynomial, by (1.11) and (1.22),

(1.25) Iγα(Dγαf) = Dγ

α(Iγαf) = Π0f.

6 IRIS A. LOPEZ P. AND WILFREDO O. URBINA R.

Following H. Sugita [11] (see also [14]), Gaussian Potential spaces of or-der α ≥ 0 Lpα(γd), 1 < p < ∞, can be defined as the completion of thepolynomials with respect to the norm

(1.26) ‖f‖p,α :=∥∥∥(I − L)α/2f

∥∥∥p,γd

.

In a previous paper [6], we have obtained a characterization of the Gauss-ian Potential spaces of integer order Lpk(γd), by using the Littlewood Paleytype of functions with respect to γd. As an application of the operatorFraccional Derivative Dγ

α we will give a characterization of the GaussianPotential spaces Lpα(γd), which is simpler and more powerful, valid for any1 < p <∞ and α ≥ 0.

In the next section first we are going to give alternative expressions ofthe operator Dγ

α that allow to handle better this operator and lead to twoversions of Calderon’s reproduction formula for the Gaussian measure.

We wish to express our deep gratitude to Prof. A. Eduardo Gatto for hisuseful conversations, suggestions and for his enthusiastic encouragement.We also want to thanks Prof. J. L. Torrea for an important observation andProf. P. Graczyk for uncountable discussions on the subjet that improvedradically this paper.

2. The results

We start giving an alternative representation of Dγα and Iγα that as we

already mention are very useful in what follows. Before that, we need thefollowing technical result of the asymptotic behavior of {Pt}t>0 at infinity.

Lemma 2.1. The Poisson-Hermite semigroup {Pt}t>0, has exponential de-cay on C⊥0 =

⊕∞n=1Cn. More precisely, if

∫Rd f(y)γd(dy) = 0,

(2.1) |Ptf(x)| ≤ Cd,f (d+ |x|)e−t.

Proof. Since {Pt}t>0, is an strongly continuos semigroup we have

(2.2) limt→0+

Ptf(x) = f(x)

and by hypothesis, since we are assuming that∫

Rd f(y)γd(dy) = 0,

(2.3) limt→∞

Ptf(x) = 0.

Let us prove that ∣∣∣∣ ∂∂tPtf(x)∣∣∣∣ ≤ Cd,f (d+ |x|)e−t.

Since∂

∂tTtf(x) = LTtf(x),

taking derivatives in (1.6), we have

∇xTtf(x) =(e−tTt

(∂f

∂x1

)(x), . . . , e−tTt

(∂f

∂xd

)(x))

FRACTIONAL DIFFERENTATION FOR γd 7

and

4xTtf(x) =d∑j=1

e−2tTt

(∂2f

∂x2j

)(x).

Therefore, taking f ∈ C2B(Rd) and using (1.7) we have that

∂

∂tPtf =

1√π

∫ ∞

0

e−u√u

t

2uLTt2/4ufdu

and carrying on the computations as in [5], we get∣∣∣∣ ∂∂tPtf(x)∣∣∣∣ ≤ Cd

∫ ∞

0

e−u√u

t

u

d∑j=1

e−t2/2u

2+ |xj |e−t

2/4u

fdu≤ Cd,f (d+ |x|)e−t,

then

|Ptf(x)| ≤∫ ∞

t

∣∣∣∣ ∂∂sPsf(x)∣∣∣∣ ds ≤ Cd,f (d+ |x|)e−t.

�

Now, let us give the alternate representation of Dγα and Iγα,

Proposition 2.1. Suppose f ∈ C2B(Rd) such that

∫Rd f(y)γd(dy) = 0, then

(2.4) Dγαf =

1αcα

∫ ∞

0t−α

∂

∂tPtfdt, 0 < α < 1,

(2.5) Iγαf = − 1αΓ(α)

∫ ∞

0tα∂

∂tPtfdt, α > 0.

