E extracta mathematicae Vol. 33, Núm. 2, 229 – 249 (2018)

Sobolev Spaces and Potential Spaces Associated to
Hermite Polynomials Expansions

Iris A. López P.

Universidad Simón Bolivar, Departamento de Matemáticas Puras y Aplicadas,

Aptd 89000, Caracas 1080-A, Venezuela

iathamaica@usb.ve

Presented by Carmen Calvo Received June 14, 2017

Abstract: The aim of this paper is to study the relation existing between potential spaces
and Sobolev spaces, induced by the Ornstein-Uhlenbeck differential operator and associated
to Hermite polynomials expansions, where we consider the multidimensional Gaussian mea-
sure. By means of analytical methods we prove that potential spaces and Sobolev spaces
are spaces with equivalent norms.

Key words: Hermite polynomials, Gaussian measure, Sobolev spaces, potential spaces,
Meyer’s multiplier theorem.

AMS Subject Class. (2000): 42C10, 26A33.

1. Introduction

In the classical harmonic analysis theory the study of the differentiability
and smoothness of functions, f : Rd → R, can be described in the context of
Banach spaces of functions. Thus, Sobolev spaces W

p
k (λd) and potential spaces

L
p
s(λd) are taking into consideration, where we denote λd as the Lebesgue

measure, s > 0, k ∈ N and 1 < p < ∞. On the one hand, Sobolev spaces
W

p
k (λd) allows us to consider functions, such that, f ∈ L

p(λd) and its partial
derivatives ∂αf ∈ Lp(λd), with |α| ≤ k, where the derivatives are understood
in a suitable weak sense to make the space complete. On the other hand,
potential spaces L

p
s(λd) are defined by using fractional powers of the Laplacean

(−△)−s/2, 0 < s < d, and its variants (I − △)−s/2, s > 0. These fractional
powers are known as Riesz potentials and Bessel potentials respectively and
we have that the potential spaces L

p
s(λd) are subspaces of L

p(λd), such that,
f = (I − △)−s/2g, with g ∈ Lp(λd). Riesz potentials, Bessel potentials and
their properties have been deeply studied and by means of these operators it
has been obtained that

W
p
k (λd) = L

p
k(λd) with k ∈ N and 1 < p < ∞,

229



230 i. a. lópez p.

(see [15, 6]). This identity is very meaningful because the condition defining
to W

p
k (λd) spaces is not easy to check for any function given, nevertheless, the

condition defining to L
p
k(λd) can be described in terms of Bessel potentials

which have integral representations.

Similar identity holds in the Hermite setting and a probabilistic proof of
this fact has been given by H. Sugita in [16] and S. Watanabe in [17], where the
notion of Sobolev spaces of Wiener functionals has been introduced to develop
Malliavin’s calculus. This question has been studied by B. Bongioanni and J.
L Torrea in [1, 2], where they considered Sobolev spaces associated to Hermite
functions expansions and Laguerre functions expansions. However, in [7] we
introduced the fractional derivative for the gaussian measure γd and by means
of analytical methods we obtained that

W
p
k (γ1) = L

p
k(γ1), with k ∈ N and 1 < p < ∞,

but only in the unidimensional case.

Therefore, the purpose of this paper is to extend this identity to the mul-
tidimensional case. If d ≥ 1, k ∈ N and 1 < p < ∞, we shall prove that

W
p
k (γd) = L

p
k(γd)

and once more, Bessel potentials and adjoint Gaussian-Riesz transforms are
key tools in the proof of this fact. In the Laguerre polynomials and Jacobi
polynomials setting, some similar results have been obtained in [5] and [8],
respectively. Particularly, if 1 < p < ∞, 0 < s < 1 and k ≥ 1 we obtain that
W

p
k (γd) ⊂ L

p
s(γd) and the inclusion is proper.

The paper is organized as follows. Section 2 contains some basic facts
notation and we obtain the result of this paper in Section 3.

2. Preliminary definitions.

Let β = (β1, . . . ,βd) ∈ Nd ∪ {⃗0} be a multi-index, so β! =
∏d
i=1 βi!

and |β| =
∑d

i=1 βi. Let us denote by ∂i =
∂
∂xi

, for each 1 ≤ i ≤ d, and
∂β = ∂

β1
1 · · ·∂

βd
d . Let P be the subspace of polynomials functions on R

d.

