Acta Mathematica Sinica, English Series Aug., 2016, Vol. 32, No. 8, pp. 925–942 Published online: July 15, 2016 DOI: 10.1007/s10114-016-5274-0 Http://www.ActaMath.com

Acta Mathematica Sinica, English Series Springer-Verlag Berlin Heidelberg & The Editorial Office of AMS 2016

Lp Estimates of Rough Maximal Functions Along Surfaces with Applications Ahmad AL-SALMAN

Abdulla M. JARRAH

Department of Mathematics, Yarmouk University, Irbid, Jordan E-mail : [email protected] [email protected] Abstract In this paper, we study the Lp mapping properties of certain class of maximal oscillatory singular integral operators. We prove a general theorem for a class of maximal functions along surfaces. As a consequence of such theorem, we establish the Lp boundedness of various maximal oscillatory singular integrals provided that their kernels belong to the natural space L log L(Sn−1 ). Moreover, we highlight some additional results concerning operators with kernels in certain block spaces. The results in this paper substantially improve previously known results. Keywords

Oscillatory singular integrals, rough kernels, maximal functions, highly monotone curves

MR(2010) Subject Classification

1

42B20, 42B15, 42B25

Introduction and Statement of Results

Let Rn , n ≥ 2 be the n-dimensional Euclidean space and Sn−1 be the unit sphere in Rn equipped with the normalized Lebesgue measure dσ. For nonzero y ∈ Rn , we shall let y  = |y|−1 y. Let Ω ∈ L1 (Sn−1 ) be a homogeneous function of degree zero on Rn which satisfies the cancellation property  Ω(y  )dσ(y  ) = 0.

(1.1)

Sn−1

For suitable mappings Ψ : Rn → R and ψ : (0, ∞) → R, we consider the maximal function  |Ω(y  )| |f (x − Ψ(y), xd+1 − ψ(|y|))| (1.2) MΨ,ψ,Ω (f )(x, xd+1 ) = sup n dy. |y| j∈Z 2j <|y|≤2j+1 The study of maximal functions in the form (1.2) has a long history. The Lp boundedness of such operators plays an indispensable role in the theory of singular integrals. By specializing to the case Ψ(y) = y, the maximal function MΨ,ψ,Ω reduces to the maximal function along surfaces Mψ introduced by Kim et al. [11]. It is shown in [11] that the maximal function Mψ is bounded on Lp for 1 < p < ∞ provided that Ω ∈ C ∞ (Sn−1 ) and ψ is a convex increasing function such that ψ(0) = 0. Later, as a consequence of a more general result, Al-Salman and Pan showed in [2] that Mψ is bounded on Lp for 1 < p < ∞ under the weaker condition Ω ∈ L log+ L(Sn−1 ). Here, it should be remarked that the condition Ω ∈ L log+ L(Sn−1 ) is known to be optimal size condition in the sense that the classical Calder´ on–Zygmund singular + 1−∈ n−1 (S )\ L log+ L(Sn−1 ) operator TΩ may fail boundedness if Ω is assumed to be in L(log L) for some ∈> 0. Received April 28, 2015, accepted January 4, 2016

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The study of the maximal function MΨ,ψ,Ω depends heavily on the one dimensional maximal function Mγ,ψ which is given by  2j+1 dt Mγ,ψ (f )(x, xd+1 ) = sup (1.3) |f (x − γ(t), xd+1 − ψ(t))| , t j∈Z 2j where γ(t) = (γ1 (t), . . . , γd (t)) : R → Rd is a polynomial mapping. The class of maximal functions in the form (1.3) generalizes the maximal functions along the parabola introduced by Stein [16]. In Section 2 (see Theorem 2.4), we shall show that the operator Mγ,ψ is bounded on Lp provided that the function ψ that shares certain properties of polynomials and convex functions (see Theorem 1.1 below). The following is our first main result: Theorem 1.1 Suppose that Ω is a homogeneous function of degree zero on Rn and that Ω ∈ Lq (Sn−1 ) for some q > 1. Suppose also that Ψ = (Ψ1 , . . . , Ψd ) is a polynomial mapping and that ψ : (0, ∞) → R is a function satisfying ψ(t) = θ(t) + ϕ(t), where θ is a polynomial, ϕ(j) (0) = 0 for 0 ≤ j ≤ N and ϕ(j) is positive nondecreasing on (0, ∞) for 0 ≤ j ≤ N + 1 where N = max{deg(Ψ), deg(θ)}. Let MΨ,ψ,q be the maximal function  |Ω(y  )| MΨ,ψ,q (f )(x, xd+1 ) = sup |f (x − Ψ(y), xd+1 − ψ(|y|))| dy, (1.4) |y|n j∈Z 2δq (Ω)j <|y|≤2δq (Ω)(j+1) where δq (Ω) = log(e + Ωq ). Then for 1 < p < ∞, there exists a constant Cp > 0 independent of the coefficients of the polynomial mappings {Ψj : 1 ≤ j ≤ d} and the polynomial θ such that MΨ,ψ,q (f )p ≤ δq (Ω)Cp ΩL1 f p .

