ON THE Lp BOUNDEDNESS OF ROUGH
PARAMETRIC MARCINKIEWICZ FUNCTIONS
AHMAD AL-SALMAN
HUSSAIN AL-QASSEM
Department of Mathematics and Statistics
Sultan Qaboos University
P.O. Box 36, Al-Khod 123 Muscat
Sultanate of Oman.
EMail: alsalman@squ.edu.om
Department of Mathematics and Physics
Qatar University,
Qatar
Boundedness of Rough Parametric
Marcinkiewicz Functions
Ahmad Al-Salman and
Hussain Al-Qassem
vol. 8, iss. 4, art. 108, 2007
Title Page
EMail: husseink@qu.edu.qa
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25 November, 2007
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Communicated by:
L. Debnath
Page 1 of 17
2000 AMS Sub. Class.:
Primary 42B20; Secondary 42B15, 42B25.
Key words:
Parametric Marcinkiewicz operators, rough kernels, Fourier transforms, Parametric maximal functions.
Abstract:
In this paper, we study the Lp boundedness of a class of parametric
Marcinkiewicz integral operators with rough kernels in L(log+ L)(Sn−1 ). Our
result in this paper solves an open problem left by the authors of ([6]).
Acknowledgements:
This paper was written during the authors’ time in Yarmouk University.
Received:
07 November, 2006
Accepted:
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Contents
1
Introduction
3
2
Preparation
6
3
Rough Parametric Maximal Functions
11
4
Proofs of The Main Results
14
Boundedness of Rough Parametric
Marcinkiewicz Functions
Ahmad Al-Salman and
Hussain Al-Qassem
vol. 8, iss. 4, art. 108, 2007
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1.
Introduction
Let n ≥ 2 and Sn−1 be the unit sphere in Rn equipped with the normalized Lebesgue
measure dσ. Suppose that Ω is a homogeneous function of degree zero on Rn that
satisfies Ω ∈ L1 (Sn−1 ) and
Z
(1.1)
Ω(x)dσ(x) = 0.
Sn−1
Boundedness of Rough Parametric
Marcinkiewicz Functions
Ahmad Al-Salman and
Hussain Al-Qassem
ρ
In 1960, Hörmander ([9]) defined the parametric Marcinkiewicz function µΩ of
higher dimension by
(1.2)
ρ
Z
∞
µΩ f (x) =
−ρt
−n+ρ
f (x − y) |y|
2
−∞
2
Z
! 12
Ω(y)dy dt
vol. 8, iss. 4, art. 108, 2007
Title Page
,
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|y|≤2t
ρ
where ρ > 0. It is clear that if ρ = 1, then µΩ is the classical Marcinkiewicz
integral operator introduced by Stein ([11]) which will be denoted by µΩ . When
Ω ∈ Lip α (Sn−1 ), (0 < α ≤ 1), Stein proved that µΩ is bounded on Lp for all
1 < p ≤ 2. Subsequently, Benedek-Calderón-Panzone proved the Lp boundedness
of µΩ for all 1 < p < ∞ under the condition Ω ∈ C 1 (Sn−1 ) ([4]). Recently,
under various conditions on Ω, the Lp boundedness of µΩ and a more general class
of operators of Marcinkiewicz type has been investigated (see [1] – [2], [5], among
others).
ρ
In ([9]), Hörmander proved that µΩ is bounded on Lp for all 1 < p < ∞, provided
that Ω ∈ Lipα (Sn−1 ), (0 < α ≤ 1) and ρ > 0.
ρ
A long standing open problem concerning the operator µΩ is whether there are
ρ
some Lp results on µΩ similar to those on µΩ when Ω satisfies only some size con-
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ditions. In a recent paper, Ding, Lu, and Yabuta ([6]) studied the operator
(1.3)
Z
ρ
∞
2−ρt
µΩ,h f (x) =
2
Z
! 12
f (x − y) |y|−n+ρ h(|y|)Ω(y)dy dt
,
|y|≤2t
−∞
where ρ is a complex number, Re(ρ) = α > 0, and h is a radial function on Rn
satisfying h(|x|) ∈ l∞ (Lq )(R+ ), 1 ≤ q ≤ ∞, where l∞ (Lq )(R+ ) is defined as
follows: For 1 ≤ q < ∞,


! 1q
Z 2j


dr
l∞ (Lq )(R+ ) = h : khkl∞ (Lq )(R+ ) = sup
<C
|h(r)|q


r
j∈Z
2j−1
∞
∞
∞
+
Boundedness of Rough Parametric
Marcinkiewicz Functions
Ahmad Al-Salman and
Hussain Al-Qassem
vol. 8, iss. 4, art. 108, 2007
Title Page
+
and for q = ∞, l (L )(R ) = L (R ).