Proof. Let us start proving (2.4). Integrating by parts in (1.23) we get

Dγαf(x) =

1cα

lima→0+

b→∞

∫ b

at−α−1 (Ptf(x)− f(x)) dt

=1cα

lima→0+

b→∞

{t−α

−α(Ptf(x)− f(x))

∣∣ba

+1α

∫ b

at−α

∂

∂tPtf(x)dt

}

=1αcα

∫ ∞

0t−α

∂

∂tPtf(x)dt

since, by (2.2) and (2.3), we have

limb→∞

(Pbf(x)− f(x)

bα

)= 0

and

lima→0+

∣∣∣∣Paf(x)− f(x)aα

∣∣∣∣ ≤ lima→0+

1aα

∫ a

0

∣∣∣∣ ∂∂sPsf(x)∣∣∣∣ ds

≤ Cd,f (d+ |x|) lima→0+

1− e−a

aα= 0.

8 IRIS A. LOPEZ P. AND WILFREDO O. URBINA R.

Let us prove now (2.5). Again, by integrating by parts, we have

Iγαf(x) =1

Γ(α)lima→0+

b→∞

∫ b

atα−1Ptf(x)dt

=1

Γ(α)lima→0+

b→∞

{tα

αPtf(x)

∣∣ba− 1α

∫ b

atα∂

∂tPtf(x)dt

}

= − 1αΓ(α)

∫ ∞

0tα∂

∂tPtf(x)dt,

since, by the previous result

limb→∞

|Pbf(x)bα| ≤ Cd,f (d+ |x|) limb→∞

bαe−b = 0

andlima→0+

|Paf(x)aα| = 0.

�

Observe that the previous proposition is also true for f = hβ, the Hermitepolynomial of order β, |β| > 0, and therefore is true for any nonconstantpolynomial f such that

∫Rd f(y)γd(dy) = 0.

By using (1.8) and (2.4), Dγα can be expressed explicitly as

Dγαf(x) =

∫Rd

Kα(x, y)f(y)dy,

where,

Kα(x, y) = Cd

∫ ∞

0

∫ 1

0t−αet

2/4logr(−logr)1/2 e− |y−rx|2

1−r2

(1− r2)d/2

×

(2r2 |y − rx|2 − 2r(1− r2)〈y − rx, x〉 − dr2(1− r2)

(1− r2)2

)dr

rdt.

Now let us write

(2.6) qt(x, y) = −t ∂∂t

∫ 1

0texp

(t2/4 log r

)(− log r)3/2

exp(−|y−rx|2

1−r2

)(1− r2)d/2

dr

r

,

and define the operator Qt as

(2.7) Qtf(x) = −t ∂∂tPtf(x) =

∫Rd

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

Following [3] we get immediately from (2.4) and (2.5), the following formulas

Corollary 2.1. Suppose f ∈ C2B(Rd) such that

∫Rd f(y)γd(dy) = 0, then we

have

(2.8) −αDγαf =

1cα

∫ ∞

0t−α−1Qtfdt, 0 < α < 1,

FRACTIONAL DIFFERENTATION FOR γd 9

(2.9) αIγαf =1

Γ(α)

∫ ∞

0tα−1Qtfdt, α > 0.

An interesting use of the family {Qt} is that it allows to give a version ofCalderon’s reproduction formula for the Gaussian measure.

Theorem 2.1. i) Suppose f ∈ L1(γd) such that∫

Rd f(y)γd(dy) = 0, thenwe have

(2.10) f =∫ ∞

0Qtf

dt

t.

ii) Suppose f a polynomial such that∫

Rd f(y)γd(dy) = 0, then we have

(2.11) f = Cα

∫ ∞

0

∫ ∞

0t−αsαQt (Qsf)

ds

s

dt

t0 < α < 1.

Also,

(2.12)∫ ∞

0

∫ ∞

0t−αsαQt (Qsf)

ds

s

dt

t= dα

∫ ∞

0u∂2

∂u2Pufdu.

Formula (2.11) is the aftermentioned version of Calderon’s reproductionformula for the Gaussian measure.

Proof. i) Using (2.2) and (2.3) we have,∫ ∞

0Qtf

dt

t= lim

a→0+

b→∞

(−∫ b

a

∂

∂tPtfdt) = lim

a→0+

b→∞

(−Ptf)∣∣ba

= f.

