Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2010, Article ID 204509, 17 pages
doi:10.1155/2010/204509
Research Article
Convolution Operators and Bochner-Riesz Means
on Herz-Type Hardy Spaces in the Dunkl Setting
A. Gasmi and F. Soltani
Department of Mathematics, Faculty of Sciences of Tunis, University of EL-Manar, Tunis 2092, Tunisia
Correspondence should be addressed to F. Soltani, fethisoltani05@yahoo.fr
Received 12 February 2010; Accepted 20 October 2010
Academic Editor: Ricardo Estrada
Copyright q 2010 A. Gasmi and F. Soltani. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
p
We study the Dunkl convolution operators on Herz-type Hardy spaces Hα,2 and we establish a
version of multiplier theorem for the maximal Bochner-Riesz operators on the Herz-type Hardy
p
spaces Hα,∞ .
1. Introduction
The classical theory of Hardy spaces on n has received an important impetus from the
work of Fefferman and Stein, Lu and Yang 1, 2. Their work resulted in many applications
involving sharp estimates for convolution and multiplier operators.
By using the technique of Herz-type Hardy spaces for the Dunkl operator Λα , we are
attempting in this paper to study the Dunkl convolution operators, and we establish a version
of multiplier theorem for the maximal Bochner-Riesz operators on these spaces.
The Dunkl operator Λα , α > −1/2, associated with the reflection group 2 on :
Λα fx :
2α 1 fx − f−x
d
fx ,
dx
x
2
1.1
is the operator devised by Dunkl 3 in connection with a generalization of the classical theory
of spherical harmonics. The Dunkl analysis with respect to α ≥ −1/2 concerns the Dunkl
operator Λα , the Dunkl transform Fα , the Dunkl convolution ∗α , and a certain measure μα on
.
p
In this paper we define a Herz-type Hardy spaces Hα,q , 0 < p ≤ 1 < q ≤ ∞, in the
Dunkl setting. Next, we consider the Dunkl convolution operators Tk f : k∗α f, where k is
2
International Journal of Mathematics and Mathematical Sciences
a locally integrable function on . We use the atomic decomposition of the Herz-type Hardy
p
p
q
spaces Hα,q to study the Hα,2 − Hα,2 -bounded and the H1α,2 − L1 -bounded of the operators Tk .
Finally, we establish a version of multiplier theorem for the maximal Bochner-Riesz operators
η
σα , t > 0, η > α 1/2:
η η
σα f : supΦα,t∗α f ,
1.2
t>0
where
η
Φα,tx
∞
Γ η 1 2α2 −1n
tx2n ,
t
:
2n n!Γ n α η 2
2α1
2
n0
p
1.3
p
on the Herz-type Hardy spaces Hα,∞ . In this version we prove the Hα,∞ − Lp -bounded of the
η
operators σα , for α 1/2 < η < α 3/2.
The content of this work is the following. In Section 2, we recall some results about
p
harmonic analysis and we define a Herz-type Hardy spaces Hα,q , 0 < p ≤ 1 < q ≤ ∞ for the
p
q
Dunkl operator Λα . In Section 3, we study the Hα,2 −Hα,2 -bounded and the H1α,2 −L1 -bounded
p
of the convolution operators Tk . In Section 4, we prove the Hα,∞ −Lp -bounded of the maximal
η
Bochner-Riesz operators σα .
Throughout the paper we use the classic notation. Thus S and S are the
Schwartz space on and the space of tempered distributions on , respectively. Finally, C
will denote a positive constant not necessary the same in each occurrence.
2. The Dunkl Harmonic Analysis on
Ê
For α ≥ −1/2 and λ ∈ , the initial problem
Λα fx λfx,
f0 1,
2.1
has a unique analytic solution Eα λx called Dunkl kernel 4–6 given by
Eα λx α λx λx
α1 λx,
2α 1
x∈ ,
2.2
where
∞
α λx : Γα 1
n0
22n n!
is the modified spherical Bessel function of order α.
λx2n
,
Γn α 1
2.3
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3
We notice that, the Dunkl kernel Eα λx can be expanded in a power series 7 in the
form
Eα λx ∞
λxn
bn α
n0
2.4
,
where
b2n α 22n n!
