The Hardy space H1 in the rational Dunkl setting

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The Hardy space H1 in the rational Dunkl setting

Jean-Philippe Anker, Néjib Ben Salem, Jacek Dziubanski, Nabila Hamda

To cite this version:

Jean-Philippe Anker, Néjib Ben Salem, Jacek Dziubanski, Nabila Hamda. The Hardy space H1 in the rational Dunkl setting. Constructive Approximation, Springer Verlag, 2015, 42, pp.93-128.

�10.1007/s00365-014-9254-2�. �hal-00864457�

JEAN-PHILIPPE ANKER, N´EJIB BEN SALEM, JACEK DZIUBA ´NSKI, AND NABILA HAMDA

Abstract. This paper consists in a first study of the Hardy spaceH¹ in the rational Dunkl setting. Following Uchiyama’s approach, we characterizeeH¹atomically and by means of the heat maximal operator. We also obtain a Fourier multiplier theorem for H¹. These results are proved here in the one-dimensional case and in the product case.

1. Introduction

Dunkl theory is a far reaching generalization of Euclidean Fourier analysis, which includes most special functions related to root systems, such as spherical functions on Riemannian symmetric spaces. It started in the late eighties with Dunkl’s seminal article [7] and developed extensively afterwards. We refer to the lecture notes [18] for the rational Dunkl theory, to the lecture notes [15] for the trigonometric Dunkl theory, and to the books [4, 11] for the generalized quantum theories.

This paper deals with the real Hardy space H

in the rational Dunkl setting, where the underlying space is of homogeneous type in the sense of Coifman-Weiss. In such a setting, the theory of Hardy spaces goes back to the seventies [6, 12]. Here we follow Uchyama’s approach [25] and we characterize the Hardy space H

in two ways, by means of the heat maximal operator and atomically. The first characterization, which requires precise heat kernel estimates, has lead us to a seemingly new observation, namely that the heat kernel has a rather slow decay in certain directions and is in particular not Gaussian in the present setting (see Remark 2.4). The second characterization is used to prove a Fourier multiplier theorem for H

.

Throughout the paper we shall restrict to the one-dimensional case and to the product case.

This restriction is due to our present lack of knowledge in general about the behavior of the Dunkl kernel on the one hand and about generalized translations on the other hand.

After this informal introduction, let us introduce some notation and state our main results.

On R

we consider the Dunkl operators D

f (x) =

j

f (x) +

j

f (x) − f (σ

x)

(j = 1, 2, . . . , n) associated with the reflections

(1.1) σ

(x

, x

, . . . , x

, . . . , x

) = (x

, x

, . . . , − x

, . . . , x

)

Date: September 21, 2013.

2010 Mathematics Subject Classification. Primary : 42B30. Secondary : 33C52, 33C67, 33D67, 33D80, 35K08, 42B25, 42C05.

Key words and phrases. Dunkl theory, heat kernel, Hardy space, maximal operator, atomic decomposition, multiplier.

JPA and JD partially supported by the European Commission (European program ToKHANAP2005-2009, Polish doctoral program SSDNM 2010-2013) and by a French-Polish cooperation program (PHC Polonium 22529RF 2010-2011). JPA, NBS, NHK partially supported by a French-Tunisian cooperation program (PHC Utique / CMCU 10G1503 2010-2013). JD partially supported by the Polish National Science Centre (Narodowe Centrum Nauki, grant DEC-2012/05/B/ST1/00672) and by the University of Orl´eans.

1

and the multiplicities k

≥ 0. Their joint eigenfunctions constitute the Dunkl kernel

(1.2) E(x, y) = Y

j=1

E

(x

, y

) , where

(1.3)

E

(x, y) =

Γ(k) Γ(¹₂)

Z

−1

du (1 − u)

(1+ u)

e

= e

Z

dv v

(1 − v)

e

| {z }

1

F

(k; 2 k +1; − 2 xy)

(see for instance [18, Example 2.34]). Here

F

(a; b; z) is the confluent hypergeometric func- tion, which is also known as the Kummer function and denoted by M (a, b, z). Notice that E(x, y) = e

if all multiplicities k

vanish.