Let us prove (2.11), given f a polynomial such that∫

Rd f(y)γd(dy) = 0, byCorollary 2.1, we have

Dγα (Iγαf) =

1αcα

∫ ∞

0t−α−1Qt (Iγαf) dt.

Now, using the definition of Qt and Fubini’s Theorem, we have

Qt (Iγαf) =1

αΓ(α)

∫Rd

∫ ∞

0sα−1Qs(f)(y)dsdy.

Again using the definition of Qs we obtain

f = Dγα (Iγαf) = dα

∫ ∞

0

∫ ∞

0t−α−1sα−1Qt (Qsf) dsdt.

To show (2.12) we see that from (2.7)

Qt (Qsf) (x) = ts∂

∂t

∂

∂sPt+sf(x).

But∂

∂t

∂

∂sPt+sf(x) =

∂2

∂u2Puf(x)

∣∣u=t+s

,

10 IRIS A. LOPEZ P. AND WILFREDO O. URBINA R.

then∫ ∞

0

∫ ∞

0t−α−1sα−1Qt (Qsf) dsdt =

∫ ∞

0

∫ ∞

0t−αsα

∂2

∂u2Puf

∣∣u=t+s

dsdt

= dα

∫ ∞

0u∂2

∂u2Pufdu,

where dα = B(−α+1,α+1)aαcα

, B(−α + 1, α + 1) being the Beta function of pa-rameter (−α+ 1, α+ 1). �

Now let us consider the Gaussian Potential spaces. As we already mentionthe Gaussian Potential spaces, Lpα(γd), 1 < p <∞, α ≥ 0, can be defined asthe completion of the polynomials with respect to the norm

(2.13) ‖f‖p,α :=∥∥∥(I − L)α/2f

∥∥∥p,γd

.

Remember that the Gaussian Bessel potentials are defined as

(2.14) (I − L)−α/2f =∞∑n=0

(1 + n)−α/2Jnf.

Since

(I − L)−α/2f =∞∑n=0

(1 + n)−α/2Jnf = J0f +∞∑n=1

(n

1 + n)α/2

1nα/2

Jnf,

the Lp(γd) continuity of them follows from P.A. Meyer’s Multipliers theoremtwice, (see [7]).

First let us consider the inclusion properties among the Gaussian Potentialspaces, (see [14])

Proposition 2.2. i) If p < q then Lqα(γd) ⊂ Lpα(γd) for each α ≥ 0.ii) If 0 ≤ α < β then Lpβ(γd) ⊂ Lpα(γd) for each 1 < p <∞.

Proof. i) is immediate by Holder’s inequality.ii) Given f a polynomial, and consider

ψ = (I − L)β/2f =∑n≥0

(1 + n)β/2 Jnf,

which is trivially in Lpβ(γd), then as (I −L)(α−β)/2ψ = (I −L)α/2f, we have

‖f‖p,α = ‖(I − L)α/2f‖p,γ = ‖(I − L)(α−β)/2ψ‖p,γ≤ C‖ψ‖p,γ = C‖f‖p,β,

then, by the density of the polynomials, the inclusion of Lpβ(γd) in Lpα(γd) iscontinuos. �

Now, let us establish a relation among different norms of Potential spaces,(see[10, pages 136–138]).

FRACTIONAL DIFFERENTATION FOR γd 11

Proposition 2.3. Given 1 < p < ∞ and α ≥ 1, if f ∈ Lpα(γd) then f∈ Lpα−1(γd), and for each j = 1 . . . d, ∂jf ∈ Lpα−1(γd).Moreover

‖f‖p,α−1 +d∑j=1

‖∂jf‖p,α−1 ≤ Ap,α ‖f‖p,α .

Proof. Let f be a polynomial and considering

ψ = (I − L)α/2f =∑n≥0

(1 + n)α/2 Jnf,

which is also a polynomial, f can be written as

f =∑n≥0

(1

1 + n

)α/2Jnψ.

Now,

f =∑n≥0

(1

1 + n

)(α−1)/2( 11 + n

)1/2

Jnψ = (I − L)−(α−1)/2(I − L)−1/2ψ,

and also

∂jf =∑n≥0

(1

1 + n

)α/2 ∑|β|=n

cψβ√

2βjhβ−ej

=∑n≥0

(1

1 + n

)(α−1)/2( n

1 + n

)1/2

Rj (Jnψ)

= (I − L)−(α−1)/2T (Rjψ) ,

with Tψ =∑

n≥0

(n

1+n

)1/2Jnψ, a Meyer’s multiplier with h(z) = (z + 1)−1/2 .