(1.5)

We remark here that Theorem 1.1 serves as a basis for various Lp boundedness results. In the following, we present two main applications of this theorem: 1.1 Rough Maximal Functions Along Surfaces The first immediate consequence of Theorem 1.1 is the following substantial improvement (in terms of surfaces) of Theorem 4.3 in [2]: Theorem 1.2 Suppose that Ω is a homogeneous function of degree zero on Rn and Ω ∈ L log+ L(Sn−1 ). Suppose also that Ψ = (Ψ1 , . . . , Ψd ) is a polynomial mapping, ψ : (0, ∞) → R is a function satisfying ψ(t) = θ(t) + ϕ(t), where θ is a polynomial ϕ(j) (0) = 0 for 0 ≤ j ≤ N and ϕ(j) is positive nondecreasing on (0, ∞) for 0 ≤ j ≤ N + 1 where N = max{deg(Ψ), deg(θ)}. Then MΨ,ψ,Ω (f )p ≤ Cp ΩL log+ L f p

(1.6)

for 1 < p < ∞. Here, Cp is a positive constant independent of the coefficients of the polynomial mappings {Ψj : 1 ≤ j ≤ d} and the polynomial θ. The proof of Theorem 1.2 will be presented in Section 3. It depends on a suitable application of Theorem 1.1 along with suitable decomposition of the function Ω.

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1.2 Truncated Maximal Oscillatory Singular Operators The second main application of Theorem 1.1 is establishing Lp estimates of truncated maximal oscillatory singular integral operators. For suitable mappings Φ : Rn → Rd , Ψ : Rn → R, and ψ : (0, ∞) → R, we let TΦ,Ψ,ψ,Ω be the oscillatory singular integral operator along the surface (Ψ(y), ψ(|y|)) given by  −n eiΦ(y) f (x − Ψ(y), xd+1 − ψ(|y|))Ω(y  ) |y| dy. (1.7) TΦ,Ψ,ψ,Ω (f )(x, xd+1 ) = Rn

We let

∗ TΦ,Ψ,ψ,Ω

be the associated truncated maximal operator   ε ∗ TΦ,Ψ,ψ,Ω (f )(x, xd+1 ) = sup TΦ,Ψ,ψ,Ω (f )(x, xd+1 ) ,

(1.8)

ε>0

where

 ε TΦ,Ψ,ψ,Ω (f )(x, xd+1 ) =

|y|>ε

eiΦ(y) f (x − Ψ(y), xd+1 − ψ(|y|))Ω(y  ) |y|−n dy.

∗ is bounded By a well-known limiting argument, it can be shown that if the operator TΦ,Ψ,ψ,Ω p p on L , then the operator TΦ,Ψ,ψ,Ω is bounded on L too. However, it is well known that the ∗ is very involved and it is more treatment of maximal truncated operators in the form TΦ,Ψ,ψ,Ω delicate than that for the corresponding operators TΦ,Ψ,ψ,Ω . For background information and related results concerning the study of truncated singular integral operators, we refer readers to [14–16], among others. Historically, when Φ(y) ≡ 0, Ψ(y) = y, and ψ ≡ 0, the corresponding operator TΦ,Ψ,ψ,Ω reduces to the classical Calder´ on–Zygmund singular integral operator TΩ = T0,I,0,Ω [4, 5]. For background information regarding Calder´ on–Zygmund operator, we refer readers to consult [2, 4, 5, 14–16] among others. On the other hand, when Φ(y) ≡ 0, Ψ(y) = y, and ψ is convex increasing, the operator TΦ,Ψ,ψ,Ω reduces to the singular integral operator along surfaces of revolution Tψ,Ω = T0,I,ψ,Ω which has been introduced by Kim et al. [11]. As for the case of the maximal function Mψ , Kim et al. showed in [11] that the special operator Tψ,Ω is bounded on Lp for 1 < p < ∞ provided that Ω ∈ C ∞ (Sn−1 ) and φ is a convex increasing function such that φ(0) = 0. Subsequently, Al-Salman and Pan showed that Kim et al.’s result still holds under the weaker condition Ω ∈ L log+ L(Sn−1 ). When Ψ = (Ψ1 , . . . , Ψd ) is a polynomial mapping and Φ is a homogeneous function that satisfies β

