ON CERTAIN ROUGH INTEGRAL OPERATORS AND EXTRAPOLATION

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Rough Integral Operators and Extrapolation Hussain Al-Qassem, Leslie Cheng

and Yibiao Pan vol. 10, iss. 3, art. 78, 2009

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ON CERTAIN ROUGH INTEGRAL OPERATORS AND EXTRAPOLATION

HUSSAIN AL-QASSEM LESLIE CHENG

Department of Mathematics and Physics Department of Mathematics

Qatar University,Doha-Qatar Bryn Mawr College, Bryn Mawr, PA 19010, U.S.A.

EMail:husseink@yu.edu.jo EMail:lcheng@brynmawr.edu

YIBIAO PAN

Department of Mathematics

University of Pittsburgh, Pittsburgh, PA 15260, U.S.A.

EMail:yibiao@pitt.edu

Received: 24 January, 2009

Accepted: 31 August, 2009

Communicated by: S.S. Dragomir

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

Key words: Maximal operator, Rough kernel, Singular integral, Orlicz spaces, Block spaces, Extrapolation,L^pboundedness.

Abstract: In this paper, we obtain sharpL^pestimates of two classes of maximal operators related to rough singular integrals and Marcinkiewicz integrals.These estimates will be used to obtain similar estimates for the related singular integrals and Marcinkiewicz integrals. By the virtue of these estimates and extrapolation we obtain theL^pboundedness of all the aforementioned operators under rather weak size conditions. Our results represent significant improvements as well as natural extensions of what was known previously.

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1. Introduction and Main Results

Throughout this paper, let Rⁿ, n ≥ 2, be the n-dimensional Euclidean space and Sⁿ⁻¹be the unit sphere inRⁿequipped with the normalized Lebesgue surface mea- suredσ. Also, we let ξ⁰ denote ξ/|ξ|for ξ ∈ Rⁿ\{0}and p⁰ denote the exponent conjugate top,that is1/p+ 1/p⁰ = 1.

Let KΩ,h(x) = Ω(x⁰)h(|x|)|x|⁻ⁿ, where h : [0, ∞) −→ C is a measurable function andΩis a function defined onSⁿ⁻¹ withΩ∈L¹(Sⁿ⁻¹)and

(1.1)

Z

Sⁿ⁻¹

Ω (x)dσ(x) = 0.

For 1 ≤ γ ≤ ∞, let ∆_γ(R₊) denote the collection of all measurable functions h: [0,∞)−→Csatisfying sup

R>0

1 R

RR

0 |h(t)|^γdt1/γ

<∞.

Define the singular integral operatorS_Ω,h,the parametric Marcinkiewicz integral operatorM_Ω,hand their related maximal operatorsS_Ω^(γ,∗)andM^(γ,∗)_Ω by

(1.2) SΩ,hf(x) = p.v.

Z

Rⁿ

f(x−u)KΩ,h(u)du,

(1.3) M_Ω,hf(x) = Z ∞

0

t^−ρ Z

|u|≤t

f(x−u)|u|^ρK_Ω,h(u)du

2 dt t

!¹₂ ,

(1.4) S_Ω^(γ,∗)f(x) = sup

h

|S_Ω,hf(x)|,

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(1.5) M^(γ,∗)_Ω f(x) = sup

h

|M_Ω,hf(x)|,

wheref ∈ S(Rⁿ), ρ = σ+iτ (σ, τ ∈ Rwithσ > 0)and the supremum is taken over the set of all radial functionshwithh∈L^γ(R+, dt/t)andkhk_Lγ(R+,dt/t)≤1.

Ifh(t)≡1andρ = 1, we shall denoteS_Ω,hbyS_ΩandM_Ω,hbyM_Ω,which are respectively the classical singular integral operator of Calderón-Zygmund and the classical Marcinkiewicz integral operator of higher dimension.

The study of the mapping properties of SΩ,MΩ and their extensions has a long history. Readers are referred to [7], [3], [4], [16], [18], [5], [10], [22], [24], [25]

and the references therein for applications and recent advances on the study of such operators.

Let us now recall some results which will be relevant to our current study. We start with the following results on singular integrals:

Theorem 1.1. IfΩsatisfies one of the following conditions, then S_Ω,h is bounded onL^p(Rⁿ)for1< p <∞.

(a) Ω ∈ L(logL)(Sⁿ⁻¹)andh ∈ ∆_γ(R₊)for someγ > 1.Moreover, the condi- tion Ω∈ L(logL)(Sⁿ⁻¹)is an optimal size condition for theL^p boundedness ofS_Ω (see [13] in the caseh(t)≡1and see [10]).

(b) Ω ∈ L(logL)^1/γ⁰(Sⁿ⁻¹) andh ∈ L^γ(R+, dt/t)for some γ > 1(see [2] and see [8] in the caseγ = 2).