Then, by using the Lp(γd)-continuity of the Gaussian Riesz transform, theLp(γd)-continuity of the Bessel potentials and Meyer’s Multipliers theorem,we obtain

‖f‖p,α−1 +d∑j=1

∥∥∥∥ ∂

∂xjf

∥∥∥∥p,α−1

≤∥∥∥(I − L)−1/2ψ

∥∥∥p,γd

+d∑j=1

‖T (Rjψ)‖p,γd

≤ Ap,d ‖ψ‖p,γ = Ap,d ‖f‖p,α .

In the general case the density of the polynomials in Lpα(γd) is used.�

In [10], Stein is able to prove, in the classical case, that actually that thosetwo norms are equivalents and then, Sobolev spaces and Potential spaces areequivalent.

Now following Stein [9] we will start considering the Gaussian Potentialspaces of order α ≥ 0 and p = 2, by using the operator Dγ

α.

12 IRIS A. LOPEZ P. AND WILFREDO O. URBINA R.

Proposition 2.4. Let α ≥ 0, if f ∈ L2α(γd) then Dγ

αf ∈ L2(γd) and

‖Dγαf‖2,γd

≤ ‖f‖2,α .

Proof. Let f be a polynomial and considering ψ = (I−L)α/2f , which is alsoa polynomial, then Jnψ = (1 + n)α/2 Jnf and f can be written as

f =∑n≥0

(1

1 + n

)α/2Jnψ.

Trivially f, ψ ∈ L2(γd).Now, by definition, we have,

‖f‖2,α = ‖ψ‖2,γd=

∑n≥0

(1 + n)α||Jnf ||22,γd

1/2

,

then by Parseval’s identity we obtain,∫Rd

|Dγαf(x)|2 γd(dx) =

∑n≥0

nα||Jnf ||22,γd≤ Cα

∑n≥0

(1 + n)α||Jnf ||22,γd

= Cα

∥∥∥(I − L)α/2f∥∥∥2

2,γd

therefore,‖Dγ

αf‖2,γd≤ ‖f‖2,α .

For the general case, the density of the polynomial functions in Lp(γd) isthen used. �

Actually we can go further. In the following theorem we first extendthe fractional derivative operator Dγ

α to all the Gaussian Potential spacesLαp (γd), 1 < p <∞. The union of these spaces

Lα(γd) :=⋃p>1

Lpα(γd)

is a natural domain of Dγα. Next we show that if f ∈ Lα(γd) then f ∈ Lpα(γd)

is equivalent to Dγαf ∈ Lp(γd). The main tool is Meyer’s multiplier theorem.

Let us underline that the definition of Dγα on all the spaces Lpα(γd), 1 < p <

∞, is also based on an application of Meyer’s theorem.

Theorem 2.2. Let α ≥ 0 and 1 < p <∞.i) If {Pn}n is a sequence of polynomials such that limn→∞ Pn = f inLpα(γd), then limnD

γαPn exists in Lpα(γd) and does not depend on the

choice of a sequence {Pn}n.If f ∈ Lpα(γd)∩Lrα(γd), then the limit does not depend on the choiceof p or r. ThusDγαf = limn→∞Dγ

αPn in Lpα(γd), limn→∞ Pn = f in Lpα(γd), f ∈Lα(γd), is well defined.

FRACTIONAL DIFFERENTATION FOR γd 13

ii) f ∈ Lpα(γd) if and only if Dγαf ∈ Lp(γd). Moreover,

(2.15) Bp,α ‖f‖p,α ≤ ‖Dγαf‖p,γd

≤ Ap,α ‖f‖p,α .