Γn α 1,
Γα 1
Note 1. Let μα be the measure on
b2n1 α 2α 1b2n α 1.
2.5
given by
dμα x :
1
2α1 Γα
1
|x|2α1 dx.
2.6
We denote by Lp , μα , p ∈0, ∞, the space of measurable functions f on , such that
f p
Lα
:
Ê
fxp dμα x
f L∞
α
1/p
< ∞,
p ∈ 0, ∞,
2.7
: ess supfx < ∞.
Ê
x∈
The Dunkl kernel gives rise to an integral transform, called Dunkl transform on
, which was introduced by Dunkl in 8, where already many basic properties were
established. Dunkl’s results were completed and extended later on by de Jeu in 5.
The Dunkl transform of a function f ∈ L1 , μα , is given by
Fα f λ :
Ê
Eα −iλxfxdμα x,
λ∈ .
2.8
For T ∈ S , we define the Dunkl transform Fα T of T, by
Fα T, ϕ : T, Fα ϕ ,
ϕ ∈ S .
2.9
Note 2. For all x, y, z ∈ , we put
Wα x, y, z : 1 − σx,y,z σz,x,y σz,y,x Δα |x|, y, |z| ,
2.10
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where
⎧ 2
x y2 − z2
⎪
⎪
, if x, y ∈ \ {0}
⎨
2xy
σx,y,z :
⎪
⎪
⎩
0,
otherwise
⎧ 2 α−1/2
2
⎪
⎪
|x| y − z2 z2 − |x| − y
⎪
⎪
⎪
⎪d
⎨ α
xyz2α
Δα |x|, y, |z| :
⎪
⎪
⎪
⎪
⎪
⎪
⎩
0,
if |z| ∈ Ax,y ,
2.11
otherwise,
Ax,y |x| − y, |x| y .
21−α Γα 12
dα √
,
π Γα 1/2
We denote by νx,y the following signed measure:
⎧ ⎪
⎪
⎪Wα x, y, z dμα z
⎪
⎪
⎨
dνx,y z : dδx z
⎪
⎪
⎪
⎪
⎪
⎩
dδy z
The Dunkl translation operators τx , x ∈
continuous functions on , by
τx f y :
|x||y|
||x|−|y||
if x, y ∈
\ {0},
if y 0,
2.12
if x 0.
see 6 are defined for f ∈ C the space of
fzdνx,y z −||x|−|y||
−|x||y|
fzdνx,y z.
2.13
Let f and g be two functions in S . We define the Dunkl convolution product ∗α of
f and g by
f∗α gx :
Ê
τx f −y g y dμα y ,
x∈ .
2.14
For T ∈ S and f ∈ S , we define the Dunkl convolution product T∗α f by
T∗α fx : T y , τx f −y ,
x∈ .
2.15
We begin by recalling the definition of the Herz-type Hardy space in the Dunkl setting.
Firstly we introduce a class of fundamental functions that we will call atoms.
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5
is called a p, q atom, if a satisfies
Let 0 < p ≤ 1 < q ≤ ∞. A measurable function a on
the following conditions:
i there exists r > 0 such that suppa ⊂ −r, r;
ii a
Lqα ≤ r −2α11/p−1/q , where r is given in i;
iii Ê axxj dμα x 0, for all j 0, 1, . . . , 2s 1,
where s α 11/p − 1, the integer part of α 11/p − 1.
p
Let 0 < p ≤ 1 < q ≤ ∞. Our Herz-type Hardy space Hα,q is constituted by all those
f ∈ S that can be represented by
f
∞
2.16
λj aj ,
j0
where λj ∈ and aj is a p, q atom, for all j ∈
2.16 converges in S .
p
We define on Hα,q the norm · Hpα,q by
, such that
∞
j0
|λj |p < ∞ and the series in
⎛
⎞1/p
∞
f p : inf ⎝ |λj |p ⎠ ,
Hα,q
2.17
j0
where the infimum is taken over all those sequences {λj }j∈Æ ⊂ such that f is given by 2.16
for certain p, q atoms aj , j ∈ .