Let us first define the Hardy space H

by means of the heat maximal operator. The Dunkl laplacian

Lf (x) = X

j=1

D

f (x) = X

n

∂ xj

f (x) +

j

∂

∂ xj

f (x) −

j

f (x) − f (σ

x) o is the infinitesimal generator of the heat semigroup

e

( t > 0),

which acts by linear self-adjoint operators on L

( R

, dµ) and by linear contractions on L

( R

, dµ), for every 1 ≤ p ≤ ∞ , where

(1.4) dµ(x) = dµ

(x

) . . . dµ

(x

) = | x

|

. . . | x

|

dx

. . . dx

The heat semigroup consists of integral operators e

f(x) =

Z

Rⁿ

dµ(y) h

(x, y) f (y) associated with the heat kernel [17]

(1.5) h

(x, y) = c

t

e

2

4t

E

2t

,

, where

(1.6) N = n + X

j=1

2 k

is the homogeneous dimension and

c

= 2

Z

Rⁿ

dµ(x) e

2

2

= 2

Y

j=1

Γ(k

+

) .

From this point of view, the Hardy space H

consists of all functions f ∈ L

( R

, dµ) whose maximal heat transform

(1.7) h

f (x) = sup

Z

Rⁿ

dµ(y) h

(x, y) f (y) belongs to L

( R

, dµ) and the norm is given by

k f k

= k h

f k

.

Let us turn next to the atomic definition of the Hardy space H

. Notice that R

, equipped

with the Euclidean distance d(x, y) = | x − y | and with the measure µ, is a space of ho-

mogeneous type in the sense of Coifman-Weiss (see Appendix A). Recall that an atom is a

measurable function a : R

→ C such that

• a is supported in a ball B ,

• k a k

. µ(B)

,

• Z

Rⁿ

dµ(x) a(x) = 0.

By definition, the atomic Hardy space H

consists of all functions f ∈ L

( R

, dµ) which can be written as f = P

ℓ

λ

a

, where the a

’s are atoms and P

ℓ

| λ

| < + ∞ , and the norm is given by

k f k

= inf X

ℓ

| λ

| , where the infimum is taken over all atomic decompositions of f .

Our first main result is the following theorem.

Theorem 1.8. The spaces H

and H

coincide and their norms are equivalent, i.e., there exists a constant C > 0 such that

C

k f k

≤ k f k

≤ C k f k

. The Fourier transform in the Dunkl setting is given by

(1.9) F f (ξ) = c

Z

Rn

dµ(x) f (x) E(x, − iξ) .

It is an isometric isomorphism of L

( R

, dµ) onto itself and the inversion formula reads f (x) = F

f ( − x) .

Notice that, if all multiplicities k

vanish, then (1.9) boils down to the classical Fourier trans- form

f b (ξ) = (2π)

Z

Rⁿ

dx f (x) e

.

Our second main result is the following H¨ ormander type multiplier theorem (see [10] for the original multiplier theorem on L

spaces).

Theorem 1.10. Let χ = χ(ξ) be a smooth radial function on R

such that χ(ξ) =

( 1 if | ξ |∈

2

, 2 , 0 if | ξ | ∈ /

, 4

. If a function m = m(ξ) on R

satisfies

M = sup

k χ m(t . ) k

2

< + ∞ , for some ε > 0, then the multiplier operator

T

f = F

{ m ( F f ) } is bounded on the Hardy space H

and

k T

k

. M .

Here W

( R

) denotes the classical L

Sobolev space on R

, whose norm is given by k g k

2

= n Z

Rⁿ

dx (1+ | x |

)

|b g(x) |

o

.

Notice that the multiplier m is continuous and bounded, as

+ ε >

.