We denote the Gaussian measure γd(dx) =
e−|x|

2

πd/2
dx, with x ∈ Rd and the

Ornstein-Uhlenbeck differential operator is defined as

L =
1

2
△x − ⟨x,∇x⟩ .



sobolev spaces and potential spaces 231

Let us consider the normalized Hermite polynomials of order β, in d vari-
ables, defined as

hβ(x) =
1(

2|β|β!
)1/2

d∏
i=1

(−1)βiex
2
i
∂βi

∂x
βi
i

(
e−x

2
i
)
,

with h0(x) = 1, (see [18, pages 105–106]) and it is well known that the Hermite
polynomials are eigenfunctions of L; this way,

Lhβ(x) = − |β|hβ(x). (2.1)

Also,
∂jhβ(x) =

√
2βjhβ−ej, for each 1 ≤ j ≤ d

and if α is a multi-index,

∂αhβ(x) =



2|α|/2

(
d∏
j=1

βj(βj − 1) · · · (βj − αj + 1)

)1/2
hβ−α(x), if αj ≤ βj,

∀j = 1, . . . ,d,

0, otherwise.

Now, given a function f ∈ L1(γd) its β-Fourier-Hermite coefficient is de-
fined by

c
f
β =

∫
Rd
f(x)hβ(x)γd(dx)

and let Cn be the closed subspace of L
2(γd) generated by the linear combina-

tions of {hβ : |β| = n}. The orthogonality of the Hermite polynomials with
respect to γd, lets us see that {Cn} is an orthogonal decomposition of L2(γd)

L2(γd) =
∞⊕
n=0

Cn

which is called the Wiener chaos decomposition, see [17]. Now, we denote Jn
the orthogonal projection of L2(γd) onto Cn. If f ∈ L2(γd), we have that

Jnf =
∑
|β|=n

c
f
βhβ

and its Hermite expansion is given by f =
∑

n≥0 Jnf. Then, following [16]
there exists a positive constant Cp,n, such that,

∥Jnf∥p,γd ≤ Cp,n∥f∥p,γd for 1 < p < ∞.



232 i. a. lópez p.

Also, if f ∈ L2(γd) the operator

Lf =
∑
n≥0

−nJnf,

defined on the domain D2(L) = {f ∈ L2(γd) :
∑

n≥0
∑

|β|=n |c
f
β|

2 < ∞} is
a self-adjoint extension of L considered on dense subspace of L2(γd). More
precisely, L has a clousure wich also will denote by L.

In this context, the Ornstein-Uhlenbeck semigroup, {Tt}t≥0, is defined 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), (2.2)

where its kernel is given by Mehler formula

1

(1 − e−2t)d/2
e
− e

−2t(|x|2+|y|2)−2e−t⟨x,y⟩
1−e−2t =

∑
n≥0

∑
|β|=n

e−|β|thβ(x)hβ(y).

It is well known that {Tt}t≥0 is a symmetric diffusion semigroup, with in-
finitesimal generator L, see [14, 16] and moreover

∥Tt(I − J0 − · · · − Jn−1)f∥p,γd ≤ Cp,ne
−nt∥f∥p,γd, 1 < p < ∞. (2.3)

By means of Bochner subordination formula the Poisson-Hermite semi-
group {Pt}t≥0 is defined as

Ptf(x) =
1
√
π

∫ ∞
0

e−u
√
u
Tt2/4uf(x)du (2.4)

and similarly, {Pt}t≥0 is a strongly continuous semigroup on L
p(γd) with in-

finitesimal generator (−L)1/2, (see [14]). From (2.1) we obtain that

Tthβ(x) = e
−t|β|hβ(x) and Pthβ(x) = e

−t
√

|β|hβ(x) (2.5)

Moreover,

Tt(Jnf) = e
−ntJnf and Pt(Jnf) = e

−
√
ntJnf. (2.6)

Now, if s > 0, similar to the classical case, the gaussian fractional integral
of order s, I

γ
s , is defined by

Iγs := (−L)
−s/2Π0, where Π0 = I − J0.



sobolev spaces and potential spaces 233

This (formal) definition is correct for all Hermite polynomials hβ, since by
using (2.1) we have that

Iγs hβ(x) =
1

|β|s/2
hβ(x), ∀|β| > 0, (2.7)

and define I
γ
s h0(x) = 0, see [3, 13]. In the case that f ∈ L1(γd), an integral

representation of I
γ
s is obtained in [13], which is given by

Iγs f =
1

Γ(s)

∫ ∞
0

ts−1Pt(I − J0)fdt. (2.8)

By using (2.5), we can see that (2.8) coincides with (2.7), if f = hβ, ∀ |β| > 0
and consequently, (2.8) coincides with (2.7) if f is a nonconstant polynomial
or f ∈ L2(γd).
Again, from (2.3) and (2.4) we obtain that

∥Pt(I − J0)f∥p,γd ≤ Cpe
−t∥f∥p,γd 1 < p < ∞, (2.9)

since

e−t =
1
√
π

∫ ∞
0

e−u
√
u
e−t

2/4udu.

