ON THE L

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Boundedness of Rough Parametric Marcinkiewicz Functions

Ahmad Al-Salman and Hussain Al-Qassem vol. 8, iss. 4, art. 108, 2007

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ON THE L

^p

BOUNDEDNESS OF ROUGH PARAMETRIC MARCINKIEWICZ FUNCTIONS

AHMAD AL-SALMAN HUSSAIN AL-QASSEM

Department of Mathematics and Statistics Department of Mathematics and Physics

Sultan Qaboos University Qatar University,

P.O. Box 36, Al-Khod 123 Muscat Qatar

Sultanate of Oman.

EMail:alsalman@squ.edu.om EMail:husseink@qu.edu.qa

Received: 07 November, 2006 Accepted: 25 November, 2007 Communicated by: L. Debnath

2000 AMS Sub. Class.: Primary 42B20; Secondary 42B15, 42B25.

Key words: Parametric Marcinkiewicz operators, rough kernels, Fourier transforms, Para- metric maximal functions.

Abstract: In this paper, we study the L^p boundedness of a class of parametric Marcinkiewicz integral operators with rough kernels inL(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.

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1. Introduction

Letn≥2andSⁿ⁻¹be the unit sphere inRⁿequipped with the normalized Lebesgue measuredσ. Suppose thatΩis a homogeneous function of degree zero onRⁿ that satisfiesΩ∈L¹(Sⁿ⁻¹)and

(1.1)

Z

Sⁿ⁻¹

Ω(x)dσ(x) = 0.

In 1960, Hörmander ([9]) defined the parametric Marcinkiewicz functionµ^ρ_Ω of higher dimension by

(1.2) µ^ρ_Ωf(x) = Z ∞

−∞

2^−ρt Z

|y|≤2^t

f(x−y)|y|^−n+ρΩ(y)dy

2

dt

!¹₂ ,

where ρ > 0. It is clear that if ρ = 1, then µ^ρ_Ω is the classical Marcinkiewicz integral operator introduced by Stein ([11]) which will be denoted by µ_Ω. When Ω ∈ Lip_α(Sⁿ⁻¹), (0 < α ≤ 1), Stein proved that µ_Ω is bounded on L^p for all 1 < p ≤ 2.Subsequently, Benedek-Calderón-Panzone proved the L^p boundedness of µ_Ω for all 1 < p < ∞ under the condition Ω ∈ C¹(Sⁿ⁻¹) ([4]). Recently, under various conditions onΩ,theL^p boundedness ofµ_Ω and a more general class of operators of Marcinkiewicz type has been investigated (see [1] – [2], [5], among others).

In ([9]), Hörmander proved thatµ^ρ_Ωis bounded onL^pfor all1< p <∞,provided thatΩ∈Lip_α(Sⁿ⁻¹),(0< α≤1)andρ >0.

A long standing open problem concerning the operator µ^ρ_Ω is whether there are someL^p results onµ^ρ_Ω similar to those onµΩ whenΩsatisfies only some size con-

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ditions. In a recent paper, Ding, Lu, and Yabuta ([6]) studied the operator (1.3) µ^ρ_Ω,hf(x) =

Z ∞

−∞

2^−ρt Z

|y|≤2^t

f(x−y)|y|^−n+ρh(|y|)Ω(y)dy

2

dt

!¹₂ , where ρ is a complex number, Re(ρ) = α > 0, andh is a radial function onRⁿ satisfying h(|x|) ∈ l^∞(L^q)(R⁺), 1 ≤ q ≤ ∞, where l^∞(L^q)(R⁺) is defined as follows: For1≤q <∞,

l^∞(L^q)(R⁺) =







h:khk_l∞(L^q)(R⁺)= sup

j∈Z

Z 2^j 2^j−1

|h(r)|^qdr r

!¹_q

< C





 and forq=∞,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 degree zero onRⁿsatisfying (1.1) andh(|x|) ∈ l^∞(L^q)(R⁺)for some1 < q ≤ ∞.

IfRe(ρ) = α >0, then

µ^ρ_Ω,hf

2 ≤ C/√

αkfk₂, whereC is independent ofρand f.