As the same in 7, we prove the following theorem.
p
Theorem 2.1. Let 0 < p ≤ 1 < q ≤ ∞ and f ∈ Hα,q . Then
Fα f y ≤ Cy2α11/p−1 f p
Hα,q
,
y∈ .
2.18
p
3. Dunkl Convolution Operators on Hα,2
p
In the following, we study on Hα,2 , 0 < p ≤ 1 the Dunkl convolution operators defined by
Tk f : k∗α f, where k is a locally integrable function on .
Theorem 3.1. Let 0 < p ≤ q ≤ 1. Assume that for every n ∈ we are given ξn > 0 and a function gn
such that
i gn x 0, |x| ≥ 2−n ,
ii gn L1α ≤ ξn 22α11/q−1/pn ,
iii x2s2 Fα gn L2α ≤ ξn 22α11/q−1/2n , s α 11/p − 1.
∞
q
Suppose also that ∞
n0 ξn < ∞ and define k n0 gn . Then Tk defines a bounded linear
p
q
mapping from Hα,2 into Hα,2 .
To prove this theorem the following lemma is needed.
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International Journal of Mathematics and Mathematical Sciences
Lemma 3.2. For y, z ∈
and j ∈ , there are a constants Aj,α , such that
j
τy xj z Aj,α
i0
j!
yi zj−i ,
bi αbj−i α
3.1
where bj α are the constants given by 2.4.
Proof. Let y, z ∈ . By dominated convergence theorem, we can write
τy xj z lim
xj Eα txdνy,z x,
t→0
Ê
3.2
and by derivation under the integral sign, we get
j
τy xj z lim Λα, t
Eα txdνy,z x.
Ê
t→0
3.3
Then, from Theorem 2.4, 6 we obtain
j τy xj z lim Λα, t Eα ty Eα tz .
t→0
3.4
Let Ft Eα tyEα tz for |t| < ξ, where ξ > 0. According to 2.11, 9,
j
Λα, t Ft F j t
j−1 j i
0
0
j
0
0
j
1
1
1
1
t · t1 . . . ti 2 Qj−1 t1 , . . . , tj−1 F
t · t1 . . . tj ,
2 Pj−1 t1 , . . . , ti F
i1
3.5
where Pj−1 t01 , . . . , t0i , i 1, 2, . . . , j − 1, and Qj−1 t11 , . . . , t1j−1 are polynomials of degree at most
j − 1 with respect to each variable. Moreover,
j
F t j
j
i0
i
!
i j−i
yi zj−i Eα ty Eα tz.
3.6
Then, from 2.4 we deduce
j j
i! j − i ! i j−i j!
j
y
yi zj−i .
z
lim F t i
t→0
bi αbj−i α
b αbj−i α
i0
i0 i
j
3.7
Therefore,
j
j
τy xj z lim Λα, t Ft Aj,α
t→0
j!
yi zj−i ,
b
αb
α
i
j−i
i0
3.8
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7
where
Aj,α 1 j−1 "
#
2i Pj−1 t01 , . . . , t0i 2j Qj−1 t11 , . . . , t1j−1 .
3.9
i1
This finishes the proof of the lemma.
Proof of Theorem 3.1. Firstly, notice that gn L1α ≤ ξn , n ∈ . Hence, the series defining k
converges in L1 , μα and k ∈ L1 , μα .
Let a be a p, 2 atom. Suppose that ax 0, |x| > r and that a
L2α ≤ r −2α11/p−1/2 ,
where r > 0. We can write
Tk a ∞
gn ∗α a.
3.10
n0
Step 1. Let n ∈ . From 2.13, for |y| − |z| ≥ 2−n , we have
τy gn z |y||z|
||y|−|z||
gn xdνy,z x −||y|−|z||
−||y||z||
gn xdνy,z x 0.
3.11
Hence, for |y| > r 2−n , we deduce
g n ∗α a y r
−r
a−zτy gn zdμα z 0.
3.12
Step 2. Firstly, let us consider that r ≥ 2−n . From Proposition 3i, 10 and condition ii of
the theorem, we have
gn ∗α a 2 ≤ 4gn 1 a
2 ≤ 4ξn r 2−n −2α11/q−1/2 .