The theory of classical real Hardy spaces in R

originates from the study of holomorphic

functions of one variable in the upper half-plane. We refer the reader to the original works

of Stein-Weiss [22], Burkholder-Gundy-Silverstein [3] and Fefferman-Stein [9]. An important

contribution to this theory lies in the atomic decomposition introduced by Coifman [5] and extended to spaces of homogeneous type by Coifman-Weiss [6] (see also [12]). More information can be found in the book [21] and references therein.

Our paper is organized as follows. Section 2 is devoted to the heat kernel in dimension 1.

There we analyze its behavior thoroughly and we remove a small part, in order to get Gaussian estimates similar to the Euclidean setting. These results are extended to the product case in Section 3. Section 4 is devoted to the proof of Theorem 1.8 and Section 5 to the proof of Theorem 1.10. Section 6 consists of 3 appendices. Appendix A contains information about the measure of balls, which is used throughout the paper. Appendices B and C are devoted to so-called folklore results in connection with Uchiyama’s Theorem, which have been used for instance in [8].

This paper results from two independent research works, which were carried out by the first and third authors, respectively by the second and fourth authors, and which have been merged into a joint article.

2. Heat kernel estimates in dimension 1

Consider first the one-dimensional Dunkl kernel E(x, y) = E

(x, y). As the case k = 0 is trivial, we may assume that k > 0.

Lemma 2.1. (a) E(x, y) is a holomorphic function of (x, y) ∈ C

. (b) E(x, y) > 0 for every x, y ∈ R .

(c) E(x, y) has the following symmetry and rescaling properties : ( E(x, y) = E(y, x) ∀ x, y ∈ C ,

E(λx, y) = E(x, λy) ∀ λ, x, y ∈ C .

(d) For every y ∈ C , x 7→ E(x, y) is an eigenfunction of the Dunkl operator Df (x) = f

(x) +

{ f (x) − f ( − x) }

and of the Dunkl laplacian

Lf (x) = D

f (x) = f

(x) +

f

(x) −

{ f (x) − f ( − x) } . More precisely

D

E(x, y) = y E (x, y) and L

E(x, y) = y

E(x, y) . (e) As xy → 0,

E(x, y) = 1 + O( | xy | ) . (f) As xy → + ∞ ,

E(x, y) =

π

e

(xy)

1 −

+ O(

) . (g) As xy → −∞ ,

E(x, y) =

π

e

( − xy)

1 +

+ O(

) .

Proof. The first four properties are known to hold in general. In dimension 1, they can be also deduced from the explicit expression (1.3), as does (e). As already observed in [20, Section 2]

(see also [18, Example 5.1]), the asymptotics of E(x, y) at infinity follow from the asymptotics of the confluent hypergeometric function, which read, let say for 0 < a < b,

1

F

(a; b; z) ∼

e

z

X

(1−a)_ℓ(b−a)_ℓ ℓ!

z

as z → + ∞ and

1

F

(a; b; z) ∼

| z |

X

(a)ℓ(a−b+1)ℓ

ℓ!

| z |

as z → −∞ (see for instance [1, (13.5.1)] or [14, (13.7.2)]).

Consider next the one-dimensional heat kernel (2.2) h

(x, y) = c

t

e

2 4t

E(

2t

,

2t

) = c

t

e

F

(k ; 2k +1; −

) , where c

= 2

Γ(k+

).

Proposition 2.3. (a) h

(x, y) is a C

function of (t, x, y) ∈ (0, + ∞ ) × R

. (b) h

(x, y) > 0 for every t > 0 and x, y ∈ R .

(c) h

(x, y) has the following symmetry and rescaling properties : ( h

(x, y) = h

(y, x) ∀ x, y ∈ R ,

h

(λx, λy) = | λ |

h

(x, y) ∀ λ ∈ R

, ∀ t > 0 , ∀ x, y ∈ R . (d) h

(x, y) satisfies the heat equation

( ∂

h

(x, y) = L

h

(x, y),

lim

h

(x, y) | y |

dy = δ

(y).