TheL^pboundedness ofµ^ρ_Ω,hforp6= 2was left open by the authors of ([6]). The main purpose of this paper is to establish theL^pboundedness ofµ^ρ_Ω,hforp6= 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^∞(L^q)(R⁺), 1 < q ≤ ∞, and Re(ρ) = α > 0, then

µ^ρ_Ω,hf

p ≤ C/αkfk_p for all 1 < p < ∞, where C is independent ofρandf.

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Also, in this paper, we establish the L^p 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 onRⁿ. Ifh(|x|)∈l^∞(L^q)(R⁺),1< q ≤ ∞, andα >0, then

kM_αfk_p ≤ C α kfk_p

for all1 < p < ∞ with a constantC independent ofα, where M_α is the operator defined by

(1.4) M_αf(x) = sup

t∈R

2^−αt

Z

|y|≤2^t

Ω(y)|y|^−n+ρh(|y|)f(x−y)dy

.

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.

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2. Preparation

Suppose a ≥ 1. For a suitable family of measures τ = {τ_t : t ∈ R} onRⁿ and a suitable family of C^∞ functions Φ_a ={ϕ_t : t ∈ R}on Rⁿ, define the family of operators{Λ_τ,Φ_a_,s :t, s∈R}by

(2.1) Λ_τ,Φ,s,a(f)(x) = Z ∞

−∞

|τ_at∗ϕ_t+s∗f(x)|²dt ¹₂

. Also, define the operatorτ^∗ by

(2.2) τ^∗(f)(x) = sup

t∈R

(|τ_t| ∗ |f|)(x).

The proof of our result will be based on the following lemma:

Lemma 2.1. Suppose that for someB >0,ε >0, andβ >0, we have

(i) kτtk ≤βfort∈R;

(ii) |ˆτ_t(ξ)| ≤β(2^t|ξ|)^±^ε^a forξ∈Rⁿandt ∈R; (iii) kτ^∗(f)k_q ≤Bkfk_qfor someq >1;

(iv) The functions ϕ_t, t ∈ R satisfy the properties that ϕˆ_t is supported in {ξ ∈ Rⁿ : 2^−(t+1)a ≤ |ξ| ≤ 2^−(t−1)a}and

d^γϕˆt

dξ^γ (ξ)

≤C_γ|ξ|^−|γ|for any multi-index γ ∈ (N∪(0))ⁿ with constants C_γ depend only on γ and the dimension of the underlying spaceRⁿ.

Then for _q+1^2q < p < _q−1^2q , there exists a constantC_p independent ofa, β, B,s, and εsuch that

(2.3) kΛ_τ,Φ,s,a(f)k_p ≤C_p(βB)¹²(βB⁻¹)^θ(p)² 2^(ε+1)θ(p)2^{−εθ(p)|s|}kfk_p

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for all f ∈ L^p(Rⁿ), where θ(p) = ^2q−pq+p_p if p ∈

2,_q−1^2q

and θ(p) = ^pq+p−2q_p if p∈

2q q+1,2

.

Proof. We start with the case p = 2. By Plancherel’s formula and the conditions (i)-(ii), we obtain

(2.4) kΛ_τ,Φ,s,a(f)k₂ ≤β2^ε+12^−ε|s|kfk₂ for allf ∈L²(Rⁿ).

Next, setp₀ = 2q⁰and choose a non-negative functionv ∈L^q₊(Rⁿ)withkvk_q = 1 such that

kΛτ,Φ,s,a(f)k²_p

0 = Z

Rⁿ

Z ∞

−∞

|τat∗ϕt+s∗f(x)|²v(x)dtdx.

Now it is easy to see that

(2.5) kΛ_τ,Φ,s,a(f)k_p

0 ≤p

βkg_a,s(f)k_p

0kτ^∗(v)k_q¹² whereg_a,s is the operator

(2.6) g_a,s(f)(x) =

Z ∞

−∞

|ϕ_t+s∗f(x)|²dt ¹₂

.