Lα
Lα
Lα
3.13
Assume now that r < 2−n . Since Ê axxj dμα x 0, j 0, 1, . . . , 2s 1, with s α 11/p − 1, we have
⎡
⎤
2s1
yj
j
gn ∗α ax a −y ⎣τy gn x −
Λα gn x⎦dμα y ,
b α
Ê
j0 j
where bj α are the constants given by 2.4.
x∈ ,
3.14
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International Journal of Mathematics and Mathematical Sciences
Using the properties of the Dunkl transform established by de Jeu 5 see also 7, 10,
we deduce
gn ∗α a
L2α
2s1
yj
j
≤ Ê a −y τy gn −
Λα gn dμα y
2
j0 bj α
Lα
j 2s1
yj
Ê a −y Fα τy gn −
Fα Λα gn dμα y
2
j0 bj α
Lα
(
j )
2s1
ixy
Êa −y Eα ixy −
Fα gn dμα y .
b
α
j
j0
2
3.15
Lα
According to page 302, 7
ixy j 2s2
2s1
1
xy
Eα ixy −
≤
.
2α
2
Γα
1
b
α
j
j0
3.16
Using condition iii of the theorem and Hölder’s inequality, we get
gn ∗α a
L2α
r 1
2s2 a y y2s2 dμα y
≤ 2α
Fα g n 2
x
L
2 Γα 1
α
−r
≤
r
1/2
1
2s2 4s4
2
g
y
F
2
y
dμ
a
x
α
n
α
Lα
L2α
22α Γα 1
0
3.17
≤ γn 22α11/q−1/2n r 2s−2α11/p−12 ,
where
γn :
ξn
cξn .
*
22α Γα 1 2s α 32α1 Γα 1
3.18
Using the fact that s − α 11/p − 1 1 > 0 and r < 2−n , we obtain
gn ∗α a
L2α
−2α11/q−1/2
≤ γn 22α11/q−1/2n ≤ γn r 2−n
.
3.19
Step 3. We now prove for all j 0, . . . , 2s 1; s α 11/p − 1, that
Ê
xj gn ∗α a xdμα x 0.
3.20
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9
Fubini’s theorem and 6 see also page 20, 10 lead to
Ê
xj gn ∗α a xdμα x
Ê
xj
Ê
Ê
Ê
a y
a y
a y
Ê
Ê
Ê
gn zdνx,−y z dμα y dμα x
3.21
j
gn z
x dνy,z x dμα zdμα y
Ê
gn zτy xj zdμα zdμα y .
Hence, by Lemma 3.2 and by taking into account that Ê axxj dμα x 0, j 0, 1, . . . , 2s 1,
with s α 11/p − 1, we get
Ê
j
xj gn ∗α a xdμα x Aj,α
i0
j!
bi αbj−i α
Ê
yi a y dμα y
Ê
gn zzj−i dμα z 0.
3.22
According to the previous three steps, we conclude that 1/γn gn ∗α a is a q, 2 atom.
q
Then, Tk a ∈ Hα,2 , and
Tk a
q
Hα,2
∞
≤c
ξn q
!1/q
3.23
.
n0
p
Let now f be in Hα,2 . Assume that f ∞
j0 λj aj , where λj ∈ and aj is a p, 2 atom,
∞
p
for every j ∈ , and such that j0 |λj | < ∞. The series defining f converges in L1 , μα .
*
In fact, it is sufficient to note that a
L1α ≤ 1/ 2α Γα 1r 2α11−1/p . Hence f ∈ L1 , μα .
Moreover, k ∈ L1 , μα . Then by Proposition 3i, 10, the operator Tk is bounded from
L1 , μα into itself, and from this, we deduce that Tk f ∞
j0 λj Tk aj . Using the fact that
∞
∞
p 1/p
, we obtain
j0 |λj | ≤ j0 |λj | Tk f Hα,2 ≤
q
c
∞
n0
!1/q
q
ξn f Hα,2 .
p
3.24
This completes the proof of the Theorem 3.1.
H1α,2 .