Therefore, from (2.8) and (2.9) we can conclude

∥Iγs f∥p,γd ≤ Cp∥f∥p,γd 1 < p < ∞. (2.10)

Now, for α ∈ Nd, we consider the Riesz transform of order |α|, associated
to L, defined as, (see [10])

Rα|α| := ∂
αI

γ
|α| (2.11)

and if f ∈ L1(γd) with c
f
0 = 0, then

Rα|α|f(x)

= Cd,α

∫
Rd

∫ 1
0
r|α|−1

(
−logr
1 − r2

) |α|−2
2

hα

(
y − rx
√
1 − r2

)
e
− |y−rx|

2

1−r2

(1 − r2)d/2+1
drf(y)dy.

Lp(γd) estimates for 1 < p < ∞ of the Gaussian-Riesz transform, have been
showed by several authors using probabilistic and analytic methods (see for



234 i. a. lópez p.

example [4, 10, 11] among other authors). Particularly, if f = hβ, for each
i = 1, . . . ,d and βi ≥ αi, we have

Rα|α|hβ(x) =

(
2|α|

|β||α|

)1/2 [ d∏
i=1

βi(βi − 1) · · · (βi − αi + 1)

]1/2
hβ−α(x)

and the j-th Gaussian-Riesz transform of first order, R
j
1 = ∂jI

γ
1 , j = 1, . . . ,d,

with respect to Hermite polynomials can be expressed as

R
j
1hβ =

√
2βj
|β|

hβ−ej.

In [7], the j-th adjoint operator, (R
j
1)

∗ of the Gaussian-Riesz transform R
j
1,

has been defined as ⟨
(R

j
1)

∗f,g
⟩
γd

= ⟨f,Rj1g⟩γd
and we can observe that I

γ
1 = (−L)

−1/2 is a self-adjoint operator, then inte-
grating by parts with respect xj, we obtain

⟨f,Rj1g⟩γd = ⟨f,∂jI
γ
1 g⟩γd = ⟨δjf,I

γ
1 g⟩γd = ⟨I

γ
1 δjf,g⟩γd,

where δj(·) = −∂j(·) + 2xj(·). This way, we can express, for each j = 1, . . . ,d,

(R
j
1)

∗ := I
γ
1 δj = (−L)

−1/2δj.

By means of the identity√
2(βj + 1)hβ+ej − 2xjhβ +

√
2βjhβ−ej = 0,

where h−ej = 0, ∀j = 1, . . . ,d, (see [18, pages 105–106]), we have that

δjhβ =
√
2(βj + 1)hβ+ej (2.12)

and therefore,

(R
j
1)

∗hβ =

√
2(βj + 1)

|β| + 1
hβ+ej. (2.13)

Also, in [7] we obtain the boundedness of j-th adjoint operator of the
Gaussian-Riesz transform (R

j
1)

∗ for 1 < p < ∞. This follows easily from
Hölder’s inequality and the Lp(γd) continuity of the Riesz transform. So,∥∥∥(Rj1)∗f∥∥∥

p,γd
= Sup

∥g∥q≤1

∣∣∣⟨f,Rj1g⟩γd∣∣∣ ≤ Cq∥f∥p,γd.



sobolev spaces and potential spaces 235

Similarly, the j-th adjoint Gaussian-Riesz operator (R
j
k)

∗ of higher order k,
with k ≥ 1, is defined by

(R
j
k)

∗ := I
γ
kδ

k
j = (−L)

−k/2δkj , for each j = 1, . . . ,d,

If α ∈ Nd, we can define the higher order adjoint operator of the Riesz trans-
form by

(Rα|α|)
∗ := I

γ
|α|δ

αd
d ◦ · · · ◦ δ

α1
1 ,

thus, we get

(Rα|α|)
∗hβ(x) =

(
2|α|

|β + α||α|

)1/2 [ d∏
i=1

(βi + 1) · · · (βi + αi)

]1/2
hβ+α(x)

and if 1 < p < ∞ with f ∈ Lp(γd), we have that

∥(Rα|α|)
∗f∥p,γd ≤ Cp∥f∥p,γd.

Now, in [7] the Gaussian fractional derivative of order s > 0, D
γ
s , is defined

formally as
Dγs := (−L)

s/2

and for Hermite polynomials, from (2.1), we have that

Dγshβ(x) = |β|
s/2

hβ(x). (2.14)

Particularly, from (2.14), we have that D
γ
sh0(x) = 0 and, similar to the clas-

sical case, the Gaussian fractional derivative of a constant function is equal to
zero, see [14, 12]. In the case of 0 < s < 1, we can write

Dγsf =
1

cs

∫ ∞
0

t−s−1(Ptf − f)dt, (2.15)

where

cs =

∫ ∞
0

u−s−1(e−u − 1)du,

for f ∈ P and f ∈ L2(γd) (see [7]). This way, (2.15) is an integral repre-
sentation of D

γ
sf. By using (2.5) we get that (2.15) coincides with (2.14), if

f = hβ, ∀|β| > 0 and 0 < s < 1. Similarly, by means of the property Pt1 = 1
we conclude that (2.15) coincides with (2.14), if f = h0. Moreover, if f ∈ P,
by (2.7) and (2.14) we obtain that

Iγs (D
γ
sf) = D

γ
s (I

γ
s f) = Π0f. (2.16)



236 i. a. lópez p.