(e) The heat kernel has the following global behavior :

h

(x, y) ≍

 

 



 

 

t

e

2

4t

if | xy |≤ t,

t

(xy)

e

if xy ≥ t, t

( − xy)

e

if − xy ≥ t, and the following asymptotics :

h

(x, y) =

 

 



 

 

c

t

e

2

4t

1 + O

if

→ 0,

1 2√

π

e

t

(xy)

1 − k

+ O(

) if

→ + ∞ ,

k

2√π

e

t

( − xy)

1 + O( −

) if

→−∞ . (f) The following gradient estimates hold for the heat kernel :

h

(x, y) .

 

 



 

 

t

( | x | + | y | ) e

2

4t

if | xy |≤ t,

t

| x − y | + t

| y |

(xy)

e

if xy ≥ t, t

| x +y | + t

( | x |

+ | y |

) ( − xy )

e

if − xy ≥ t.

Proof. The first five properties follow from the expression (2.2) and from Lemma 2.1. Let us turn to the proof of (f). By differentiating (2.2) with respect to y and by using the well-known formula

d

dz 1

F

(a; b; z) =

F

(a +1; b +1; z) (see for instance [1, (13.4.8)] or [14, (13.3.15)]), we get

∂

∂y

h

(x, y) = c

t

e

2t 1

F

(k; 2k +1; −

) −

F

(k +1; 2k + 2; −

) .

We conclude by using again the behavior of the confluent hypergeometric function.

Remark 2.4. It follows from Proposition 2.3.(e) and Appendix A that h

(x, x) ≍ µ(B (x, √

t ))

and h

(x, − x) ≍ µ(B(x, √

t ))

for every t > 0 and x ∈ R . Observe in particular that the heat kernel has no global Gaussian behavior and decays rather slowly in certain directions. This phenomenon is even more striking in the product case (3.1), where

h

(x, y) ≍ µ(B (x, √

t ))

if t > 0 , x ∈ R

and y = ( − x

, x

, . . . , x

) .

Let us eventually introduce a variant of the heat kernel with a Gaussian behavior. Given two smooth bump functions χ

and χ

on R such that

 



 

0 ≤ χ

≤ 1, χ

= 1 on

− 1, +

, supp χ

⊂

− 2 , +

,

and

 



 

0 ≤ χ

≤ 1, χ

= 1 on

0, +

, supp χ

⊂

− 1, +1 , consider the smooth cutoff function

χ

(x, y) =

( χ

χ

if x 6 = 0 , 0 if x = 0 , and the truncated heat kernel

H

(x, y) = { 1 − χ

(x, y) } h

(x, y) ∀ t > 0, ∀ x, y ∈ R .

Remark 2.5. The truncated heat kernel H

(x, y) inherits the following properties of the heat kernel h

(x, y) :

(a) Smoothness : H

(x, y) is a C

function of (t, x, y) ∈ (0, + ∞ ) × R

. (b) Non-negativity : H

(x, y) ≥ 0 for every t > 0 and x, y ∈ R .

(c) Rescaling : H

(λx, λy) = | λ |

H

(x, y) for every λ ∈ R

, t > 0 and x, y ∈ R . (d) Approximation of identity : lim

H

(x, y) | y |

dy = δ

(y) for every x, y ∈ R . Theorem 2.6. The following estimates hold for the truncated heat kernel H

(x, y).

(a) On-diagonal estimate :

H

(x, x) ≍ µ(B (x, √

t ))

∀ t > 0, ∀ x ∈ R . (b) Off-diagonal Gaussian estimate :

0 ≤ H

(x, y) . µ(B(x, √

t ))

e

∀ t > 0, ∀ x, y ∈ R . (c) Gradient estimate :

H

(x, y) . t

µ(B (x, √

t ))

e

∀ t > 0, ∀ x, y ∈ R . (d) Lipschitz estimates :

| H

(x, y) − H

(x, y

) | . µ(B(x, √

t ))

t

∀ t > 0, ∀ x, y, y

∈ R , with the following improvement, if | y − y

| ≤

| x − y | :

| H

(x, y) − H

(x, y

) | . µ(B (x, √

t ))

e

.