We now study the Dunkl convolution operators Tk on the Herz-type Hardy spaces
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International Journal of Mathematics and Mathematical Sciences
Theorem 3.3. Let k be a locally integrable function on . Assume that the following three conditions
are satisfied:
i Tk defines a bounded linear operator from L2 , μα into itself.
ii Tk defines a bounded linear operator from L1 , μα into S .
iii There exist A and c > 1 such that
|z|>cR |τx
kz − kz|dμα z ≤ A,
|x| ∈ 0, R , R > 0.
3.25
Then Tk defines a bounded linear mapping from H1α,2 into L1 , μα .
Proof. Let a be a 1, 2 atom. We choose r > 0 such that suppa ⊂ −r, r and a
L2α ≤ r −α1 .
We can write
(
|Tk ax|dμα x Ê
)
|x|<cr
|x|≥cr
|Tk ax|dμα x : I1 I2 .
3.26
Here c > 1 is the one given in iii.
From condition i of the theorem and Hölder’s inequality, we deduce that
I1 ≤
1/2
1/2 cr
2
|Tk ax| dμα x
dμα x
≤ C
a
L2α r α1 ≤ C.
2
Ê
3.27
0
Also, by taking into account that Ê aydμα y 0, the condition iii of the theorem allows
us to write
I2 |x|≥cr
≤
|x|≥cr
r
−r
≤C
τx k y a −y dμα y dμα x
Ê
τx k y − kx a −y dμα y dμα x
Ê
a −y r
−r
|x|≥cr
τy kx − kxdμα xdμα y
a y dμα y ≤ C
a
L2α
3.28
r
1/2
≤ C.
2 dμα y
0
Hence, it concludes that
Tk a
L1α ≤ C.
3.29
Note that the positive constant C is not depending on the 1, 2 atom a.
Let now f be in H1α,2 . Then f ∈ S and f ∞
j0 λj aj , where λj ∈ and aj is a 1, 2
∞
atom, for every j ∈ and j0 |λj | < ∞.
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11
The series defining f converges in L1 , μα . In fact, it is sufficient to note that a
L1α ≤
*
1/ 2α Γα 1, for every 1, 2 atom a. Hence f ∈ L1 , μα . Then the condition ii of the
theorem implies that
Tk f ∞
3.30
λj Tk aj .
j0
By 3.29 the series in 3.30 converges in L1 , μα and Tk f
L1α ≤ C
Tk f
L1α ≤ C
f
H1α,2 .
∞
j0
|λj |. Hence
p
4. Maximal Bochner-Riesz Operators on Hα,∞
η
The Bochner-Riesz mean σα,t , for t > 0 and η > α 1/2 associated to the Dunkl transform Fα
is defined by
η σα,t f x
t
:
y2
1− 2
t
−t
!η
Eα ixy Fα f y dμα y ,
4.1
f ∈ S .
η
η
The maximal operator σα , η > α 1/2 associated to the Bochner-Riesz means σα,t, t > 0, is
defined by
η η
σα f : supσα,t f .
t>0
4.2
Lemma 4.1. For t > 0 and η > α 1/2, one has
η
η
i σα,t f Φα,t ∗α f, where
η
Φα,tx
Γ η1
t2α2 αη1 itx.
: α1 2 Γ αη2
4.3
Here α is the modified spherical Bessel function given by 2.3.
η
ii The operator σα is bounded from Lp , μα , 1 ≤ p ≤ ∞ into itself.
Proof. Let t > 0 and η > α 1/2.
i By taking into account that the functions |z|α1/2 α iz and α iz are bounded on
it is not hard to see that
η Φα,t
L1α
η Φα,1 L1α
Γ η1
α1 αη1 ixdμα x < ∞.
2 Γ αη2 Ê
4.4
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On the other hand, from 11, we have
1 1−y
2
Γα 1Γ η 1
2α1
α itxy y dy αη1 itx.