Proof. Let f be a polynomial and considering ψ = (I−L)α/2f , which is alsoa polynomial, then f can be written as

Dγαf =

∑n≥0

nα/2Jnf =∑n≥0

(n

1 + n

)α/2Jnψ.

since ‖f‖p,α = ‖ψ‖p,γd, by Meyer’s Multipliers Theorem, using the multiplier

h(z) = (z + 1)−α/2, we obtain that

‖Dγαf‖p,γd

≤ Ap,α ‖ψ‖p,γd= Ap,α ‖f‖p,α .

To prove the converse, suppose f polynomial, then Dγαf is also a polynomial

and therefore Dγαf ∈ Lp(γd), consider

ψ = (I − L)α/2f =∑n≥0

1 + nα/2Jnf =∑n≥0

(1 + n

n

)α/2Jn (Dγ

αf) ,

so by Meyer’s Multipliers Theorem, using the multiplier h(z) = (z + 1)α/2,we have

‖f‖p,α = ‖ψ‖p,γd≤ Bp,α ‖Dγ

αf‖p,γd.

Thus we get (2.15) for polynomials. Then we can prove part i) using (2.15),the completness of Lpα(γd) and Proposition 2.2 ii). Finally we get (2.15) forany f ∈ Lp(γd). �

Corollary 2.2. i) Lαp (γd) = {f : Dγαf ∈ Lp(γd)} and if α = k ∈ N

then we can also write

Lpk(γd) ={f ∈ Lk(γd) : Dγ

j f ∈ Lp(γd), j ≤ k

}.

ii) If f ∈ Lp|α|(γd), then

(2.16) ‖∂αf‖p,γd≤ Cp ‖f‖p,|α| .

Proof. i) Immediate. For the case α = k ∈ N we use Proposition 2.2 ii).ii) By (1.25) we have that if f is a non-constant polynomial

(2.17) ∂αf = Rα|α|Dγ|α|f,

then, by Theorem 2.2 and using the Lp(γd)-continuity of the Riesz transformwe have

(2.18) ‖∂αf‖p,γd≤ Cp ‖f‖p,|α| .

If f ∈ Lp|α|(γd) the result follows by the density of the polynomials in Lp|α|(γd).�

14 IRIS A. LOPEZ P. AND WILFREDO O. URBINA R.

Finally, let us consider the Gaussian Sobolev spaces in the unidimensionalcase,

W pk (γ1) = {f :

dj

dxjf ∈ LP (γ1), 0 ≤ j ≤ k}

with the classical norm,

‖f‖W pk

=k∑j=0

∥∥∥∥ djdxj f∥∥∥∥p,γ1

.

In the next Proposition, we will give an easy analytic proof of the fact thatif f ∈ Lpk(γ1), with 1 < p <∞ and |α| = k, then

Lpk(γ1) = W pk (γ1).

The fractional Derivate and the adjoint operators of the Gaussian Riesztransform, will be one of key ingredients of the next result.

Proposition 2.5. Let 1 < p < ∞ and k ≥ 1. Suppose f ∈ Lpk(γ1). Thenthere exist a positive constant Bp,k such that

(2.19) ‖f‖p,k ≤ Bp,k

∥∥∥∥ dkdxk f∥∥∥∥p,γ1

Proof. If f is a polynomial function, we define the operator Tk as

Tkf =∑n≥0

(n+ k)k

2k(n+ k)...(n+ 1)cfnhn

and using Meyer’s Multipliers Theorem with

h(z) =2−k/2(1 + kz)k/2

((1 + kz) · · · (1 + z))1/2,

we obtain that Tk has a Lp(γd)-continuous extension,

‖Tkf‖p,γd≤ Cp ‖f‖p,γd

.

Now, let us define the operator Uk as

Ukhn =(n+ k

2

)k/2((n+ k)...(n+ 1))−1/2 hn+k

and if f is a polynomial function,

Ukf =∑n≥0

cfnUkhn.

We denote (Rk)∗ = (−L)−k/2δk, the unidimensional adjoint Gaussian Riesztransform of order k, for k ≥ 1, then by using (1.20) it is clear that

(2.20) (Rk)∗hn(x) = 2k/2[(n+ 1)...(n+ k)]1/2

(n+ k)k/2hn+k.

FRACTIONAL DIFFERENTATION FOR γd 15

Therefore, we get for each k ≥ 1, and n ≥ 1

Ukhn = Tk ◦ (Rk)∗hn.