Here c denotes some positive constant and the ball measure has the following behavior, ac- cording to Appendix A :

µ(B(x, √ t )) ≍

( t

if | x | ≤ √ t ,

| x |

√

t if | x | ≥ √ t .

Proof. As far as (a), (b), (c) are concerned, the case x = 0 follows immediately from the previous heat kernel estimates. Thus we may assume that x 6 = 0 and reduce furthermore to x = 1 by rescaling.

(a) is immediate :

H

(1, 1) = h

(1, 1) ≍

( t

if t ≤ 1 t

if t ≥ 1

)

≍ µ(B(1, √ t ))

.

y t

0

1 2

−

− 1

2

− 2

Figure 1. Cases and subcases considered in the proofs of (b) and (c) Let us next prove (b).

• Case 1. Assume that | y |≤ t.

◦ Subcase 1.1. Assume that t is bounded above, say t ≤

. Then H

(1, y) ≤ h

(1, y) ≍ t

e

2

4t

= t

e

t

e

2 8t

e

is bounded above by

µ(B(1, √

t ))

e

as t

≍ µ(B(1, √

t )), t

. e

≤ e

2

8t

and e

≍ 1.

◦ Subcase 1.2. Assume that t is bounded below, say t ≥

. Then H

(1, y) ≤ h

(1, y) ≍ t

e

2

4t

= t

e

e

with t

≍ µ(B(1, √

t )) and e

≍ 1.

• Case 2. Assume that y is close to − x = − 1, say − 2 ≤ y ≤−

.

◦ Subcase 2.1. If t ≤

( ≤− y ), then

H

(1, y) = 0 .

◦ Subcase 2.2. If t is bounded below, say t ≥

, we argue as in Subcase 1.2.

• Case 3. Assume that y ≥ t.

◦ Subcase 3.1. Assume that t is bounded below, say (y ≥ )t ≥

. Then H

(1, y) ≤ h

(1, y) ≍ t

y

e

≤ t

e

with t

≍ µ(B (1, √

t )).

◦ Subcase 3.2. Assume that y ≥

≥ t. Then

H

(1, y) ≤ h

(1, y) ≍ t

y

e

. t

e

with t

≍ µ(B (1, √

t )).

◦ Subcase 3.3. Assume that t ≤ y ≤

. Then

H

(1, y) ≤ h

(1, y) ≍ t

y

e

is bounded above by

µ(B(1, √

t ))

e

as t

≍ µ(B(1, √

t )) and y

≤ t

. e

≤ e

.

• Case 4. Assume that y ≤ − t (< 0) and that y stays away from − 1, say y / ∈ − 2, −

. Notice that (1+y)

≥

if and only if y / ∈ − 2, −

.

◦ Subcase 4.1. Assume that 2 ≤ t ≤− y . Then

H

(1, y) ≤ h

(1, y) ≍ t

( − y)

e

≤ t

e

with t

≍ µ(B (1, √

t )).

◦ Subcase 4.2. Assume that t ≤ 2 ≤− y . Then

H

(1, y) ≤ h

(1, y) ≍ t

( − y)

e

. t

e

with t

≍ µ(B (1, √

t )).

◦ Subcase 4.3. Assume that t ≤− y ≤

. Then

H

(1, y) ≤ h

(1, y) ≍ t

( − y)

e

≤ t

e

e

is bounded above by

µ(B(1, √

t ))

e

as t

≍ µ(B(1, √

t )) and t

. e

≤ e

.

The proof of (c) follows the same pattern. To begin with, observe that the derivative

∂

∂y

1 − χ

(1+ y) χ

(t)

| {z }

χt(1,y)

= − χ

(1+ y) χ

(t)

of the cut-off is bounded and vanishes unless y ∈ − 3 , − 2

∪ −

, 0

and t ≤ 1. According to the subcases 1.1, 4.2 and 4.3 above, the contribution of

{ 1 − χ

(1, y) } h

(1, y) to

H

(1, y) is bounded by

µ(B(1, √

t ))

e

≤ t

µ(B(1, √

t ))

e

Thus it remains for us to estimate the contribution of { 1 − χ

(1, y) }

h

(1, y).