2Γ α η 2
η
0
4.5
Thus,
η
Φα,t x
1 η t2α2
1 − y2 α itxy y2α1 dy
α
2 Γα 1 0
!η
t
y2
1− 2
Eα ixy dμα y .
t
−t
4.6
Applying Inversion Theorem5, we obtain
η
Fα Φα,t
y y2
1− 2
t
!η
χ−t,t y ,
4.7
where χ−t,t is the characteristic function of the set −t, t. Thus,
η σα,t f x
Ê
η Eα ixy Fα Φα,t y Fα f y dμα y ,
4.8
and from Proposition 3 ii, 10, we deduce that
η η
σα,t f x Φα,t ∗α fx.
4.9
ii Using i and Proposition 3 i, 10, we obtain
η
σα f p
Lα
η ≤ 4Φα,1 1 f Lpα .
4.10
Lα
This clearly yields the result.
p
η
Theorem 4.2. Let 0 < p ≤ 1 < q ≤ ∞ and f ∈ Hα,q . For t > 0 and η > α 1/2, the operator σα,t
p
extended to a bounded operator from Hα,q into S .
p
η
Proof. According to Theorem 2.1, if f ∈ Hα,q , then σα,tf is in S and it is defined by
+ η , t
σα,t f , ϕ −t
y2
1− 2
t
!η
Fα f y Fα ϕ y dμα y ,
ϕ ∈ S .
4.11
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13
Moreover,
+ ,
2α11/p−1 η
y Fα ϕ y dμα y ,
σα,t f , ϕ ≤ C f Hpα,q
Ê
η
ϕ ∈ S .
4.12
p
Hence σα,t is a bounded operator from Hα,q into S .
η
p
We now study the behavior of the maximal Bochner-Riesz operator σα on Hα,∞ .
η
Theorem 4.3. Let 2α 1/α η 3/2 < p ≤ 1. Then the maximal Bochner-Riesz operator σα ,
p
α 1/2 < η < α 3/2 is bounded from Hα,∞ into Lp , μα .
To prove this theorem we need the following lemma.
Lemma 4.4. i For x, y ∈
and η > α 1/2,
−αη3/2
η .
τx Φα,t y ≤ Ctα−η1/2 |x| − y
4.13
ii For 0 < |y| < |x| and η > α 1/2,
−αη3/2
η η
.
τx Φα,t y − Φα,tx ≤ Cytα−η3/2 |x| − y
4.14
Proof. i Since the function |z|α1/2 α iz is bounded on , it follows that
η
Φα,t z≤ Ctα−η1/2 |z|−αη3/2 ;
t > 0, z ∈ .
4.15
η
According to 6 and the fact that Φα,t is even, we obtain
η τx Φα,t y
π
0
η
Φα,t
x, y
dρx,y θ,
θ
4.16
where
Γα 1
1 − sgn xy cos θ sin2α θdθ ,
dρx,y θ : √
π Γα 1/2
x, y
θ
: x2 y2 − 2xy cos θ.
4.17
By 4.15 and using the fact that x, yθ ≥ ||x| − |y||, we deduce that
π
−αη3/2
η dρx,y θ
x, y θ
τx Φα,t y ≤ Ctα−η1/2
0
−αη3/2
≤ Ctα−η1/2 |x| − y
.
4.18
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International Journal of Mathematics and Mathematical Sciences
ii From 11, we have
Γ η1
d η
Φ z − α2 t2α4 zαη2 itz.
dz α,t
2 Γ αη3
4.19
Then, similarly to the proof in i, we have
d η
Φ z ≤ Ctα−η3/2 |z|−αη3/2 ;
dz α,t η η
τx Φα,t y − Φα,tx π
0
η
Φα,t
t > 0, z ∈ ,
η
x, y θ − Φα,tx, 0θ dρx,y θ,
4.20
4.21
which can be written as
η τx Φα,t y
−
η
Φα,t x
π 1
0
≤ y 0
d η Φα,t x, sy θ dsdρx,y θ
ds
1 π
0
0
4.22
d η Φα,t x, sy θ dρx,y θds.
ds
Then from 4.20, it follows
1 π
−αη3/2
η η
dρx,y θds
x, sy θ
τx Φα,t y − Φα,tx ≤ Cytα−η3/2
0
≤ Cytα−η3/2
1
0
|x| − sy−αη3/2 ds.
4.23
0
Hence, if 0 < |y| < |x|, we obtain
−αη3/2
η η
,
τx Φα,t y − Φα,tx ≤ Cytα−η3/2 |x| − y
4.24
which completes the proof of the lemma.
