(1)ON THE L BOUNDEDNESS OF ROUGH PARAMETRIC MARCINKIEWICZ FUNCTIONS AHMAD AL-SALMAN AND HUSSAIN AL-QASSEM DEPARTMENT OFMATHEMATICS ANDSTATISTICS SULTANQABOOSUNIVERSITY P.O

ON THE L BOUNDEDNESS OF ROUGH PARAMETRIC MARCINKIEWICZ FUNCTIONS

AHMAD AL-SALMAN AND HUSSAIN AL-QASSEM DEPARTMENT OFMATHEMATICS ANDSTATISTICS

SULTANQABOOSUNIVERSITY

P.O. BOX36, AL-KHOD123 MUSCAT

SULTANATE OFOMAN. alsalman@squ.edu.om

DEPARTMENT OFMATHEMATICS ANDPHYSICS

QATARUNIVERSITY, QATAR

husseink@qu.edu.qa

Received 07 November, 2006; accepted 25 November, 2007 Communicated by L. Debnath

ABSTRACT. In this paper, we study theL^pboundedness 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]).

Key words and phrases: Parametric Marcinkiewicz operators, rough kernels, Fourier transforms, Parametric maximal func- tions.

2000 Mathematics Subject Classification. Primary 42B20; Secondary 42B15, 42B25.

1. INTRODUCTION

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

(1.1)

Z

Sⁿ⁻¹

Ω(x)dσ(x) = 0.

In 1960, Hörmander ([9]) defined the parametric Marcinkiewicz function µ^ρ_Ω of higher di- mension 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 op- erator introduced by Stein ([11]) which will be denoted by µ_Ω. When Ω ∈ Lip_α(Sⁿ⁻¹),

This paper was written during the authors’ time in Yarmouk University.

285-06

(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 conditionΩ ∈ 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 that µ^ρ_Ω is bounded on L^p for all 1 < p < ∞, provided that Ω∈Lip_α(Sⁿ⁻¹),(0< α≤1)andρ >0.

A long standing open problem concerning the operator µ^ρ_Ω is whether there are some L^p results onµ^ρ_Ω similar to those on µΩ when Ωsatisfies only some size conditions. 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, and h is a radial function on Rⁿ satisfying h(|x|)∈l^∞(L^q)(R⁺),1≤q≤ ∞, wherel^∞(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 onRⁿ satisfying (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 ofρandf.

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

p ≤C/αkfk_pfor all1< p <∞, whereCis independent ofρandf.

Also, in this paper, we establish theL^p boundedness of the related parametric maximal func- tion. In fact, we have the following result:

Theorem 1.3. Suppose thatΩ∈L(log⁺L)(Sⁿ⁻¹)is a homogeneous function of degree zero on Rⁿ. Ifh(|x|)∈l^∞(L^q)(R⁺),1< q ≤ ∞, andα >0, then

kM_αfk_p ≤ C α kfk_p

for all1< p <∞with a constantCindependent ofα, whereM_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 opera- tors of Marcinkiewicz type. An extensive discussion of more general operators will appear in forthcoming papers.

Throughout the rest of the paper the letterC will stand for a constant but not necessarily the same one in each occurrence.

2. PREPARATION

Supposea ≥ 1. For a suitable family of measures τ = {τt : t ∈ R}on Rⁿ and a suitable family ofC^∞ functions Φa ={ϕt : t ∈ R} on Rⁿ, define the family of operators {Λτ,Φa,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) kτ_tk ≤βfort∈R;

(ii) |ˆτ_t(ξ)| ≤β(2^t|ξ|)^±aε 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-indexγ ∈(N∪(0))ⁿ with constantsC_γ depend only onγand the dimension of the underlying spaceRⁿ. Then for _q+1^2q < p < _q−1^2q , there exists a constantCpindependent ofa, β, B,s, andεsuch 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 for allf ∈L^p(Rⁿ), whereθ(p) = ^2q−pq+p_p ifp∈

2,_q−1^2q

andθ(p) = ^pq+p−2q_p ifp∈

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 = 1such that kΛ_τ,Φ,s,a(f)k²_p

0 = Z

Rⁿ

Z ∞

−∞

|τ_at∗ϕ_t+s∗f(x)|²v(x)dtdx.

Now it is easy to see that

(2.5) kΛτ,Φ,s,a(f)k_p

0 ≤p

βkga,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 constantC_p0 independent ofaands. 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.

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 that Ω ∈ L^∞(Sⁿ⁻¹) is a homogeneous function of degree zero on Rⁿ satisfying (1.1) and h(|x|) ∈ l^∞(L^q)(R⁺), 1 < q ≤ 2. Then for a complex number ρ with Re(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

e^−iξ·yΩ(y)|y|^−n+ρh(|y|)dy

≤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 easy to see that

(2.12)

2^−αt Z

|y|≤2^t

e^−iξ·yΩ(y)|y|^−n+ρh(|y|)dy

≤

∞

X

j=0

2^−αj Z 2^t−j

2^t−j−1

|h(r)| |G(ξ, r)|r⁻¹dr.