(c) Ω ∈ Bq^(0,0)(Sⁿ⁻¹)for some q > 1 andh ∈ ∆_γ(R+) for some γ > 1. More- over, the condition Ω ∈ Bq^(0,0)(Sⁿ⁻¹) is an optimal size condition for the L^p boundedness ofS_Ω(see [4] and [5]).

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(d) Ω∈Bq^(0,−1/2)(Sⁿ⁻¹)for someq >1andh∈L²(R+, dt/t)([9]).

HereL(logL)^α(Sⁿ⁻¹)(forα > 0)is the class of all measurable functionsΩon Sⁿ⁻¹which satisfy

kΩk_L(log_L)^α_(Sn−1) = Z

Sⁿ⁻¹

|Ω(y)|log^α(2 +|Ω(y)|)dσ(y)<∞

andBq^(0,υ)(Sⁿ⁻¹), υ > −1,is a special class of block spaces whose definition will be recalled in Section2.

Theorem 1.2. IfΩsatisfies one of the following conditions, thenM_Ω,h is bounded onL^p(Rⁿ)for1< p <∞.

(a) Ω ∈ L(logL)^1/2(Sⁿ⁻¹)and h = 1. Moreover, the exponent 1/2 is the best possible ([26], [7]).

(b) Ω ∈ Bq^(0,−1/2)(Sⁿ⁻¹) for some q > 1 and h = 1. Moreover, the condition Ω ∈ Bq^(0,−1/2)(Sⁿ⁻¹) is an optimal size condition for the L² boundedness of MΩ([3]).

(c) Ω ∈ L(logL)^1/γ⁰(Sⁿ⁻¹), h ∈ L^γ(R₊, dt/t), 1 < γ ≤ 2and γ⁰ ≤ p < ∞ or Ω∈L(logL)^1/2(Sⁿ⁻¹, h∈L^γ(R₊, dt/t),2< γ <∞and2≤p <∞([6]).

On the other hand, the study of the related maximal operator S_Ω^(γ,∗) has attracted the attention of many authors. See for example, [14], [1], [5], [6], [8], [9], [27].

Theorem 1.3. IfΩsatisfies one of the following conditions, then S_Ω^(γ,∗) is bounded onL^p(Rⁿ).

(a) Ω∈C(Sⁿ⁻¹),(nγ)/(nγ−1)< p <∞and1≤γ ≤2. Moreover, the range ofpis the best possible [14].

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(b) Ω∈Bq^(0,−1/2)(Sⁿ⁻¹)for someq > 1, γ = 2,2≤p <∞.Moreover, the condi- tionΩ ∈ Bq^(0,−1/2)(Sⁿ⁻¹)is an optimal size condition for theL² boundedness ofS_Ω^(2,∗)to hold ([1], [9]).

(c) Ω∈L(logL)^1/γ

0

(Sⁿ⁻¹), γ⁰ ≤p <∞.Moreover, the exponent1/2 is the best possible for theL²boundedness ofS_Ω^(2,∗)to hold (see [8] forγ = 2and [2] for γ 6= 2).

In view of the above results, the following question is very natural:

Problem 1. Is there any analogue of Theorem1.1(d) and Theorem1.3(b) in the case γ 6= 2? Is there an analogy of Theorem1.2(c)? Is there any room for improvement of the range ofpin both Theorem1.2(c) and Theorem1.3(c)?

The purpose of this paper is two-fold. First, we answer the above questions in the affirmative. Second, we present a unified approach different from the ones em- ployed in previous papers (see for example, [1], [2], [6], [8], [9]) in dealing with the operatorsS_Ω,h,M_Ω,h, S_Ω^(γ,∗) andM^(γ,∗)_Ω when the kernel functionΩbelongs to the block spaceBq^(0,υ)(Sⁿ⁻¹) (forυ > −1)or Ωbelongs to the classL(logL)^α(Sⁿ⁻¹) (for α > 0). This approach will mainly rely on obtaining some delicate sharp L^p estimates and then applying an extrapolation argument.

Now, let us state our main results.

Theorem 1.4. Suppose thatΩ∈L^q(Sⁿ⁻¹)for some1< q ≤ ∞.Then (1.6)

S_Ω^(γ,∗)(f) L^p(Rⁿ)

≤C_p q

q−1 1/γ⁰

kΩk_Lq(Sⁿ⁻¹)kfk_Lp(Rⁿ)

holds for(nαγ⁰)/(γ⁰n+nα−γ⁰)< p <∞ and1≤γ ≤2,whereα= max{2, q⁰}.

Moreover, the exponent1/γ⁰is the best possible in the caseγ = 2.

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Theorem 1.5. LetαandΩbe as in Theorem1.4and let1≤γ <∞. Then (1.7)

M^(γ,∗)_Ω (f)

L^p(Rⁿ)≤C_p q

q−1 1/β

kΩk_Lq(Sⁿ⁻¹)kfk_Lp(Rⁿ)

holds for(nαβ)/(βn+nα−β)< p < ∞,whereβ = max{2, γ⁰}.Moreover, the exponent1/β is the best possible in the caseγ = 2.