By the condition (iv) and a well-known argument (see [12, p. 26-28]), it is easy to see that

(2.7) kg_a,s(f)k_p

0 ≤C_p₀kfk_p

0

for allf ∈ L^p⁰(Rⁿ)with constant C_p₀ independent of aand s. Thus, by (2.5) and (2.7), we have

(2.8) kΛ_τ,Φ,s,a(f)k_p

0 ≤C_p₀p

βBkfk_p

0.

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By duality, we get

(2.9) kΛ_τ,Φ,s,a(f)k_(p

0)⁰ ≤C_(p₀₎⁰p

βBkfk_(p

0)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 thatΩ∈L^∞(Sⁿ⁻¹)is a homogeneous function of degree zero onRⁿ satisfying (1.1) and h(|x|) ∈ l^∞(L^q)(R⁺), 1 < q ≤ 2. Then for a complex numberρwithRe(ρ) =α >0, we have

(2.10)

2^−αt Z

|y|≤2^t

e^−iξ·yΩ(y)|y|^−n+ρh(|y|)dy

≤2C

α khk_l∞(L^q)(R⁺)kΩk^1−2/q₁ ⁰kΩk^2/q_∞⁰(2^t|ξ|)^−ε, and

(2.11)

2^−αt Z

|y|≤2^t

≤2C

α khk_l∞(L^q)(R⁺)kΩk₁(2^t|ξ|)^ε

for all0< ε < min{1/2, α}. The constantC is independent ofΩ,α, andt.

Proof. Forξ ∈ Rⁿandr ∈ R⁺, letG(ξ, r) = R

Sⁿ⁻¹e^−irξ·y⁰Ω(y⁰)dσ(y⁰). Then it is

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easy to see that (2.12)

2^−αt Z

|y|≤2^t

≤

∞

X

j=0

2^−αj Z 2^t−j

2^t−j−1

|h(r)| |G(ξ, r)|r⁻¹dr.

Using the assumption that 1 < q ≤ 2, it is straightforward to show that the right hand side of (2.12) is dominated by

(2.13) 2khk_l∞(L^q)(R⁺)kΩk^1−2/q₁ ⁰

∞

X

j=0

2^−αj

Z 2^t−j 2^t−j−1

|G(ξ, r)|²r⁻¹dr

!_q¹0

. Now, forξ ∈Rⁿ,y⁰, z⁰ ∈Sⁿ⁻¹,j ≥0, andt∈R, set

I_j,t(ξ, y⁰, z⁰) = Z 2^t−j

2^t−j−1

e^−irξ·(y⁰^−z⁰⁾r⁻¹dr.

Then, we have (2.14)

Z 2^t−j 2^t−j−1

!_q¹0

≤ kΩk^2/q_∞ ⁰ Z

Sⁿ⁻¹×Sⁿ⁻¹

|I_j,t(ξ, y⁰, z⁰)|dσ(y⁰)dσ(z⁰) _q¹0

. By integration by parts, we have

(2.15) |Ij,t(ξ, y⁰, z⁰)| ≤(2^t−j−1|ξ| |ξ⁰·(y⁰ −z⁰)|)⁻¹.

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On the other hand, we have

(2.16) |I_j,t(ξ, y⁰, z⁰)| ≤ln 2.

Thus, by combining (2.15) and (2.16), we get

(2.17) |I_j,t(ξ, y⁰, z⁰)| ≤(2^t−j−1|ξ| |ξ⁰·(y⁰−z⁰)|)^−ε

for0< ε <min{1/2, α}. Therefore, by (2.14) and (2.17), we obtain that (2.18)

Z 2^t−j 2^t−j−1

!_q¹0

≤ kΩk^2/q_∞ ⁰C(2^t−j−1|ξ|)^−ε,

where the constantC is independent ofΩ,j, andt. Moreover, sinceε≤ 1/2, it can be shown thatC 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 (2.19)

2^−αt Z

|y|≤2^t

≤ 2(ln 2)^q¹⁰

α khk_l∞(L^q)(R⁺)kΩk₁2^t|ξ|. On the other hand, we have

(2.20)

2^−αt Z

|y|≤2^t

≤ 2(ln 2)^q¹⁰

α khk_l∞(L^q)(R⁺)kΩk₁. Thus, by interpolation between (2.19) and (2.20), we get (2.11). This completes the proof of Lemma2.2.