Ding, Lu, and Yabuta ([6]) proved the following:
Theorem 1.1. Suppose that Ω ∈ L(log+ L)(Sn−1 ) is a homogeneous function of
∞
q
+
degree zero on Rn satisfying (1.1) and h(|x|)
√ ∈ l (L )(R ) for some 1 < q ≤ ∞.
ρ
If Re(ρ) = α > 0, then µΩ,h f 2 ≤ C/ α kf k2 , where C is independent of ρ and
f.
ρ
The Lp boundedness of µΩ,h for p 6= 2 was left open by the authors of ([6]). The
ρ
main purpose of this paper is to establish the Lp boundedness of µΩ,h for p 6= 2. Our
main result of this paper is the following:
Theorem 1.2. Suppose that Ω ∈ L(log+ L)(Sn−1 ) is a homogeneous function of
degree zero on Rn satisfying (1.1). If h(|x|) ∈ l∞ (Lq )(R+ ), 1 < q ≤ ∞, and
ρ
Re(ρ) = α > 0, then µΩ,h f p ≤ C/α kf kp for all 1 < p < ∞, where C is
independent of ρ and f .
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Also, in this paper, we establish the Lp boundedness of the related parametric
maximal function. In fact, we have the following result:
Theorem 1.3. Suppose that Ω ∈ L(log+ L)(Sn−1 ) is a homogeneous function of
degree zero on Rn . If h(|x|) ∈ l∞ (Lq )(R+ ), 1 < q ≤ ∞, and α > 0, then
kMα f kp ≤
C
kf kp
α
for all 1 < p < ∞ with a constant C independent of α, where Mα is the operator
defined by
Z
−n+ρ
−αt
(1.4)
Mα f (x) = sup 2
Ω(y) |y|
h(|y|)f (x − y)dy .
t∈R
|y|≤2t
The method employed in this paper is based in part on ideas from [1], [2] and [3],
among others. A variation of this method can be applied to deal with more general
integral operators of Marcinkiewicz type. An extensive discussion of more general
operators will appear in forthcoming papers.
Throughout the rest of the paper the letter C will stand for a constant but not
necessarily the same one in each occurrence.
Boundedness of Rough Parametric
Marcinkiewicz Functions
Ahmad Al-Salman and
Hussain Al-Qassem
vol. 8, iss. 4, art. 108, 2007
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2.
Preparation
Suppose a ≥ 1. For a suitable family of measures τ = {τt : t ∈ R} on Rn and
a suitable family of C ∞ functions Φa = {ϕt : t ∈ R} on Rn , define the family of
operators {Λτ,Φa ,s : t, s ∈ R} by
Z ∞
21
2
(2.1)
Λτ,Φ,s,a (f )(x) =
|τat ∗ ϕt+s ∗ f (x)| dt .
−∞
Also, define the operator τ ∗ by
Boundedness of Rough Parametric
Marcinkiewicz Functions
Ahmad Al-Salman and
Hussain Al-Qassem
vol. 8, iss. 4, art. 108, 2007
τ ∗ (f )(x) = sup(|τt | ∗ |f |)(x).
(2.2)
t∈R
The proof of our result will be based on the following lemma:
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Lemma 2.1. Suppose that for some B > 0, ε > 0, and β > 0, we have
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(i) kτt k ≤ β for t ∈ R;
ε
(ii) |τ̂t (ξ)| ≤ β(2t |ξ|)± a for ξ ∈ Rn and t ∈ R;
(iv) The functions ϕt , t ∈ R satisfy the properties that ϕ̂t is supported in {ξ ∈
γ
Rn : 2−(t+1)a ≤ |ξ| ≤ 2−(t−1)a } and ddξϕ̂γ t (ξ) ≤ Cγ |ξ|−|γ| for any multi-index
γ ∈ (N∪(0))n with constants Cγ depend only on γ and the dimension of the
underlying space Rn .