Theorem 1.6. Suppose that Ω∈ L^q(Sⁿ⁻¹), 1 < q ≤ 2, and h ∈L^γ(R₊, dt/t) for some1< γ ≤ ∞.Then

(1.8) kS_Ω,h(f)k_Lp(Rⁿ)≤C_p(q−1)^−1/γ⁰khk_Lγ(R+,dt/t)kΩk_Lq(Sⁿ⁻¹)kfk_Lp(Rⁿ)

for1< p <∞and

(1.9) kM_Ω,h(f)k_Lp(Rⁿ) ≤C_p(q−1)^−1/βkhk_Lγ(R+,dt/t)kΩk_Lq(Sⁿ⁻¹)kfk_Lp(Rⁿ)

for(nβq⁰)/(βn+nq⁰ −β)< p <∞.

By the conclusions from Theorems1.4–1.6and applying an extrapolation method, we get the following results:

Theorem 1.7.

(a) If Ω ∈ L(logL)^1/γ

0

(Sⁿ⁻¹)and1 < γ ≤ 2,the operatorS_Ω^(γ,∗) is bounded on L^p(Rⁿ) for2≤p < ∞;

(b) If Ω ∈ Bq^(0,1/γ⁰⁻¹⁾(Sⁿ⁻¹) and 1 < γ ≤ 2,the operator S_Ω^(γ,∗) is bounded on L^p(Rⁿ) for2≤p < ∞;

(c) If Ω ∈ L(logL)^1/γ

0

(Sⁿ⁻¹)andh ∈ L^γ(R₊, dt/t)for some1 < γ ≤ ∞,the operatorSΩ,his bounded onL^p(Rⁿ) for1< p <∞;

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(d) If Bq^(0,1/γ⁰⁻¹⁾(Sⁿ⁻¹)for someq >1andh∈ L^γ(R₊, dt/t)for some1 < γ ≤

∞,the operatorS_Ω,his bounded onL^p(Rⁿ) for1< p <∞.

Theorem 1.8. Let1< γ < ∞andβ = max{2, γ⁰}.

(a) If Ω∈L(logL)^1/β(Sⁿ⁻¹)and1< γ < ∞,the operatorM^(γ,∗)_Ω is bounded on L^p(Rⁿ)for2≤p <∞;

(b) If Ω ∈ Bq^(0,1/β−1)(Sⁿ⁻¹)and1< γ < ∞,the operator M^(γ,∗)_Ω is bounded on L^p(Rⁿ)for2≤p <∞.

(c) If Ω ∈ L(logL)^1/β(Sⁿ⁻¹) andh ∈ L^γ(R₊, dt/t)for some 1 < γ < ∞, the operatorM_Ω,his bounded onL^p(Rⁿ)for2≤p <∞;

(d) If B^(0,1/β−1)q (Sⁿ⁻¹)for someq > 1andh ∈ L^γ(R₊, dt/t)for some1< γ <

∞,the operatorM_Ω,his bounded onL^p(Rⁿ)for2≤p <∞.

Remark 1.

1. For anyq >1,0< α < β and−1 < υ,the following inclusions hold and are proper:

L^q(Sⁿ⁻¹)⊂L(logL)^β(Sⁿ⁻¹)⊂L(logL)^α(Sⁿ⁻¹), [

r>1

L^r(Sⁿ⁻¹)⊂B^(0,υ)_q (Sⁿ⁻¹)for any −1< υ, B_q^(0,υ²⁾(Sⁿ⁻¹)⊂B_q^(0,υ¹⁾(Sⁿ⁻¹)for any −1< υ₁ < υ₂,

L^γ(R+, dt/t)⊂∆_γ(R+) for1≤γ <∞.

The question regarding the relationship between Bq^(0,υ−1) and L(logL)^υ over Sⁿ⁻¹ (forυ >0)remains open.

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2. The L^p boundedness of S_Ω^(γ,∗) for (nαγ⁰)/(γ⁰n +nα− γ⁰) < p < ∞ and Ω∈ L^q(Sⁿ⁻¹) was proved in [1] only in the case γ = 2, but the importance of Theorem 1.4 lies in the fact that the estimate (1.6) in conjunction with an extrapolation argument will play a key role in obtaining all our results and will allow us to obtain theL^p boundedness ofS_Ω^(γ,∗) under optimal size conditions onΩ.

3. We point out that it is still an open question whether the L^p boundedness of S_Ω^(γ,∗) holds for 2 < γ < ∞.In the caseγ = ∞,the authors of [14] pointed out that the maximal operator S_Ω^(∞,∗)(f) is not bounded on all L^p spaces for 1 ≤ p≤ ∞.On the other hand, we notice that Theorem1.5gives thatM^(γ,∗)_Ω is bounded onL^p for any1≤γ <∞.

4. Theorem 1.7 (a)(c) and Theorem 1.8 (c) represent an improvement over the main results in [8] and improves the range ofpin [2]. Also, Theorem1.7(b)(d) and Theorem1.8(d) represent an improvement over the main results in [9] and [1].