Following S. Watanabe [17] and H. Sugita [16], we consider the Gaussian-
Bessel potentials defined by

(I − L)−s/2f =
∞∑
n=0

(1 + n)−s/2Jnf, for f ∈ P.

By means of the Gamma function we obtain that

1

Γ(s/2)

∫ ∞
0

t
s
2
−1e−(1+n)tdt = (1 + n)−s/2,

thus, (2.6) lets us write

(I − L)−s/2f =
1

Γ(s/2)

∫ ∞
0

t
s
2
−1e−tTtfdt, for f ∈ P (2.17)

and by use of the contraction property of the semigroup {Tt}t≥0, we obtain
that

∥(I − L)−s/2f∥p,γd ≤ ∥f∥p,γd.

Then, the Gaussian-Bessel potential spaces of order s ≥ 0, Lps(γd), with 1 <
p < ∞, can be defined as the completion of the polynomials with respect to
the norm

∥f∥p,s :=
∥∥∥(I − L)s/2f∥∥∥

p,γd
;

in other words, L
p
s(γd) is a subspace of L

p(γd) consisting of all f which can
be written in the form

f = (I − L)−s/2ψ, with ψ ∈ Lp(γd),

where
∥f∥p,s = ∥ψ∥p,γd.

These potential spaces present the following inclusion properties (see [7, 17]
for more details).

i) If p ≤ q then Lqs(γd) ⊆ L
p
s(γd), for each s > 0.

ii) If 0 < s ≤ r then Lpr(γd) ⊆ L
p
s(γd), for each 1 < p < ∞.

Then, in [7] the following Theorem has been obtained

Theorem 2.1. Let s ≥ 0 and 1 < p < ∞. Then f ∈ Lps(γd) if and only if
D
γ
sf ∈ Lp(γd). Moreover,

Bp,s ∥f∥p,s ≤ ∥D
γ
sf∥p,γd ≤ Ap,s ∥f∥p,s .



sobolev spaces and potential spaces 237

Particularly, we can observe that if {Pn} ∈ P, such that, limn→∞ Pn = f
in L

p
s(γd), then limn D

γ
sPn exists in L

p
s(γd) and does not depend on the choice

of a sequence {Pn}n.
Theorem 2.1 is a corollary of the following theorem, obtained by P.A.

Meyer, see [9, 16]. This theorem shall be an important tool to develop our
result in Section 3.

Theorem 2.2. (Meyer’s multiplier theorem) Let Tϕ be given by

Tϕ =
∑
n≥0

ϕ(n)Jn,

where {ϕ(n)}n≥0 is a real sequence. Assume that h(z) is a function, which
is analytic on some neighborhood of the origin. If there are n0 ∈ N, and a
positive constant s, such that h(n−s) = ϕ(n) ∀n ≥ n0, then Tϕ can uniquely
extend to a bounded linear operator on Lp(γd), for each 1 < p < ∞.

As an application of Meyer’s multiplier theorem, in [7], the following result
has been obtained.

Proposition 2.1. Given 1 < p < ∞ and s ≥ 1. If f ∈ Lps(γd), then
f ∈ Lps−1(γd) and for each j = 1, . . . ,d, ∂jf ∈ L

p
s−1(γd). Moreover,

∥f∥p,s−1 +
d∑
j=1

∥∂jf∥p,s−1 ≤ Ap,s,d ∥f∥p,s .

Finally, let us consider the Gaussian Sobolev space defined as

W
p
k (γd) := {f : ∂

αf ∈ Lp(γd), α ∈ Nd, |α| ≤ k},

where ∂0f = f, equipped with the norm

∥f∥Wp
k
:=

∑
|α|≤k

∥∂αf∥p,γd

(see [15, pages 121–122]; where W
p
k (λd) spaces are considered).

Then for each 1 < p < ∞ and k ≥ m, we can see that ∥f∥Wpm ≤ ∥f∥Wpk
and therefore, W

p
k (γd) ⊆ W

p
m(γd).

Also, from the definition of W
p
k (γd) spaces we can see that f ∈ W

p
k (γd), if

and only if, f and ∂jf ∈ W
p
k−1(γd) for each j = 1, . . . ,d. Moreover, the two

norms

∥f∥Wp
k

and ∥f∥
W

p
k−1

+
d∑
j=1

∥∂jf∥
W

p
k−1



238 i. a. lópez p.

are equivalent.