(2.3)
2q
q+1
<p<
2q
,
q−1
there exists a constant Cp independent of a, β, B, s, and
1
kΛτ,Φ,s,a (f )kp ≤ Cp (βB) 2 (βB −1 )
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(iii) kτ ∗ (f )kq ≤ B kf kq for some q > 1;
Then for
ε such that
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θ(p)
2
2(ε+1)θ(p) 2−εθ(p)|s| kf kp
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p
n
for all f ∈ L (R ), where θ(p) =
2q
p ∈ q+1
,2 .
2q−pq+p
p
if p ∈ 2,
2q
q−1
and θ(p) =
pq+p−2q
p
if
Proof. We start with the case p = 2. By Plancherel’s formula and the conditions
(i)-(ii), we obtain
(2.4)
kΛτ,Φ,s,a (f )k2 ≤ β2ε+1 2−ε|s| kf k2
for all f ∈ L2 (Rn ).
Next, set p0 = 2q 0 and choose a non-negative function v ∈ Lq+ (Rn ) with kvkq = 1
such that
Z Z ∞
2
kΛτ,Φ,s,a (f )kp0 =
|τat ∗ ϕt+s ∗ f (x)|2 v(x)dtdx.
Rn
−∞
Boundedness of Rough Parametric
Marcinkiewicz Functions
Ahmad Al-Salman and
Hussain Al-Qassem
vol. 8, iss. 4, art. 108, 2007
Title Page
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Now it is easy to see that
(2.5)
p
1
β kga,s (f )kp0 kτ ∗ (v)kq2
kΛτ,Φ,s,a (f )kp0 ≤
where ga,s is the operator
Z
(2.6)
∞
ga,s (f )(x) =
12
|ϕt+s ∗ f (x)| dt .
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2
−∞
By the condition (iv) and a well-known argument (see [12, p. 26-28]), it is easy to
see that
(2.7)
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kga,s (f )kp0 ≤ Cp0 kf kp0
for all f ∈ Lp0 (Rn ) with constant Cp0 independent of a and s. Thus, by (2.5) and
(2.7), we have
p
(2.8)
kΛτ,Φ,s,a (f )kp0 ≤ Cp0 βB kf kp0 .
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By duality, we get
kΛτ,Φ,s,a (f )k(p0 )0 ≤ C(p0 )0
(2.9)
p
βB kf k(p0 )0 .
Therefore, by interpolation between (2.4), (2.8), and (2.9), we obtain (2.3). This
concludes the proof of the lemma.
Now we establish the following oscillatory estimates:
Lemma 2.2. Suppose that Ω ∈ L∞ (Sn−1 ) is a homogeneous function of degree zero
on Rn satisfying (1.1) and h(|x|) ∈ l∞ (Lq )(R+ ), 1 < q ≤ 2. Then for a complex
number ρ with Re(ρ) = α > 0, we have
(2.10)
−αt
Z
e−iξ·y Ω(y) |y|−n+ρ h(|y|)dy
2
Boundedness of Rough Parametric
Marcinkiewicz Functions
Ahmad Al-Salman and
Hussain Al-Qassem
vol. 8, iss. 4, art. 108, 2007
Title Page
|y|≤2t
Contents
0
0
C
t
−ε
≤ 2 khkl∞ (Lq )(R+ ) kΩk1−2/q
kΩk2/q
1
∞ (2 |ξ|) ,
α
and
(2.11)
−αt
Z
2
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−iξ·y
e
−n+ρ
Ω(y) |y|
h(|y|)dy
|y|≤2t
≤2
C
ε
khkl∞ (Lq )(R+ ) kΩk1 (2t |ξ|)
α
for all 0 < ε < min{1/2, α}. The constant C is independent of Ω, α, and t.
R
0
Proof. For ξ ∈ Rn and r ∈ R+ , let G(ξ, r) = Sn−1 e−irξ·y Ω(y 0 )dσ(y 0 ). Then it is
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easy to see that
(2.12)
−αt
Z
e−iξ·y Ω(y) |y|−n+ρ h(|y|)dy
2
|y|≤2t
≤
∞
X
−αj
Z
2t−j
|h(r)| |G(ξ, r)| r−1 dr.