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. Definitions and Some Basic Lemmas

Block spaces originated from the work of M.H. Taibleson and G. Weiss on the con- vergence of the Fourier series in connection with developments of the real Hardy spaces. Below we shall recall the definition of block spaces on Sⁿ⁻¹. For further background information about the theory of spaces generated by blocks and its applications to harmonic analysis, one can consult the book [20]. In [20], Lu introduced the spacesBq^(0,υ)(Sⁿ⁻¹)with respect to the study of singular integral operators.

Definition 2.1. Aq-block onSⁿ⁻¹is anL^q(1< q ≤ ∞)functionb(x)that satisfies (i) supp(b)⊂I;

(ii) kbk_Lq ≤ |I|^−1/q⁰, where |·| denotes the product measure on Sⁿ⁻¹, and I is an interval onSⁿ⁻¹,i.e.,I = {x∈Sⁿ⁻¹ :|x−x₀|< α}for someα > 0and x₀ ∈Sⁿ⁻¹.

The block spaceBq^(0,υ) =Bq^(0,υ)(Sⁿ⁻¹)is defined by B_q^(0,υ) =

(

Ω∈L¹(Sⁿ⁻¹) : Ω =

∞

X

µ=1

λ_µb_µ, M_q^(0,υ) {λ_µ}

<∞ )

, where each λ_µ is a complex number; each b_µ is a q-block supported on an intervalI_µ onSⁿ⁻¹,υ >−1and

M_q^(0,υ) {λ_µ}

=

∞

X

µ=1

λ_µ

n

1 + log^(υ+1) I_µ

−1o . Let

kΩk_B^(0,υ)

q (Sⁿ⁻¹) =N_q^(0,υ)(Ω) = inf

M_q^(0,υ) {λ_µ} , where the infimum is taken over allq-block decompositions ofΩ.

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Definition 2.2. For arbitraryθ ≥ 2andΩ : Sⁿ⁻¹ →R,we define the sequence of measures{σ_Ω,h,k : k ∈ Z} and the corresponding maximal operatorσ_Ω,h,θ^∗ onRⁿ by

(2.1)

Z

Rⁿ

f dσ_Ω,h,θ,k = Z

θ^k≤|u|<θ^k+1

h(|u|)Ω(u⁰)

|u|ⁿ f(u)du;

(2.2) σ_Ω,h,θ^∗ (f) = sup

k∈Z

||σ_Ω,h,θ,k| ∗f|.

We shall need the following lemma which has its roots in [15] and [5]. A proof of this lemma can be obtained by the same proof (with only minor modifications) as that of Lemma 3.2 in [5]. We omit the details.

Lemma 2.3. Let {σ_k :k∈Z} be a sequence of Borel measures onRⁿ. Suppose that for some a ≥ 2, α, C > 0, B > 1 and p₀ ∈ (2,∞) the following hold for k ∈Z,ξ∈Rⁿand for arbitrary functions{g_k}onRⁿ:

(i) |ˆσ_k(ξ)| ≤CB a^kξ

±_log(a)^α

;

(ii)

P

k∈Z

|σ_k∗g_k|² ¹₂

p0

≤CB

P

k∈Z

|g_k|² ¹₂

p0

.

Then forp⁰₀ < p < p₀,there exists a positive constantC_p independent ofBsuch that

X

k∈Z

σ_k∗f p

≤C_pBkfk_p

holds for allf inL^p(Rⁿ).

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Lemma 2.4. Letθandσ^∗_Ω,h,θbe given as in Definition2.2. Supposeh∈L^γ(R₊, dt/t) for some1< γ ≤ ∞andΩ∈L¹(Sⁿ⁻¹).Then

(2.3)

σ^∗_Ω,h,θ(f)

p ≤Cp(logθ)^1/γ⁰khk_Lγ(R+,dt/t)kΩk_L1(Sⁿ⁻¹)kfk_p forγ⁰ < p ≤ ∞andf ∈L^p(Rⁿ),whereC_p is independent ofΩ, θandf.

Proof. By Hölder’s inequality we have

|σ_Ω,h,θ,k∗f(x)| ≤ khk_Lγ(R+,dt/t)kΩk^1/γ_L1(Sⁿ⁻¹)

×

Z θ^k+1 θ^k

Z

Sⁿ⁻¹

|Ω(y)| |f(x−ty)|^γ⁰dσ(y)dt t

!1/γ⁰

. By Minkowski’s inequality for integrals we get

σ^∗_Ω,h,θ(f)

p ≤(logθ)^1/γ⁰khk_Lγ(R+,dt/t)kΩk^1/γ_L1(Sⁿ⁻¹)

× Z

Sⁿ⁻¹

|Ω(y)|

M_y(|f|^γ⁰) p/γ⁰

dσ(y) 1/γ⁰

, where

M_yf(x) = sup

R>0

R⁻¹ Z R

0

|f(x−sy)|ds

is the Hardy-Littlewood maximal function off in the direction ofy.By the fact that the operatorM_y is bounded onL^p(Rⁿ)for1< p <∞with a bound independent of y, by the last inequality we get (2.3). Lemma2.4is thus proved.