• Case 1. Assume that | y |≤ t.

◦ Subcase 1.1. Assume that t ≤

. Then

{ 1 − χ

(1, y) }

h

(1, y) . t

(1+ | y | ) e

2 4t

.

bounded

z }| { t

e

2

8t

e

t

e

. t

µ(B(1, √

t ))

e

◦ Subcase 1.2. Assume that t ≥

. Then

{ 1 − χ

(1, y) }

h

(1, y) . t

(1+ | y | ) e

2

4t

. t

e

bounded

z }| {

1+y² t

e

2 8t

e

. t

µ(B (1, √

t ))

e

• Case 2. Assume that − 2 ≤ y ≤−

.

◦ Subcase 2.1. If t ≤

( ≤− y ), then

{ 1 − χ

(1, y) }

h

(1, y) = 0 .

◦ Subcase 2.2. If t is bounded below, say t ≥

, we argue as in Subcase 1.2.

• Case 3. Assume that y ≥ t.

◦ Subcase 3.1. Assume that ( y ≥ )t ≥

. Then { 1 − χ

(1, y) }

h

(1, y) .

t

| 1 − y | + t

y

y

e

. t

e

n

1 +

t

e

o

| {z }

bounded

. t

µ(B(1, √

t ))

e

◦ Subcase 3.2. Assume that y ≥

≥ t. Then { 1 − χ

(1, y) }

h

(1, y) .

t

| 1 − y | + t

y

y

e

. t

e

n √

t +

t

e

o

| {z }

bounded

. t

µ(B (1, √

t ))

e

◦ Subcase 3.3. Assume that t ≤ y ≤

. Then { 1 − χ

(1, y) }

h

(1, y) .

t

| 1 − y | + t

y

y

e

. t

e

t

e

| {z }

bounded

. t

µ(B (1, √

t ))

e

• Case 4. Assume that y ≤ − t (< 0) and that y / ∈ − 2, −

. Recall that (1+y)

≥

if and only if y / ∈ − 2, −

.

◦ Subcase 4.1. Assume that 2 ≤ t ≤− y . Then { 1 − χ

(1, y) }

h

(1, y) .

t

| 1+y | + t

| y |

e

. t

e

t

e

| {z }

bounded

. t

µ(B (1, √

t ))

e

◦ Subcase 4.2. Assume that t ≤ 2 ≤− y . Then { 1 − χ

(1, y) }

h

(1, y) .

t

| 1+y | + t

| y |

e

. t

e

n

|1+√y|

t

e

+ √ t o

| {z }

bounded

. t

µ(B (1, √

t ))

e

◦ Subcase 4.3. Assume that t ≤− y ≤

. Then χ

(1, y)

h

(1, y) .

t

| 1+y | + t

| y |

e

. t

e

t

e

| {z }

bounded

. t

µ(B(1, √

t ))

e

Eventually, (d) is an immediate consequence of (c). For every y

∈ [ y, y

], we have indeed e

′′)2 c t

≤ 1 .

Moreover, if | y − y

| ≤

| x − y | , then | x − y

| ≥ | x − y | − | y − y

| ≥ | x − y | − | y − y

| ≥

| x − y | , hence

e

′′)2

c t

≤ e

.

Remark 2.7. Contrarily to h

(x, y), H

(x, y) is not symmetric in the space variables x, y.

Nevertheless, according to the following result, we may replace µ(B(x, √

t )) by µ(B(y, √ t )) in the estimates (b), (c) and in the second estimate (d).

Lemma 2.8. For every ε > 0, there exists C > 0 such that

µ(B(x,√ t)) µ(B(y,√

t))

≤ C e

∀ x, y ∈ R , ∀ t > 0.