2
j=0
2t−j−1
Using the assumption that 1 < q ≤ 2, it is straightforward to show that the right
hand side of (2.12) is dominated by
! 10
Z 2t−j
∞
q
X
0
1−2/q
2 −1
−αj
(2.13)
2 khkl∞ (Lq )(R+ ) kΩk1
2
|G(ξ, r)| r dr
.
j=0
2t−j−1
Now, for ξ ∈ Rn , y 0 , z 0 ∈ Sn−1 , j ≥ 0, and t ∈ R, set
Z 2t−j
0
0
0 0
Ij,t (ξ, y , z ) =
e−irξ·(y −z ) r−1 dr.
2t−j−1
Boundedness of Rough Parametric
Marcinkiewicz Functions
Ahmad Al-Salman and
Hussain Al-Qassem
vol. 8, iss. 4, art. 108, 2007
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Then, we have
Z
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! 10
2t−j
q
|G(ξ, r)|2 r−1 dr
(2.14)
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2t−j−1
2/q 0
≤ kΩk∞
Z
10
q
0 0
0
0
|Ij,t (ξ, y , z )| dσ(y )dσ(z ) .
Sn−1 ×Sn−1
By integration by parts, we have
(2.15)
|Ij,t (ξ, y 0 , z 0 )| ≤ (2t−j−1 |ξ| |ξ 0 · (y 0 − z 0 )|)−1 .
Close
On the other hand, we have
|Ij,t (ξ, y 0 , z 0 )| ≤ ln 2.
(2.16)
Thus, by combining (2.15) and (2.16), we get
|Ij,t (ξ, y 0 , z 0 )| ≤ (2t−j−1 |ξ| |ξ 0 · (y 0 − z 0 )|)−ε
(2.17)
for 0 < ε < min{1/2, α}. Therefore, by (2.14) and (2.17), we obtain that
! 10
Z t−j
2
q
2
−1
|G(ξ, r)| r dr
(2.18)
2t−j−1
0
t−j−1
≤ kΩk2/q
|ξ|)−ε ,
∞ C(2
where the constant C is independent of Ω, j, and t. Moreover, since ε ≤ 1/2, it can
be shown that C is also independent of α. Hence by (2.12), (2.13), and (2.18), we
get (2.10).
Now we prove (2.11). Using the cancellation property (1.1), it is clear that
Z
−αt
(2.19)
2
e−iξ·y Ω(y) |y|−n+ρ h(|y|)dy
|y|≤2t
Boundedness of Rough Parametric
Marcinkiewicz Functions
Ahmad Al-Salman and
Hussain Al-Qassem
vol. 8, iss. 4, art. 108, 2007
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≤
2(ln 2)
α
1
q0
khkl∞ (Lq )(R+ ) kΩk1 2t |ξ| .
On the other hand, we have
1
Z
2(ln 2) q0
−n+ρ
−αt
−iξ·y
khkl∞ (Lq )(R+ ) kΩk1 .
(2.20) 2
e
Ω(y) |y|
h(|y|)dy ≤
α
|y|≤2t
Thus, by interpolation between (2.19) and (2.20), we get (2.11). This completes the
proof of Lemma 2.2.
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3.
Rough Parametric Maximal Functions
In this section we shall establish the boundedness of certain maximal functions
which will be needed to prove our main result.
Theorem 3.1. Suppose that Ω ∈ L∞ (Sn−1 ) is a homogeneous function of degree
zero on Rn with kΩk1 ≤ 1 and kΩk∞ ≤ 2a for some a > 1. Suppose also that
h(|x|) ∈ l∞ (Lq )(R+ ), 1 < q ≤ ∞ and let Mα be the operator defined as in (1.4).
Then
(3.1)
kMα f kp ≤
aC
kf kp
α
Boundedness of Rough Parametric
Marcinkiewicz Functions
Ahmad Al-Salman and
Hussain Al-Qassem
vol. 8, iss. 4, art. 108, 2007
for all 1 < p < ∞ with constant C independent of a, f , and α.