By following an argument similar to that in [17] and [2], we obtain the following:

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Lemma 2.5. Let θ andΩbe as in Lemma 2.4. Assume that h ∈ L^γ(R₊, dt/t) for some γ ≥ 2. Then for γ⁰ < p < ∞, there exists a positive constantC_p which is independent ofθ such that

X

k∈Z

|σΩ,h,θ,k∗gk|²

!¹₂ p

≤Cp(logθ)^1/γ⁰kΩk_L1(Sⁿ⁻¹)

X

k∈Z

|gk|²

!¹₂ p

holds for arbitrary measurable functions{g_k}onRⁿ. Now, we need the following simple lemma.

Lemma 2.6. Letq > 1andβ = max{2, q⁰}.Suppose thatΩ∈L^q(Sⁿ⁻¹).Then for some positive constantC,we have

(2.4) Z

Sⁿ⁻¹

Ω(ξ)f(ξ)dσ(ξ)

2

≤ kΩk^min{2,q}_q Z

Sⁿ⁻¹

|Ω(ξ)|^max{0,2−q}|f(ξ)|²dσ(ξ) and

(2.5) Z

Sⁿ⁻¹

|Ω(ξ)|^max{0,2−q}|f(x−tξ)|dσ(ξ)

≤CkΩk^max{0,2−q}_q

M_Sph

|f|^β/2 (x)_β²

for all positive real numberst andx ∈Rⁿand arbitrary functions f,whereMSph

is the spherical maximal operator defined by M_Sphf(x) = sup

t>0

Z

Sⁿ⁻¹

|f(x−tu)|dσ(u).

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Proof. Let us first prove (2.4). First ifq≥2,by Hölder’s inequality we have

Z

Sⁿ⁻¹

Ω(ξ)f(ξ)dσ(ξ)

2

≤ kΩk²_q Z

Sⁿ⁻¹

|f(ξ)|^q⁰dσ(ξ) _q²0

≤ kΩk²_q Z

Sⁿ⁻¹

|f(ξ)|²dσ(ξ),

which is the inequality (2.4) in the caseq ≥ 2.Next, if 1 < q < 2, the inequality (2.4) follows by Schwarz’s inequality.

To prove (2.5) we need again to consider two cases. First ifq≥2,we notice that (2.5) is obvious. Next if1< q <2,(2.5) follows by Hölder’s inequality on noticing that

q 2−q

0

=q⁰/2. The lemma is thus proved.

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3. Proof of the Main Results

We begin with the proof of Theorem1.4.

Proof. SinceL^q(Sⁿ⁻¹)⊆L²(Sⁿ⁻¹)forq≥2,Theorem1.4is proved once we prove that

(3.1)

≤Cp(q−1)^−1/γ⁰kΩk_Lq(Sⁿ⁻¹)kfk_Lp(Rⁿ)

holds for1< q ≤2,(nq⁰γ⁰)/(γ⁰n+nq⁰−γ⁰)< p <∞ and1≤γ ≤2.

We shall prove (3.1) by first proving (3.1) for the casesγ = 1andγ = 2and then we use the idea of interpolation to cover the case1< γ <2.

Proof of (3.1) for the caseγ = 1.By duality we have S_Ω^(1,∗)f(x) =

Z

Sⁿ⁻¹

f(x−tu)Ω(u)dσ(u)

L^∞(R+,dt/t)

= Z

Sⁿ⁻¹

f(x−tu)Ω(u)dσ(u)

L^∞(R+,dt)

≤sup

t>0

Z

Sⁿ⁻¹

|f(x−tu)| |Ω(u)|dσ(u).

Using Hölder’s inequality we get

S_Ω^(1,∗)f(x)≤ kΩk_q

M_Sph

|f|^q⁰

(x)1/q⁰

.

By the results of E.M. Stein [23] and J. Bourgain [12] we know that M_Sph(f) is bounded onL^p(Rⁿ)forn ≥2andp > n/(n−1). Thus by using the last inequality we get (3.1) for the caseγ = 1.

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Proof of (3.1) for the caseγ = 2.By Hölder’s inequality we obtain

S_Ω^(2,∗)f(x)≤ Z ∞

0

Z

Sⁿ⁻¹

Ω(u)f(x−tu)dσ(u)

2 dt t

!¹₂ .