Remark. In [7], as has been mentioned previously, we proved that if k ≥ 1,
1 < p < ∞ and f ∈ Lpk(γ1), then there exist a positive constant, Bp,k, such
that,

∥f∥p,k ≤ Bp,k
∥∥∥∥ dkdxk f

∥∥∥∥
p,γ1

for the unidimensional case. Nevertheless, in order to motivate the results to
be developed in Section 3, we recall how this inequality was proved.

In fact, if f ∈ P we define the operator Tk as

Tkf =
∑
n≥0

(n + k)k

2k(n + k) · · · (n + 1)
cfnhn.

Then using Meyer’s Multipliers Theorem with the function

h(z) =
(1 + kz)k

2k(1 + kz) · · · (1 + z)
,

we obtain that
∥Tkf∥p,γd ≤ Cp ∥f∥p,γd .

Now, we introduce the operator Uk as

Ukhn =

(
n + k

2

)k/2 (
(n + k) · · · (n + 1)

)−1/2
hn+k

and if f ∈ P we get that Ukf =
∑

n≥0 c
f
nUk(hn).

Denoting (Rk)
∗ = (−L)−k/2δk as the unidimensional adjoint Gaussian

Riesz transform of order k, with k ≥ 1, we can see that

(Rk)
∗hn(x) = 2

k/2 [(n + 1) · · · (n + k)]
1/2

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

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

Uk(hn) = [(Rk)
∗ ◦ Tk](hn).

For this reason, if f ∈ P, by means of the Lp(γd)-continuity of the opera-
tors (Rk)∗ and Tk we obtain that

∥Ukf∥p,γd ≤ Cp,k ∥f∥p,γd . (2.18)



sobolev spaces and potential spaces 239

But by definition (Uk ◦ Rk)(hn) = hn and if f ∈ P with c
f
0 = 0, we get

D
γ
kf =

(
Uk ◦ Rk ◦ D

γ
k

)
f =

(
Uk ◦

dk

dxk
◦ Iγk ◦ D

γ
k

)
f = Uk

(
dk

dxk
f

)
.

Therefore, by using the Theorem 2.1 and (2.18) we have that

∥f∥p,k ≤ Bp,k
∥∥∥∥ dkdxk f

∥∥∥∥
p,γ1

,

and we can use the density of the polynomials in Lp(γ1) in the general case.
Now, let us consider the multidimensional case with d > 1 and we would

like to repeat a similar argument. In this case, we could define the operators

Tα(hβ) =
|β + α||α|

2|α|
d∏
i=1

(|β| + 1) · · · (|β| + αi)

hβ

and

Uα(hβ) =

(
|β + α|

2

)|α|/2 [ d∏
i=1

(βi + 1) · · · (βi + αi)

]−1/2
hβ+α

for α ∈ Nd. Thus, if f ∈ P we introduce

Tαf =
∑
n≥0

Tα(Jnf), and Uαf =
∑
n≥0

Uα(Jnf)

and by using Meyer’s Multipliers Theorem with

h(z) =
2−|α|(1 + |α|z)

d∏
i=1

(1 + αiz) · · · (1 + z)

and ϕ(|β|) =
2−|α|

(
1 +

|α|
|β|

)
d∏
i=1

(
1 +

1

|β|

)
· · ·
(
1 +

αi
|β|

),

we obtain that Tα has a L
p(γd)-continuous extension. However, we can see

that

[(Rα|α|)
∗ ◦ Tα](hβ) =

(
|β + α||α|

2|α|

)1/2



d∏
i=1

(βi + 1) · · · (βi + αi)

d∏
i=1

(|β| + 1) · · · (|β| + αi)



1/2

hβ+α



240 i. a. lópez p.

and therefore
Uαf ̸= [(Rα|α|)

∗ ◦ Tα]f,

if f ∈ P. In consequence, this reasoning fails in the multidimensional case.
This way, to prove that

∥f∥p,k ≃ ∥f∥Wp
k
,

we need to developed a different argument.
Consequently, we are able to present the results of this paper in the fol-

lowing section.

3. The results

By using (2.11) and (2.16) we can write

∂αf = Rα|α|D
γ
|α|f,

if f is a nonconstant polynomial. Then, by means of the Lp(γd)-continuity of
the Gaussian-Riesz transform and Theorem 2.1, we get

∥∂αf∥p,γd ≤ Cp,|α| ∥f∥p,|α| (3.1)

and therefore, the density of the polynomials in L
p
|α|(γd), with |α| ≤ k, the

fact that ∥f∥p,|α| ≤ ∥f∥p,k and (3.1), allows us to conclude that

∥f∥Wp
k
≤ Cp.k∥f∥p,k. (3.2)

Consequently,
L
p
k(γd) ⊆ W

p
k (γd) for each k ∈ N. (3.3)

Now, we shall prove the converse inequality in (3.2). First, we establish
the following Lemma, where the relation between Riesz potentials and Bessel
potentials are obtained in the gaussian context; see [3] for similar results and
compare with [15, pages 133–134].