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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_∞ ≤ 2^a for some a > 1. Suppose also that h(|x|) ∈ l^∞(L^q)(R⁺), 1 < q ≤ ∞ and letM_α be the operator defined as in (1.4).

Then

(3.1) kM_αfk_p ≤ aC

α kfk_p

for all1< p <∞with constantCindependent ofa, f, andα.

Proof. Sincel^∞(L^q¹)(R⁺) ⊂l^∞(L^q²)(R⁺)wheneverq2 ≤q1, it suffices to assume that 1 < q ≤ 2. By a similar argument as in ([2]), choose a collection of C^∞ functions Φ_a = {ϕ_t : t ∈ R} on Rⁿ that satisfies the following properties: ϕˆ_t is supported in{ξ ∈ Rⁿ : 2^−(t+1)a ≤ |ξ| ≤ 2^−(t−1)a},

d^γϕˆt

dξ^γ (ξ)

≤ C_γ|ξ|^−|γ| for any multi-index γ ∈ (N∪{0})ⁿ with constants C_γ depending only on the underlying dimension andγ, and

(3.2) X

j∈Z

ˆ

ϕt+j(ξ) = 1.

Fort ∈R, let{σ_t :t∈R}be the family of measures onRⁿdefined via the Fourier transform by

(3.3) σˆ_t(ξ) = 2^−αt Z

|y|≤2^t

e^−iξ·y|Ω(y)| |y|^−n+ρ|h(|y|)|dy

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Then it is easy to see that

(3.4) M_αf(x) = sup

t∈R

{|σ_t| ∗ |f(x)|}.

Now chooseφ∈ S(Rⁿ)such thatφ(η) = 1ˆ for|η| ≤ ¹₂, andφ(η) = 0ˆ for|η| ≥1.

Let{τ_t :t ∈R}be the family of measures onRⁿ defined via the Fourier transform by

(3.5) τˆ_t(ξ) = ˆσ_t(ξ)−φ(2ˆ ^tξ)ˆσ_t(0).

Then by Lemma2.2, the choice ofφ, the definitions of σ_t, τ_t, and the assumptions onΩ, we have

(3.6) |ˆτ_t(ξ)| ≤ C2^la

α (2^t|ξ|)^−ε for somel, ε >0. Moreover, it is easy to see that

(3.7) kτ_tk ≤ C

α

Therefore by interpolation between (3.6) and (3.7), we get

(3.8) |ˆτ_t(ξ)| ≤ C

α(2^t|ξ|)⁻^a^ε.

Now by (3.2), and the definitions ofσ_t, andτ_t, it is easy to see that M_αf(x)≤2√

aX

j∈Z

Λ_τ,Φ,j,a(f)(x) +Cα⁻¹M H(f)(x), (3.9)

τ^∗(f)(x)≤2√ aX

j∈Z

Λ_τ,Φ,j,a(f)(x) +Cα⁻¹M H(f)(x), (3.10)

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whereM H stands for the Hardy-Littlewood maximal function on Rⁿ,τ^∗ the maximal function that corresponds to{τ_t:t∈R}, andΛ_τ,Φ,s,ais the operator defined by (2.1).

By (3.8), it is easy to see that

(3.11) kΛ_τ,Φ,j,a(f)k₂ ≤C2^−ε|j|α⁻¹kfk₂ for allf ∈L²(Rⁿ). Therefore, by (3.10) and (3.11) we have (3.12) kτ^∗(f)k₂ ≤Cα⁻¹akfk₂.

Thus by (3.7), (3.8), (3.11), (3.12), and Lemma2.1withq= 2, we get (3.13) kΛ_τ,Φ,j,a(f)k_p ≤Cα⁻¹√

akfk_p

forp∈(⁴₃,4).Hence, by interpolation between (3.11) and (3.13), we obtain (3.14) kΛ_τ,Φ,j,a(f)k_p ≤Cα⁻¹√

a2^−ε⁰^|j|kfk_p forp∈(⁴₃,4). Hence by (3.10) and (3.14), we get

(3.15) kτ^∗(f)k_p ≤Cα⁻¹akfk_p

forp∈(⁴₃,4). Next, by repeating the above argument withq = ⁴₃ +ε(ε →0⁺), we get that

(3.16) kΛ_τ,Φ,j,a(f)k_p ≤Cα⁻¹√

a2^−ε⁰^|j|kfk_p

(3.17) kτ^∗(f)k_p ≤Cα⁻¹akfk_p

forp ∈(⁷₈,8). Now the result follows by successive applications of the above argument. This completes the proof.