Letθ = 2^q⁰ and let{ϕ_k}^∞_−∞be a smooth partition of unity in(0, ∞)adapted to the interval[θ^−k−1, θ^−k+1]. Specifically, we require the following:

ϕk ∈C^∞, 0≤ϕk ≤1, X

k∈Z

ϕk(t) = 1,

suppϕ_k⊆[θ^−k−1, θ^−k+1],

d^sϕ_k(t) dt^s

≤ C_s t^s ,

whereC_s is independent of the lacunary sequence{θ^k : k ∈ Z}.Define the multi- plier operatorsT_k inRⁿby(dT_kf)(ξ) =ϕ_k(|ξ|) ˆf(ξ).Then for anyf ∈ S(Rⁿ)and k ∈Zwe havef(x) = P

j∈Z(T_k+jf)(x).Thus, by Minkowski’s inequality we have

S_Ω^(2,∗)f(x)≤



 X

k∈Z

Z θ^k+1 θ^k

X

j∈Z

Z

Sⁿ⁻¹

Ω(u)T_k+jf(x−tu)dσ(u)

2

dt t





1 2

≤X

j∈Z

X_jf(x), where

X_jf(x) = X

k∈Z

Z θ^k+1 θ^k

Z

Sⁿ⁻¹

2 dt t

!¹₂ .

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We notice that to prove (3.1) for the case γ = 2, it is enough to prove that the inequality

(3.2) kX_j(f)k_p ≤C_p(q−1)⁻¹² kΩk_q2^−δ^p^|j|kfk_p

holds for 1 < q ≤ 2, (nq⁰γ⁰)/(nγ⁰ +nq⁰ −γ⁰) < p < ∞ and for some positive constantsC_pandδ_p.This inequality can be proved by interpolation between a sharp L² estimate and a cruder L^p estimate. We prove (3.2) first for the case p = 2. By using Plancherel’s theorem we have

(3.3) kX_j(f)k²₂ = Z

∆k+l

X

k∈Z

(H_k(ξ))²

fˆ(ξ)

2 dt t dξ, where∆_k=

ξ ∈Rⁿ:θ^−k−1 ≤ |ξ| ≤θ^−k+1 and (3.4) Hk(ξ) =

Z θ^k+1 θ^k

Z

Sⁿ⁻¹

Ω(x)e^−itξ·xdσ(x)

2 dt t

!¹₂ . We claim that

(3.5) |Hk(ξ)| ≤C(logθ)¹² kΩk_qmin n

θ^kξ

−^λ

q0

, θ^k+1ξ

λ q0o

, whereCis a constant independent ofk, ξandq.

Let us now turn to the proof of (3.5). First, by a change of variable and since

Z

Sⁿ⁻¹

Ω(x)e^−iθ^k^tξ·xdσ(x)

2

= Z

Sⁿ⁻¹×Sⁿ⁻¹

Ω(x)Ω(y)e^−iθ^k^tξ·(x−y)dσ(x)dσ(y), we obtain

(3.6) (H_k(ξ))² ≤ Z

Ω(x)Ω(y) Z θ

1

e^−iθ^k^tξ·(x−y)dt t

dσ(x)dσ(y).

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Using an integration by parts,

Z θ 1

≤C(logθ) θ^kξ

−1|ξ⁰ ·(x−y)|⁻¹.

By combining this estimate with the trivial estimate

Rθ

1 e^−iθ^k^tξ·(x−y)^dt_t

≤ (logθ), we get

Z θ 1

≤C(logθ) θ^kξ

−α|ξ⁰·(x−y)|^−α for any0< α≤1.

Thus, by the last inequality, (3.6) and Hölder’s inequality we get

|H_k(ξ)| ≤C(logθ)¹² kΩk_q θ^kξ

− ^α

2q0

Z

dσ(x)dσ(y)

|ξ⁰ ·(x−y)|^αq⁰

!_2q¹0

. By choosing α so thatαq⁰ < 1we obtain that the last integral is bounded in ξ⁰ ∈ Sⁿ⁻¹and hence

(3.7) |H_k(ξ)| ≤C(logθ)¹² kΩk_q θ^kξ

−^α

2q0

. Secondly, by the cancellation conditions onΩwe obtain

H_k(ξ)≤

Z θ^k+1 θ^k

Z

Sⁿ⁻¹

|Ω(x)|

e^−itξ·x−1 dσ(x)

2

dt t

!¹₂

≤C(logθ)¹² kΩk₁ θ^k+1ξ

.

By combining the last estimate with the estimate|H_k(ξ)| ≤ C(logθ)¹² kΩk₁, we get

(3.8) |Hk(ξ)| ≤C(logθ)¹² kΩk₁ θ^k+1ξ

α 2q0

.

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By combining the estimates (3.7)–(3.8), we obtain (3.5).

Now, by (3.4) and (3.5) we obtain

(3.9) kX_j(f)k₂ ≤C2^−λ|j|(q−1)⁻¹² kΩk_qkfk₂.

Let us now estimatekX_j(f)k_p for(nq⁰γ⁰)/(γ⁰n+nq⁰ −γ⁰)< p < ∞withp 6= 2.

Let us first consider the casep > 2.By duality, there is a nonnegative functiong in L^(p/2)⁰(Rⁿ)withkgk_(p/2)0 ≤1such that

kX_j(f)k²_p = Z

Rⁿ

X

k∈Z

Z θ^k+1 θ^k

Z

Sⁿ⁻¹

2 dt

t g(x)dx.