Lemma 3.1. Let s ≥ 0 and 1 < p < ∞.

i) Suppose f ∈ P, then there exists a constant Cp > 0, such that,∥∥(Dγs ◦ (I − L)−s/2)f∥∥p,γd ≤ Cp∥f∥p,γd.
ii) Suppose f ∈ P, then we have∥∥((I − L)s/2 ◦ Iγs )f∥∥p,γd ≤ Ap∥f∥p,γd.



sobolev spaces and potential spaces 241

iii) There exists a pair of operators T1 and T2, which are bounded operators
on Lp(γd), so that

(I − L)s/2 = T1 + Dγs ◦ T2.

Proof. Items i) and ii) have been obtained in [3]. However, we present it
with details for the sake of completeness.

i) If f ∈ P then f =
∑

n≥0 Jnf. Particularly, we observe that(
Dγs ◦ (I − L)

−s/2)J0f = cf0(Dγs ◦ (I − L)−s/2)h0 = 0
and we can express(

Dγs ◦ (I − L)
−s/2)f = ∑

n≥0

ns/2

(1 + n)s/2
Jnf.

Therefore, the result follows from Theorem 2.2 considering the function

h(z) = (z + 1)
−s/2

.

ii) Again, we consider f ∈ P, such that f ∈ (C0)⊥, then(
(I − L)s/2 ◦ Iγs

)
f =

∑
n>0

(1 + n)s/2

ns/2
Jnf

and similarly, by means of Meyer’s multiplier theorem with the function

h(z) = (z + 1)
s/2

,

we get that ∥∥((I − L)s/2 ◦ Iγs )f∥∥p,γd ≤ Ap∥f∥p,γd.
Particularly, if f is a constant function, then f = c

f
0h0 and since by defi-

nition I
γ
s h0 = 0, we have that(

(I − L)s/2 ◦ Iγs
)
f = c

f
0

(
(I − L)s/2 ◦ Iγs

)
h0 = 0.

iii) We choose T1 = I and

T2 =
(
(I − L)s/2 ◦ Iγs

)
− Iγs .

Then if f ∈ P, such that, f ∈ (C0)⊥, we obtain

∥T2f∥p,γd ≤
∥∥((I − L)s/2 ◦ Iγs )f∥∥p,γd + ∥Iγs f∥p,γd

and iii) follows from ii) and (2.10).
Particularly, if f is a constant function, we have that I

γ
s f = 0 and therefore,

T2f = 0.



242 i. a. lópez p.

Remark. It should be emphasized, that polynomials are dense in Lp(γd),
for 1 < p < ∞. This way, the Lemma 3.1 states that the operators defined by

Dγs ◦ (I − L)
−s/2 and (I − L)s/2 ◦ Iγs ,

are bounded operators on Lp(γd), on every 1 < p < ∞.

Now, on the one hand, by using (2.12) and (2.13) we observe that

(R
j
1)

∗(∂jhβ) = (R
j
1)

∗(√2βjhβ−ej) = 2βj|β|1/2 hβ,
for each j = 1, ...,d and |β| ̸= 0. Thus,

d∑
j=1

(R
j
1)

∗(∂jhβ) = 2|β|1/2hβ

and if g ∈ P, we obtain that

d∑
j=1

(R
j
1)

∗(∂jJng) = 2n
1/2Jng.

In consequence,

D
γ
1g =

1

2

d∑
j=1

(R
j
1)

∗(∂jg). (3.4)

Particularly, if g is a constant function we can see that D
γ
1g = 0 and since

∂jg = 0, then (3.4) is also true.

On other hand, if s ≥ 1, we claim that

(I − L)−(s−1)/2(∂jg) =
(
T0 ◦ ∂j(I − L)−(s−1)/2

)
g, (3.5)

where we set T0 := (I − L)(s−1)/2 ◦ I
γ
s−1.

In fact, suppose g ∈ P, such that, g ∈ (C0)⊥. Then g =
∑

n>0 Jng and

∂j(I − L)−(s−1)/2g =
∑
n>0

(
1

n + 1

)(s−1)/2 ∑
|β|=n

√
2βjc

g
βhβ−ej.



sobolev spaces and potential spaces 243

Also,

(I − L)−(s−1)/2∂jg =
∑
n>0

∑
|β|=n

√
2βjc

g
β

|β|(s−1)/2
hβ−ej

=
∑
n>0

(
1 + n

n

)(s−1)/2 (
1

1 + n

)(s−1)/2 ∑
|β|=n

√
2βjc

g
βhβ−ej

= ((I − L)(s−1)/2 ◦ Iγs−1 ◦ ∂j(I − L)
−(s−1)/2)g,

which proves (3.5). Particularly, if g is a constant function we have that

(I − L)−(s−1)/2(∂jg) = 0 and ∂j(I − L)−(s−1)/2g = c
g
0∂jh0 = 0

and therefore, (3.5) is also true.