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4. Proofs of The Main Results

Proof of Theorem1.2. Suppose thatΩ∈L(log⁺L)(Sⁿ⁻¹)andh(|x|)∈l^∞(L^q)(R⁺), 1 < q ≤ ∞. A key element in proving our results is decomposing the functionΩ as follows (for more information see [3]): For a natural number w, let E_w be the set of points x⁰ ∈ Sⁿ⁻¹ which satisfy 2^w+1 ≤ |Ω (x⁰)| < 2^w+2. Also, we letE0

be the set of pointsx⁰ ∈ Sⁿ⁻¹ which satisfy |Ω (x⁰)| < 2². Set bw = Ωχ_E

w. Set D={w:kb_wk₁ ≥2^−3w}and define the sequence of functions{Ω_w}_w∈D∪{0} by (4.1) Ω0(x) =b0(x)+X

w /∈D

b_w(x)−

Z

Sⁿ⁻¹

b0(x)dσ(x)−X

w /∈D

Z

Sⁿ⁻¹

b_w(x)dσ(x)v

and forw∈D,

(4.2) Ω_w(x) = (kb_wk₁)⁻¹

b_w(x)− Z

Sⁿ⁻¹

b_w(x)dσ(x)

.

Then, it is easy to see that forw∈D∪ {0},Ω_w satisfies (1.1), kΩ_wk₁ ≤C, kΩ_wk_∞≤C2^4(w+2), (4.3)

Ω(x) = X

w∈D∪{0}

θ_wΩ_w(x), (4.4)

whereθ₀ = 1, andθ_w =kb_wk₁ifw∈D.

Forw∈D∪ {0}, letµ^ρ_Ω_w_,hbe the operator defined as in (1.3) withΩreplaced by Ω_w. Then by (4.4), we have

(4.5) µ^ρ_Ω,hf(x)≤ X

w∈D∪{0}

θ_wµ^ρ_Ω

w,hf(x).

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Now, for w ∈ D∪ {0}, let τ_w = {τ_t,w :t ∈R} be the family of measures onRⁿ defined via the Fourier transform by

(4.6) τˆ_t,w(ξ) = 2^−αt Z

|y|≤2^t

e^−iξ·yΩ_w(y)|y|^−n+ρh(|y|)dy

and let Φ_w+2 ={ϕ_t : t ∈ R}be a collection of C^∞functions on Rⁿ defined as in the proof of Theorem3.1. LetΛτw,Φw+2,j,w+2,j ∈Zbe the operators given by (2.1).

Then by a simple change of variable we obtain (4.7) µ^ρ_Ω_w_,hf(x)≤√

w+ 2X

j∈Z

Λ_τ_w_,Φ_w+2_,j,w+2(f)(x).

Thus by Lemma2.2, the properties ofΩ_w, Theorem3.1, and Lemma2.1, we get

(4.8)

µ^ρ_Ω_w_,hf

p ≤ (w+ 2)C α kfk_p for all1< p <∞.

Therefore, for1< p <∞, by (4.7) and (4.8), we get µ^ρ_Ω,hf

p ≤ C α





 X

w∈D∪{0}

(w+ 2)θ_w





 kfk_p

≤ C

α kΩk_L(log_L)(Sn−1)kfk_p. Hence the proof is complete.

Proof of Theorem1.3. A proof of Theorem1.3can be obtained using the decompo- sition (4.4) and Theorem3.1. We omit the details

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[11] E.M. STEIN, On the function of Littlewood-Paley, Lusin and Marcinkiewicz, Trans. Amer. Math. Soc., 88 (1958), 430–466.

[12] E.M. STEIN, Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals, Princeton University Press, Princeton, NJ, 1993.