By Hölder’s inequality, Fubini’s theorem and a change of variable we have kX_j(f)k²_p ≤ kΩk₁

Z

Rⁿ

X

k∈Z

Z θ^k+1 θ^k

Z

Sⁿ⁻¹

|Ω(u)| |T_k+jf(x−tu)|²g(x)dσ(u)dt t dx

≤ kΩk₁ Z

Rⁿ

X

k∈Z

Z θ^k+1 θ^k

Z

Sⁿ⁻¹

|Ω(u)| |T_k+jf(x)|²g(x+tu)dσ(u)dt t dx

≤CkΩk₁ Z

Rⁿ

X

k∈Z

|T_k+jf(x)|²

!

σ^∗_Ω,1,θ(˜g)(−x)dx,

whereg(x) =˜ g(−x)andσ_Ω,h,θ^∗ is defined as in (2.2). By the last inequality, Lemma 2.4and using Littlewood-Paley theory ([24, p. 96]) we get

kX_j(f)k²_p ≤CkΩk₁

σ^∗_Ω,1,θ(˜g) (p/2)⁰

X

k∈Z

|T_k+jf|² (p/2)

≤C_p(logθ)kΩk²₁kfk²_p,

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which in turn easily implies

(3.10) kX_j(f)k_p ≤C_p(q−1)^−1/2kΩk_qkfk_p for2< p <∞.

By interpolating between (3.9) – (3.10) we get (3.2) for the case2≤ p <∞.Now, let us estimatekX_j(f)k_pfor the case(nq⁰γ⁰)/(γ⁰n+nq⁰−γ⁰)< p <2.By a change of variable we get

X_jf(x) = X

k∈Z

Z θ 1

Z

Sⁿ⁻¹

Ω(u)T_k+jf(x−θ^ktu)dσ(u)

2dt t

!¹₂ . By duality, there is a functionh=h_k,j(x, t)satisfyingkhk ≤1and

h_k,j(x, t) ∈L^p⁰

l²

L²

[1, θ],dt t

, k

, dx

such that kX_j(f)k_p =

Z

Rⁿ

X

k∈Z

Z θ 1

Z

Sⁿ⁻¹

Ω(u) (T_k+jf) (x−θ^ktu)h_k,j(x, t)dσ(u)dt t dx

= Z

Rⁿ

X

k∈Z

Z θ 1

Z

Sⁿ⁻¹

Ω(u) (T_k+jf) (x)h_k,j(x+θ^ktu, t)dσ(u)dt t dx.

By Hölder’s inequality and Littlewood-Paley theory we get kX_j(f)k_p ≤

(L(h))¹² p⁰

X

k∈Z

|T_k+jf|²

!¹₂ p

(3.11)

≤C_pkfk_p

(L(h))¹² _p₀,

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where

L(h) =X

k∈Z

Z θ 1

Z

Sⁿ⁻¹

Ω(u)h_k,j(x+θ^ktu, t)dσ(u)dt t

² .

Sincep⁰ > 2and

(L(h))^1/2

p⁰ = kL(h)k^1/2_p0/2,there is a nonnegative functionb ∈ L^(p⁰^/2)⁰(Rⁿ)such thatkbk_(p0/2)⁰ ≤1and

kL(h)k_p0/2 = Z

Rⁿ

X

k∈Z

Z θ 1

Z

Sⁿ⁻¹

Ω(u)hk,j(x+θ^ktu, t)dσ(u)dt t

²

b(x)dx.

By the Schwarz inequality and Lemma2.6, we obtain Z θ

1

Z

Sⁿ⁻¹

Ω(u)h_k,j(x+θ^ktu, t)dσ(u)dt t

²

≤C(logθ) Z θ

1

Z

Sⁿ⁻¹

Ω(u)h_k,j(x+θ^ktu, t)dσ(u) 2

dt t

≤C(logθ)kΩk^min{2,q}_Lq(Sⁿ⁻¹)

Z θ 1

Z

Sⁿ⁻¹

|Ω(u)|^max{0,2−q}

h_k,j(x+θ^ktu, t)

2dσ(u)dt t . Therefore, by the last inequality, a change of variable, Fubini’s theorem, Hölder’s inequality, and Lemma2.6we get

kL(h)k_p0/2 ≤C(logθ)kΩk^min{2,q}_Lq(Sⁿ⁻¹)kΩk^max{0,2−q}_Lq(Sⁿ⁻¹)

× Z

Rⁿ

X

k∈Z

Z θ 1

|h_k,j(x, t)|² dt t

! M_Sph

˜b

q⁰/2 (−x)

_q²0

dx

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≤C(logθ)kΩk²_Lq(Sⁿ⁻¹)

X

k∈Z

Z θ 1

|h_k,j(x, t)|² dt t

p⁰/2

×

MSph

˜b

q⁰/22/q⁰ (p⁰/2)⁰

.