Proof of Theorem 4.3. Let us first show that C > 0, exists such that
η σα a p ≤ C,
Lα
for every p, ∞ atom a.
4.25
International Journal of Mathematics and Mathematical Sciences
15
Let a be an p, ∞ atom. Suppose that ax 0, |x| > r and a
L∞α ≤ r −2α1/p . We
choose ∈ such that 2−1 < r ≤ 2 , we write
|x|≥4r
p
η
σα ax dμα x ≤ I1 I2 ,
4.26
where
I1 :
∞ i12 ≤|x|≤i22 t≥δi
i1
I2 :
p
η
supσα,t ax dμα x,
∞ i12 ≤|x|≤i22
i1
p
η
supσα,t ax dμα x,
4.27
4.28
t<δi
being δi 2− /ib , where b will be specified later.
According to Lemma 4.4 i, for |x| ∈ i 12 , i 22 , i 1, 2, . . ., we get
r η
a −y τx Φη y dμα y
σα,t ax ≤
α,t
−r
≤ Ctα−η1/2 r −2α1/p
2
|x| − y−αη3/2 dμα y
4.29
0
≤C
α−η1/2
2 t
iαη3/2 r 2α1/p
.
Then, using the fact that 2−1 < r ≤ 2 , we obtain
I1 ≤ C
∞
⎛
⎝
α−η1/2 {α−η1/2−2α1/p}
2
δi
iαη3/2
i1
⎞p
⎠ i2α1 22α1 ,
4.30
and hence, it concludes that
I1 ≤ C
∞
i2α1−{αη3/2α−η1/2b}p .
4.31
i1
Note that the last series is convergent provide that b < pαη3/2−2α1/pη−α−1/2.
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International Journal of Mathematics and Mathematical Sciences
On the other hand, since Ê aydμα y 0, from Lemma 4.4 ii we have
r η
a −y τx Φη y − Φη xdμα y
σα,t ax ≤
α,t
α,t
−r
2
≤ Ctα−η3/2 r −2α1/p
|x| − y−αη3/2 ydμα y
4.32
0
≤C
α−η3/2
2 t
iαη3/2 r 2α1/p
.
Then, for |x| ≥ 4r, we obtain
I2 ≤ C
∞
⎛
⎝
α−η3/2 {α−η3/2−2α1/p}
2
δi
iαη3/2
i1
⎞p
⎠ i2α1 22α1 ,
4.33
and in fact, we deduce that
I2 ≤ C
∞
i2α1−{αη3/2α−η3/2b}p .
4.34
i1
pα η 3/2 − 2α 1
.
pη − α − 3/2
Note that we can find b such that the series in 4.31 and 4.34 converge if and only if
p > 2α 1/α η 3/2. By combining 4.31 and 4.34 we show that
The last series converges provided that b >
|x|≥4r
p
η
σα ax dμα x ≤ C,
4.35
for a certain C > 0 that is not depending on a.
From Lemma 4.1 ii and 4.35 we deduce that
(
η p
σα a p ≤
Lα
|x|<4r
(
|x|≥4r
)
p
η
σα ax dμα x
)
p
≤ C a
L∞
k
|x|<4r
4.36
dμα x 1 ≤ C,
∞
p
that is, 4.25 holds. Let now f be in Hα,∞ . Assume that f j0 λj aj , wherethe series
p
converges in S , and for every j ∈ , aj is a p, ∞ atom and λj ∈ , such that ∞
j0 |λj | <
∞. From Theorem 4.2, for 0 < p ≤ 1, we can write
∞
η η λj σα,t aj x;
σα,t f x j0
t > 0, x ∈ .
4.37
International Journal of Mathematics and Mathematical Sciences
η
p
Hence, from 4.25 it follows σα f
Lp ≤ C
α
C
f
Hpα,∞ .
∞
j0
17
η
|λj |p . Thus we conclude that σα f
Lpα ≤
Acknowledgment
A. Gasmi and F. Soltani partially supported by DGRST project 04/UR/15-02 and CMCU
program 10G 1503.
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