Proof. By rescaling (see Appendix A), we can reduce to the case t = 1. The estimate

µ(B(x,1))

µ(B(y,1))

. e

is obvious if x and y are bounded or if | x | / | y | is bounded from above. In the remaining case, let say when | x | ≥ 1+ 2 | y | , we have | x | ≤ | x − y | + | y | ≤ | x − y | +

| x | , hence | x | ≤ 2 | x − y | . Furthermore, as | x − y | ≥ | x | − | y | ≥ 1, we have | x | ≤ 2 (x − y)

. Thus

µ(B(x,1))

µ(B(y,1))

. µ(B(x, 1)) ≍ ( | x | +1)

. e

. e

.

Next proposition, which will be used in the proof of Theorem 1.8, shows that the truncated heat kernel H

(x, y) captures the main features of the heat kernel h

(x, y).

Proposition 2.9. The maximal operator Q

f (x) = sup

Z

R

dµ(y) Q

(x, y) f (y) , associated with the error

Q

(x, y) = h

(x, y) − H

(x, y) = χ

(x, y) h

(x, y) ≥ 0 , is bounded from L

( R , dµ) into itself.

Proof. It suffices to check that sup

Z

R

dµ(x) sup

Q

(x, y) < + ∞ .

The case y = 0 is trivial, as χ

(x, 0) and hence Q

(x, 0) vanish, for every t > 0 and x ∈ R . Consider next the case y ∈ R

, which reduces to y = 1 by rescaling. Then χ

(x, 1) and Q

(x, 1) vanish, unless t < 9 and − 3 < x < −

, and in this range (see Proposition 2.3)

h

(x, 1) ≍ t

e

is bounded. Hence

Z

R

dµ(x) sup

Q

(x, 1) . Z

−3

dx sup

h

(x, 1) < + ∞ .

3. Heat kernel estimates in the product case

According to (1.5) and (1.2), the heat kernel in the product case splits up into one- dimensional heat kernels :

(3.1) h

(x, y) = Y

j=1

h

(x

, y

) . By expanding

h

(x

, y

) = { 1 − χ

(x

, y

) } h

(x

, y

)

| {z }

H^(j)_t (xj, yj)

+ χ

(x

, y

) h

(x

, y

)

| {z }

Q^(j)_t (xj, yj)

,

we get

h

(x, y) = H

(x, y) + P

(x, y) . Here

H

(x, y) = Y

j=1

H

(x

, y

) and P

(x, y) is the sum of all possible products

P e

(x, y) = Y

j=1

p

(x

, y

) ,

where each factor p

(x

, y

) is equal to H

(x

, y

) or Q

(x

, y

), and at least one factor p

(x

, y

) is equal to Q

(x

, y

). Notice the rescaling property

h

(λx, λy) = | λ |

h

(x, y) ∀ λ ∈ R

, ∀ t > 0, ∀ x, y ∈ R

,

and similarly for the other product kernels. The following estimates follow from the one- dimensional case (see Theorem 2.6 and Remark 2.7).

Theorem 3.2. (a) On-diagonal estimate : H

(x, x) ≍ µ(B( x, √

t ))

∀ t > 0 , ∀ x ∈ R

. (b) Off-diagonal Gaussian estimate :

0 ≤ H

(x, y) . max

µ(B (x, √

t )), µ( B( y, √

t ))

e

2 c t

for every t > 0 and for every x, y ∈ R

. (c) Gradient estimate :

|∇

H

(x, y) | . t

max

µ(B (x, √

t )), µ (B (y, √

t ))

e

2 c t

for every t > 0 and x, y ∈ R

. (d) Lipschitz estimates :

| H

(x, y) − H

(x, y

) | . µ(B (x, √

t ))

,

for every t > 0 and x, y, y

∈ R

, with the following improvement, if | y − y

| ≤

| x − y | :

| H

(x, y) − H

(x, y

) | . max

µ( B(x, √

t )), µ(B (y, √

t ))

e

2

c t |y√−y^′| t

. Let us turn to the analog of Proposition 2.9 in the product case.

Proposition 3.3. The maximal operator P

f (x) = sup

Z

Rⁿ

dµ(y) P

(x, y) f (y) ,

is bounded from L

( R

, µ) into itself.