Title Page
Proof. Since l∞ (Lq1 )(R+ ) ⊂ l∞ (Lq2 )(R+ ) whenever q2 ≤ q1 , it suffices to assume
that 1 < q ≤ 2. By a similar argument as in ([2]), choose a collection of C ∞
functions Φa = {ϕt : t ∈ R} on Rn that satisfies the following properties: ϕ̂t is
γ
supported in {ξ ∈ Rn : 2−(t+1)a ≤ |ξ| ≤ 2−(t−1)a }, ddξϕ̂γ t (ξ) ≤ Cγ |ξ|−|γ| for any
Contents
multi-index γ ∈ (N∪{0})n with constants Cγ depending only on the underlying
dimension and γ, and
X
(3.2)
ϕ̂t+j (ξ) = 1.
j∈Z
For t ∈ R, let {σt : t ∈ R} be the family of measures on Rn defined via the Fourier
transform by
Z
−αt
(3.3)
σ̂t (ξ) = 2
e−iξ·y |Ω(y)| |y|−n+ρ |h(|y|)| dy
|y|≤2t
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Then it is easy to see that
(3.4)
Mα f (x) = sup{|σt | ∗ |f (x)|}.
t∈R
Now choose φ ∈ S(Rn ) such that φ̂(η) = 1 for |η| ≤ 12 , and φ̂(η) = 0 for |η| ≥ 1.
Let {τt : t ∈ R} be the family of measures on Rn defined via the Fourier transform
by
(3.5)
τ̂t (ξ) = σ̂t (ξ) − φ̂(2t ξ)σ̂t (0).
Then by Lemma 2.2, the choice of φ, the definitions of σt , τt , and the assumptions
on Ω, we have
C2la t
(2 |ξ|)−ε
α
for some l, ε > 0 . Moreover, it is easy to see that
(3.6)
|τ̂t (ξ)| ≤
C
α
Therefore by interpolation between (3.6) and (3.7), we get
kτt k ≤
(3.7)
ε
C t
(2 |ξ|)− a .
α
Now by (3.2), and the definitions of σt , and τt , it is easy to see that
√ X
(3.9)
Λτ,Φ,j,a (f )(x) + Cα−1 M H(f )(x),
Mα f (x) ≤ 2 a
(3.8)
|τ̂t (ξ)| ≤
j∈Z
(3.10)
√ X
τ ∗ (f )(x) ≤ 2 a
Λτ,Φ,j,a (f )(x) + Cα−1 M H(f )(x),
j∈Z
Boundedness of Rough Parametric
Marcinkiewicz Functions
Ahmad Al-Salman and
Hussain Al-Qassem
vol. 8, iss. 4, art. 108, 2007
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where M H stands for the Hardy-Littlewood maximal function on Rn , τ ∗ the maximal function that corresponds to {τt : t ∈ R}, and Λτ,Φ,s,a is the operator defined by
(2.1).
By (3.8), it is easy to see that
(3.11)
kΛτ,Φ,j,a (f )k2 ≤ C2−ε|j| α−1 kf k2
for all f ∈ L2 (Rn ). Therefore, by (3.10) and (3.11) we have
(3.12)
Boundedness of Rough Parametric
Marcinkiewicz Functions
Ahmad Al-Salman and
Hussain Al-Qassem
kτ ∗ (f )k2 ≤ Cα−1 a kf k2 .
Thus by (3.7), (3.8), (3.11), (3.12), and Lemma 2.1 with q = 2, we get
√
(3.13)
kΛτ,Φ,j,a (f )kp ≤ Cα−1 a kf kp
for p ∈ ( 43 , 4). Hence, by interpolation between (3.11) and (3.13), we obtain
√
0
(3.14)
kΛτ,Φ,j,a (f )kp ≤ Cα−1 a2−ε |j| kf kp
for p ∈ ( 43 , 4). Hence by (3.10) and (3.14), we get
(3.15)
kτ ∗ (f )kp ≤ Cα−1 a kf kp
for p ∈ ( 43 , 4). Next, by repeating the above argument with q =
get that
√
0
(3.16)
kΛτ,Φ,j,a (f )kp ≤ Cα−1 a2−ε |j| kf kp
vol. 8, iss. 4, art. 108, 2007
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4
3
+ ε (ε → 0+ ), we
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(3.17)
kτ ∗ (f )kp ≤ Cα−1 a kf kp
for p ∈ ( 87 , 8). Now the result follows by successive applications of the above argument. This completes the proof.