By the condition on p we have (2/q⁰)(p⁰/2)⁰ > n/(n − 1) and hence by the L^p boundedness ofMSphand the choice ofbwe obtain

kL(h)k_p0/2 ≤C(q−1)⁻¹kΩk²_Lq(Sⁿ⁻¹). Using the last inequality and (3.11) we have

(3.12) kX_j(f)k_p ≤C_p(q−1)^−1/2kΩk_qkfk_p for(nq⁰γ⁰)/(γ⁰n+nq⁰−γ⁰)< p <2.

Thus, by (3.9) and (3.12) and interpolation we get (3.2) for(nq⁰γ⁰)/(γ⁰n+nq⁰−γ⁰)<

p <2. This ends the proof of (3.1) for the caseγ = 2.

Proof of (3.1) for the case1< γ <2.

We will use an idea that appeared in [19]. By duality we have

=kF(f)k_Lp(L^γ⁰(R+,^dt_t),Rⁿ,dx),

whereF :L^p(Rⁿ)→L^p(L^γ⁰(R₊,^dt_t),Rⁿ)is a linear operator defined by F(f)(x;t) =

Z

Sⁿ⁻¹

f(x−tu)Ω(u)dσ(u).

From the inequalities (3.1) (for the case γ = 2)and (3.1) (for the case γ = 1), we interpret that

kF(f)k_Lp(L²(R+,^dt_t),Rⁿ) ≤C(q−1)⁻¹² kΩk_Lq(Sⁿ⁻¹)kfk_Lp(Rⁿ)

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for(2nq⁰)/(2n+nq⁰−2)< p <∞and

(3.13) kF(f)k_Lp(L^∞(R+,^dt_t),Rⁿ) ≤CkΩk_Lq(Sⁿ⁻¹)kfk_Lp(Rⁿ)

for q⁰n⁰ ≤ p < ∞. Applying the real interpolation theorem for Lebesgue mixed normed spaces to the above results (see [11]), we conclude that

kF(f)k_Lp(L^γ⁰(R+,^dt_t),Rⁿ) ≤Cp(q−1)^−1/γ⁰kΩk_Lq(Sⁿ⁻¹)kfk_Lp(Rⁿ)

holds for1 < q ≤ 2, (nq⁰γ⁰)/(nγ⁰ +nq⁰ −γ⁰) < p < ∞ and1 ≤ γ ≤ 2. This completes the proof of Theorem1.4.

Next we shall present the proof of Theorem1.5.

Proof. We need to consider two cases.

Case 1. 1 ≤ γ ≤ 2. By an argument appearing in [6], we may, without loss of generality, in the definition of M_Ω,h, replace |y| ≤ t with ¹₂t ≤ |y| ≤ t. By Minkowski’s inequality and Fubini’s theorem we get



 Z ∞

0

Z

1 2t≤|y|≤t

f(x−y)Ω(y⁰)

|y|^n−ρh(|y|)dy

2

dt t^1+2σ





1 2

≤ Z ∞

0

Z ∞ 0

Z

Sⁿ⁻¹

f(x−sy)Ω(y)dσ(y)

|h(s)|χ

[¹2t,t](s) ds s^1−σ

2

dt t^1+2σ

!¹₂

≤ Z ∞

0

Z ∞ 0

Z

Sⁿ⁻¹

2

|h(s)|²χ

[¹2t,t](s) dt t^1+2σ

!¹₂ ds s^1−σ

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= Z ∞

0

Z

Sⁿ⁻¹

|h(s)|

Z ∞ s

dt t^1+2σ

¹₂ ds s^1−σ

= 1

√2σ Z ∞

0

Z

Sⁿ⁻¹

|h(s)|ds s . Therefore, by duality we have

(3.14) M^(γ,∗)_Ω f(x)≤ 1

√2σS_Ω^(γ,∗)f(x), and hence (1.7) holds by (1.6) for1≤γ ≤2.

Case 2.2< γ ≤ ∞.By a change of variable and duality we have

M^(γ,∗)_Ω f(x)≤



 Z ∞

0

Z 1 1/2

Z

Sⁿ⁻¹

f(x−stu)Ω (u)dσ(u)

γ⁰

ds s

!2/γ⁰

dt t





1 2

.

Using the generalized Minkowski inequality and sinceγ⁰ <2, we get (3.15) M^(γ,∗)_Ω f(x)≤

Z 1 1/2

|E_sf(x)|^γ⁰ ds s

^1/γ⁰ , where

E_sf(x) = Z ∞

0

Z

Sⁿ⁻¹

f(x−stu)Ω (u)dσ(u)

2 dt t

!¹₂ . SinceE_sf(x) = S_Ω^(2,∗)f(x),we obtain

(3.16) M^(γ,∗)_Ω f(x)≤S_Ω^(2,∗)f(x),

and again (1.7) holds for the case2< γ ≤ ∞by (1.6) in the caseγ = 2.