Then, we are able to prove the following proposition (see [15, pages 136–
138]).

Proposition 3.1. Given 1 < p < ∞ and s ≥ 1. Suppose that f ∈
L
p
s−1(γd) and ∂jf ∈ L

p
s−1(γd) for each j = 1, . . . ,d. Then, f ∈ L

p
s(γd) and

also,

∥f∥p,s ≤ Bp,s

(
∥f∥p,s−1 +

d∑
j=1

∥∂jf∥p,s−1

)
.

Proof. Let f ∈ Lps−1(γd) ∩ P. Then, there exists ψ ∈ P, such that f =
(I − L)−(s−1)/2ψ and therefore,

f =
∑
n≥0

(
1

n + 1

)(s−1)/2
Jnψ

and

∂jf =
∑
n≥0

(
1

n + 1

)(s−1)/2 ∑
|β|=n

√
2βjc

ψ
βhβ−ej.

Since ψ ∈ Lp1(γd) ∩ P, we can write ψ = (I −L)
−1/2h and according to the

Lemma 3.1, part iii), where we consider s = 1, T1 = I and

T2 =
(
(I − L)1/2 ◦ Iγ1

)
− Iγ1 ,



244 i. a. lópez p.

we obtain that

h = (I − L)1/2ψ = T1ψ + (D
γ
1 ◦ T2)ψ.

Therefore,

∥h∥p,γd ≤ ∥T1ψ∥p,γd +
∥∥(Dγ1 ◦ T2)ψ∥∥p,γd. (3.6)

Now, we can see that

∥h∥p,γd =
∥∥(I − L)1/2ψ∥∥

p,γd
=
∥∥(I − L)1/2(I − L)(s−1)/2f∥∥

p,γd
= ∥f∥p,s

and

∥T1ψ∥p,γd = ∥ψ∥p,γd =
∥∥(I − L)(s−1)/2f∥∥

p,γd
= ∥f∥p,s−1.

On the other hand, by means of (3.4) we have that

(D
γ
1 ◦ T2)ψ = (T2 ◦ D

γ
1)ψ =

1

2
T2

[
d∑
j=1

(R
j
1)

∗(∂jψ)

]

and therefore,

∥(Dγ1 ◦ T2)ψ∥p,γd =
1

2

∥∥∥∥∥T2
[

d∑
j=1

(R
j
1)

∗(∂jψ)

]∥∥∥∥∥
p,γd

≤ Ap

∥∥∥∥∥
d∑
j=1

(R
j
1)

∗(∂jψ)

∥∥∥∥∥
p,γd

≤ Cp
d∑
j=1

∥∂jψ∥p,γd.

Since f ∈ Lps−1(γd) ∩ P and ∂jf ∈ L
p
s−1(γd) ∩ P, for each j = 1, . . . ,d,

according to (3.5) with g = ψ we have that

T0(∂jf) = (I − L)−(s−1)/2∂jψ

and consequently,

∥∂jψ∥p,γd =
∥∥(T0 ◦ (I − L)(s−1)/2)(∂jf)∥∥p,γd

≤ Ap
∥∥(I − L)(s−1)/2(∂jf)∥∥p,γd

= Cp∥∂jf∥p,s−1,



sobolev spaces and potential spaces 245

because T0 ◦ (I − L)(s−1)/2 = (I − L)(s−1)/2 ◦ T0. Then, from (3.6) we obtain
that

∥f∥p,s ≤ Bp,s

(
∥f∥p,s−1 +

d∑
j=1

∥∂jf∥p,s−1

)
.

In the general case, the density of the polynomials in L
p
s(γd) is used.

Remark. In the proof of the Proposition 3.1, if f is a constant function,
we can see that f = ψ = h. Then, trivially we can write

h = (I − L)1/2ψ = T1ψ + (D
γ
1 ◦ T2)ψ

because (I − L)1/2ψ = ψ, T1ψ = ψ and T2ψ = 0. This way, we obtain that

∥f∥p,s−1 = ∥f∥p,s .

The Proposition 2.1 and the Proposition 3.1 let us obtain the following
corollary.

Corollary 3.1. Suppose 1 < p < ∞ and s ≥ 1. Then, f ∈ Lps(γd) if and
only if f ∈ Lps−1(γd) and for each j = 1, . . . ,d, ∂jf ∈ L

p
s−1(γd). Moreover, the

two norms, ∥f∥p,s and ∥f∥p,s−1 +
∑d

j=1 ∥∂jf∥p,s−1, are equivalent.