Thus if f is a polynomial function, using the Lp(γd) continuity of (Rk)∗ andTk we obtain that

(2.21) ‖Ukf‖p,γd≤ Cp,k ‖f‖p,γd

.

But by definition,UkR

khn = hn

and then,

Dγkf = UkRkD

γkf = Uk

(dk

dxkf

).

Now using ii) of the Theorem 2.2 and (2.21) we get

‖f‖p,k ≤ Bp,k

∥∥∥∥ dkdxk f∥∥∥∥p,γ1

and in the general case, we can use the density of the polynomials in Lp(γ1).�

This way Proposition 2.5 and Corollary 2.2 shows that, for the case d = 1,if f ∈ Lkp(γ1),

(2.22) ‖f‖p,k '∥∥∥∥ dkdxk f

∥∥∥∥p,γ1

.

It is seem possible to get an analytic proof of the fact that if f ∈ Lpk(γd) and|α| = k, 1 < p <∞, then

‖f‖p,k ' ‖∂αf‖p,γd.

A probabilistic proof of this fact, was given by H. Sugita [11] in the contextof the Sobolev spaces of Wiener functionals.

References

[1] Forzani L, Scotto R, Urbina W. A simple proof of the Lp-continuity of the higher orderRiesz transforms with respect to the Gaussian measure, Seminaire de ProbabilitesLecture Notes in Math 1755, Springer-Verlag. Berlin. (2001).

[2] Frazier M, Jawerth B, Weiss G. Littlewood Paley Theory and the Study of FunctionsSpaces, CBMS-Conference Lecture Notes 79. Amer. Math. Soc. Providence RI (1991).

[3] Gatto A. E, Segovia C, Vagi S. On Fractional Differentiation and Integration onSpaces of Homogeneous Type, Rev. Mat. Iberoamericana 12 (1996) 111-145.

[4] Gatto A. E, Vagi S. On Sobolev Spaces of Fractional Order and ε -families of Oper-ators on Spaces of Homogeneous Type, Studia Math. 1331 (1) (1999).

[5] Gutierrez C. On the Riesz Transforms for the Gaussian measure, J. Fourier Anal.Appl. (1) (1994) 107-134.

[6] Lopez I, Urbina W. On Some Functions of the Littlewood Paley Theory for γd andcharacterization of Gaussian Sobolev spaces of integer order. Preprint. Submitted forpublication.

[7] Meyer, P.A. Transformations de Riesz pour le lois Gaussiens. Lecture Notes in Math1059 (1984) Springer-Verlag 179-193.

16 IRIS A. LOPEZ P. AND WILFREDO O. URBINA R.

[8] Sjogren P. Operators associated with the Hermite Semigroup- A Survey J. FourierAnal. Appl. (3) (1997) 813-823.

[9] Stein E. The Characterization of Functions Arising as Potentials I, Bull. Amer. Math.Soc. 97 (1961) 102-104. II (ibid) 68 (1962) 577-582.

[10] Stein E. Singular Integrals and Differentiability Properties of Functions, PrincetonUniv. Press. Princeton, New Jersey. (1970).

[11] Sugita H. Sobolev spaces of Wiener functionals and Malliavin’s calculus J. Math.Kyoto Univ. 25-1 (1985) 31-48.

[12] Urbina W. On Singular Integrals with respect to the Gaussian Measure, Scoula Nor-male di Pisa. Clase de Science. Serie IV Vol XVIII 4 (1990) 531-567.

[13] Urbina W. Analisis Armonico Gaussiano: una vision panoramica, Trabajo de As-censo, Facultad de Ciencias, UCV. (1998).

[14] Watanabe S. Stochastic Differential Equations and Malliavin Calculus, Tata Instituteof Fundamental Research, Springer Verlag. Berlin. (1984).

Escuela de Matematicas, Facultad de Ciencias, UCV. Aptd 47195 Los Ch-aguaramos, Caracas 1041-A Venezuela.

E-mail address: [Iris Lopez]iathamai@euler.ciens.ucv.ve, cathar@cantv.net

E-mail address: [Wilfredo Urbina]wurbina@euler.ciens.ucv.ve