Using the assumption that1 < 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

|G(ξ, r)|²r⁻¹dr

!_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) |I_j,t(ξ, y⁰, z⁰)| ≤(2^t−j−1|ξ| |ξ⁰·(y⁰ −z⁰)|)⁻¹. 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

|G(ξ, r)|²r⁻¹dr

!_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 ofα. 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

e^−iξ·yΩ(y)|y|^−n+ρh(|y|)dy

≤ 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

e^−iξ·yΩ(y)|y|^−n+ρh(|y|)dy

≤ 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

Lemma 2.2.

3. ROUGHPARAMETRIC MAXIMALFUNCTIONS

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}onRⁿ 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-indexγ ∈ (N∪{0})ⁿ with constants C_γ depending only on the underlying dimension andγ, and

(3.2) X

j∈Z

ˆ

ϕ_t+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 Then it is easy to see that

(3.4) M_αf(x) = sup

t∈R

{|σ_t| ∗ |f(x)|}.

Now choose φ ∈ S(Rⁿ) such thatφ(η) = 1ˆ for |η| ≤ ¹₂, andφ(η) = 0ˆ for |η| ≥ 1.Let {τt:t∈R}be the family of measures onRⁿdefined via the Fourier transform by

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

Then by Lemma 2.2, the choice of φ, 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|)^−aε.

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)

whereM H stands for the Hardy-Littlewood maximal function onRⁿ,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 Lemma 2.1 withq= 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) kΛτ,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.

4. PROOFS OFTHEMAIN RESULTS

Proof of Theorem 1.2. 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 functionΩas follows (for more information see [3]): For a natural number w, let E_w be the set of points x⁰ ∈ Sⁿ⁻¹ which satisfy2^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_Ωw,hbe the operator defined as in (1.3) withΩreplaced byΩ_w. Then by (4.4), we have

(4.5) µ^ρ_Ω,hf(x)≤ X

w∈D∪{0}

θ_wµ^ρ_Ωw,hf(x).

Now, forw∈D∪ {0}, letτ_w ={τ_t,w :t ∈R}be the family of measures onRⁿdefined 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 letΦ_w+2 ={ϕ_t : t ∈ R} be a collection ofC^∞ functions on Rⁿ defined as in the proof of Theorem 3.1. Let Λ_{τw,Φw+2,j,w+2}, j ∈ Z be the operators given by (2.1). Then by a simple change of variable we obtain

(4.7) µ^ρ_Ωw,hf(x)≤√

w+ 2X

j∈Z

Λ_{τw,Φw+2,j,w+2}(f)(x).

Thus by Lemma 2.2, the properties ofΩ_w, Theorem 3.1, and Lemma 2.1, we get

(4.8)

µ^ρ_Ωw,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(logL)(Sn−1)kfk_p.

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

REFERENCES

[1] A. AL-SALMANANDH. AL-QASSEM, Flat Marcinkiewicz integral operators, Turkish J. Math., 26(3) (2002), 329–338.

[2] A. AL-SALMAN AND H. AL-QASSEM, Integral operators of Marcinkiewicz type, J. Integral Equations Appl., 14(4) (2002), 343–354.

[3] A. AL-SALMANANDY. PAN, Singular Integrals with Rough Kernels inLlog⁺L(Sⁿ⁻¹), J. Lon- don Math. Soc., (2) 66 (2002) 153–174.

[4] A. BENEDEK, A. CALDERÓNANDR. PANZONE, convolution operators on Banach space val- ued functions, Proc. Nat. Acad. Sci. USA, 48 (1962), 356–365.

[5] Y. DING, D. FAN AND Y. PAN, L^p boundedness of Marcinkiewicz integrals with Hardy space function kernel, Acta. Math. Sinica (English Series), 16 (2000), 593–600.

[6] Y. DING, S. LUAND K. YABUTA, A problem on rough parametric Marcinkiewicz functions, J.

Austral. Math. Soc., 72 (2002), 13–21.

[7] J. DUOANDIKOETXEAANDJ.L. RUBIO DE FRANCIA, Maximal and singular integral opera- tors via Fourier transform estimates, Invent. Math., 84 (1986), 541–561.

[8] D. FAN AND Y. PAN, Singular integrals with rough kernels supported by subvarieties, Amer. J.

Math., 119 (1997), 799–839.

[9] HÖRMANDER, ‘Translation invariant operators’, Acta Math., 104 (1960), 93–139.

[10] M. SAKAMOTOANDK. YABUTA, Boundedness of Marcinkiewicz functions, Studia Math., 135 (1999), 103–142.

[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 Inte- grals, Princeton University Press, Princeton, NJ, 1993.