Also, by means of the Proposition 3.1 we get the following result.

Corollary 3.2. Given 1 < p < ∞ and k ≥ 1. Suppose f ∈ Wpk (γd),
then there exists a positive constant Ap,k such that

∥f∥p,k ≤ Ap.k

(
∥f∥p,γd +

∑
|α|≤k

∥∂αf∥p,γd

)
.

Proof. By using the Proposition 3.1, with s = 1, we have

∥f∥p,1 ≤ Bp,1

(
∥f∥p,γd +

d∑
j=1

∥∂jf∥p,γd

)
.

If s = 2 we can see that

∥f∥p,2 ≤ Bp,2

(
∥f∥p,1 +

d∑
i=1

∥∂if∥p,1

)



246 i. a. lópez p.

and for each i = 1, ...,d,

∥∂if∥p,1 ≤ Ap,1

(
∥∂if∥p,γd +

d∑
j=1

∥∥∂2ijf∥∥p,γd
)
.

Then by means of the estimates above, we get

∥f∥p,2 ≤ Cp,2

(
∥f∥p,γd +

d∑
j=1

∥∂jf∥p,γd +
d∑
i=1

d∑
j=1

∥∥∂2ijf∥∥p,γd
)
,

and the result of this corollary is obtained by induction.

Therefore, we can conclude that

W
p
k (γd) ⊆ L

p
k(γd) for each k ∈ N (3.7)

and in consequence, by using (3.3) and (3.7) we obtain the following theorem
which it is the principal result of this paper.

Theorem 3.1. Suppose k is a positive integer and 1 < p < ∞. Then

L
p
k(γd) = W

p
k (γd).

Particularly, as a final comment we observe that

Remark. If 0 < s < 1, 1 < p < ∞ and k ≥ 1, then Wpk (γd) ⊂ L
p
s(γd) and

the inclusion is proper.

In fact, if 0 < s < 1 and k ≥ 1, the inclusion properties of the potential
spaces allows us to write

W
p
k (γd) = L

p
k(γd) ⊂ L

p
s(γd).

Now, let us consider d = 1, k = 1, p = 2, 0 < s < 1 and the function f defined
as

f(x) =

{√
x if x ≥ 0,

0 if x < 0.

First, one checks that f ∈ L2(γ1) but f ′ /∈ L2(γ1) and therefore f /∈ W21 (γ1).
Now, for each s ∈ (0,1) we define the family of functions

gs(x) =

{√
2sx if x ≥ 0,

0 if x < 0,



sobolev spaces and potential spaces 247

and from (2.2) we get that the semigroup, Ttgs, can be written as

Ttgs(x) =

√
2s√

π(1 − e−2t)

∫ ∞
0

e
− |y−e

−tx|2

1−e−2t
√
ydy.

By using the change of variable u =
y−e−tx√
1−e−2t

in the above identity, we have

that

Ttgs(x) =

√
2s

π

∫ ∞
− xe

−t
(1−e−2t)1/2

(√
1 − e−2tu + e−tx

)1/2
e−u

2
du

≥ e−t/2
√
x

√
2s

π

∫ ∞
0

e−u
2
du,

so, Ttgs(x) ≥ 2s/2e−t/2
√
x, for all t > 0 and x ≥ 0. Therefore,

(I − L)−s/2gs(x) =
1

Γ(s/2)

∫ ∞
0

t
s
2
−1e−tTtgs(x)dt

≥ 2s/2
√
x

(
1

Γ(s/2)

∫ ∞
0

t
s
2
−1e−2tdt

)
and then, by use of the change of variable t = u/2, we obtain

(I − L)−s/2gs(x) ≥ f(x) for each x ≥ 0. (3.8)

From (2.17) if h1 ≥ h2, we have that

(I − L)−s/2h1 ≥ (I − L)−s/2h2, (3.9)

because Tt is linear and Tth ≥ 0, if h ≥ 0. Also, if h1 ≥ h2, we can see that

(I − L)s/2h1 ≥ (I − L)s/2h2.

In fact, suppose that h1 ≥ h2 but (I − L)s/2h1 < (I − L)s/2h2, then by using
(3.9) we obtain a contradiction. Thus, by applying the operator (I − L)s/2 in
(3.8), we get

gs(x) ≥ (I − L)s/2f(x)
and consequently,

∥f∥2,s =
∥∥(I − L)s/2f∥∥

2,γ1
≤ ∥gs∥2,γ1 = 2

s/2∥f∥2,γ1 ≤
√
2∥f∥2,γ1,

since s ∈ (0,1). Therefore, we can conclude that f ∈ L2s(γ1).
This way, we have obtained a function f, such that, f /∈ W21 (γ1) but

f ∈ L2s(γ1), if 0 < s < 1.



248 i. a. lópez p.

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