4.
Proofs of The Main Results
Proof of Theorem 1.2. Suppose that Ω ∈ L(log+ L)(Sn−1 ) and h(|x|) ∈ l∞ (Lq )(R+ ),
1 < q ≤ ∞. A key element in proving our results is decomposing the function Ω
as follows (for more information see [3]): For a natural number w, let Ew be the
w+1
set of points x0 ∈ Sn−1 which satisfy 2
≤ |Ω (x0 )| < 2w+2 . Also, we let E0
0
n−1
be the set of points x ∈ S
which satisfy |Ω (x0 )| < 22 . Set bw = ΩχEw . Set
D = {w : kbw k1 ≥ 2−3w } and define the sequence of functions {Ωw }w∈D∪{0} by
Z
X
X Z
bw (x)dσ(x)v
(4.1) Ω0 (x) = b0 (x)+
bw (x)−
b0 (x)dσ(x)−
Sn−1
w∈D
/
w∈D
/
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−1
Ωw (x) = (kbw k1 )
Z
bw (x) −
bw (x)dσ(x) .
Sn−1
Then, it is easy to see that for w ∈ D∪ {0}, Ωw satisfies (1.1),
(4.3)
(4.4)
vol. 8, iss. 4, art. 108, 2007
Sn−1
and for w ∈ D,
(4.2)
Boundedness of Rough Parametric
Marcinkiewicz Functions
Ahmad Al-Salman and
Hussain Al-Qassem
kΩw k1 ≤ C, kΩw k∞ ≤ C24(w+2) ,
X
Ω(x) =
θw Ωw (x),
w∈D∪{0}
where θ0 = 1, and θw = kbw k1 if w ∈ D.
ρ
For w ∈ D∪ {0}, let µΩw ,h be the operator defined as in (1.3) with Ω replaced by
Ωw . Then by (4.4), we have
X
ρ
ρ
(4.5)
µΩ,h f (x) ≤
θw µΩw ,h f (x).
w∈D∪{0}
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Now, for w ∈ D∪ {0}, let τw = {τt,w : t ∈ R} be the family of measures on Rn
defined via the Fourier transform by
Z
−αt
(4.6)
τ̂t,w (ξ) = 2
e−iξ·y Ωw (y) |y|−n+ρ h(|y|)dy
|y|≤2t
and let Φw+2 = {ϕt : t ∈ R} be a collection of C ∞ functions on Rn defined as in
the proof of Theorem 3.1. Let Λτw ,Φw+2 ,j,w+2 , j ∈ Z be the operators given by (2.1).
Then by a simple change of variable we obtain
X
√
ρ
(4.7)
µΩw ,h f (x) ≤ w + 2
Λτw ,Φw+2 ,j,w+2 (f )(x).
Boundedness of Rough Parametric
Marcinkiewicz Functions
Ahmad Al-Salman and
Hussain Al-Qassem
vol. 8, iss. 4, art. 108, 2007
j∈Z
Thus by Lemma 2.2, the properties of Ωw , Theorem 3.1, and Lemma 2.1, we get
(4.8)
ρ
µΩw ,h f
p
≤
(w + 2)C
kf kp
α
for all 1 < p < ∞.
Therefore, for 1 < p < ∞, by (4.7) and (4.8), we get




X
C
ρ
µΩ,h f p ≤
(w + 2)θw kf kp

α
w∈D∪{0}
C
≤ kΩkL(log L)(Sn−1 ) kf kp .
α
Hence the proof is complete.
Proof of Theorem 1.3. A proof of Theorem 1.3 can be obtained using the decomposition (4.4) and Theorem 3.1. We omit the details
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References
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Boundedness of Rough Parametric
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Ahmad Al-Salman and
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[11] E.M. STEIN, On the function of Littlewood-Paley, Lusin and Marcinkiewicz,
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Boundedness of Rough Parametric
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Ahmad Al-Salman and
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vol. 8, iss. 4, art. 108, 2007
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