Φ(ty  ) = t Φ(y  )

for t > 0,

Φ(y  ) ∈ L∞ (Sn−1 ),



and Sn−1

(1.9) |Ω(y  )|−δ dσ(y  ) < ∞

(1.10)

for some δ > 0 and for some β = 0, Fan and Yang [8] proved that the operator     ∗ iΦ(y)  −n  TΦ,Ψ,Ω (f )(x) = sup  e f (x − Ψ(y))Ω(y )|y| dy  ε>0

|y|>ε

is bounded on L for all 1 < p < ∞ provided that Ω ∈ Lq (Sn−1 ) for some q > 1 and that β = 0 is not a positive integer or β is a positive integer larger than max{deg(Ψj ) : 1 ≤ j ≤ d}. Later, p

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the condition Ω ∈ Lq (Sn−1 ) was relaxed to Ω ∈ L log+ L(Sn−1 ) by Al-Salman in [1]. In this ∗ : paper, we prove the following result concerning the class of operators TΦ,Ψ,ψ,Ω Theorem 1.3 Suppose that Ω is a homogeneous function of degree zero on Rn that satisfies (1.1) and that Ω ∈ L log L(Sn−1 ). Suppose also that Ψ = (Ψ1 , . . . , Ψd ) is a polynomial mapping. Let Φ be a homogeneous function of degree β = 0 such that Φ |Sn−1 is real analytic where β is not a positive integer or β is a positive integer larger than max{deg(Ψj ) : 1 ≤ j ≤ d}. Suppose that ψ : (0, ∞) → R is a function satisfying ψ(t) = θ(t) + ϕ(t), where θ is a polynomial, ϕ(j) (0) = 0 for 0 ≤ j ≤ N and ϕ(j) is positive nondecreasing on (0, ∞) ∗ is bounded for 0 ≤ j ≤ N + 1, where N = max{deg(Ψ), deg(θ)}. Then the operator TΦ,Ψ,ψ,Ω p on L for all 1 < p < ∞. Moreover, the operator norm is independent of the coefficients of the polynomial mappings {Ψj : 1 ≤ j ≤ d}. It should be remarked here that the result in Theorem 1.3 represents a substantial improvement of Theorem 4.2 in [2] and the corresponding result in [1]. Throughout this paper, the letter C will denote a constant that may vary at each occurrence, but it is independent of the essential variables. 2

A One-Dimensional Maximal Function

The main aim of this section is to prove the Lp boundedness of the one-dimensional maximal function in (1.3) for certain mappings ψ. In order to state our result in this section, we recall the concept of highly monotone curves in [12]. Following the notation in [12], for a smooth function α : [a, b] → (0, ∞), let Dα be the differential operator defined by   d f Dα (f )(t) = (t). dt α For n smooth functions α1 , . . . , αn , define the operators D1 , D2 , . . . , Dn by D1 = Dα1 , . . . , Dk = Dαk Dk−1 ,

2 ≤ k ≤ n.

It is proved in [12] that if |Dn f | has a positive lower bound, then f satisfies oscillatory estimates that generalize those obtained by Van der Corput’s lemma. In fact, we have the following: Lemma 2.1 ([12]) Let α1 , . . . , αn be positive nondecreasing functions defined on [a, b] and let α1 = 1. Let f be a real-valued function of class C n on [a, b]. If D1 f = f  is monotone and if |Dn f | ≥ λ > 0 for all t in [a, b], then   b   1 if (t)  −n  e dt (2.1)  ≤ Cn |λα1 (a) · · · αn (a)| .  a

It is radially observed that if αk ≡ 1, then Lemma 2.2 reduces to the classical Van der corput’s lemma. Estimates in the form (2.1) are closely related to highly monotone curves. Let γ = (γk )1≤k≤n : [0, N ] → Rn be a curve of class C n with γ(0) = 0, where C n is the class of functions with n continuous derivatives. Let α1 , . . . , αn be defined inductively by α1 = 1, αk = Dk−1 γk−1

for 2 ≤ k ≤ n.

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Here D1 , D2 , . . . , Dn are the differential operators associated to α1 , . . . , αn . Let Wγ be the upper triangular matrix given by Wγ = (Dk γj )1≤k,j≤n . We say that γ is highly monotone if the following hold: (i) If 1 ≤ k ≤ j ≤ n, then Dk γj is positive and nondecreasing on (0, N ); (ii) If 1 ≤ k < j ≤ n, then Dk γj = o(Dk γk ) as t → 0+ . Examples of highly monotone curves are widely available. A sample of interesting examples can be found in [12]. Among various properties of highly monotone curves, the following can be found in [12]: Lemma 2.2 ([12]) Let γ = (γk )1≤k≤n : [0, N ] → Rn be highly monotone curve 1 ≤ k ≤ j ≤ n and 0 < t < N . Then Dk γj (t) ≥ γj (t)/tk α1 (t) · · · αn (t). Now, we prove the following proposition: Proposition 2.3 Let ϕ be of class C n+1 (0, ∞). Then the curve γ = (t, . . . , tn , ϕ(t)) is highly monotone if, and only if ϕ(j) (0) = 0 for 1 ≤ j ≤ n and that ϕ(j) is positive nondecreasing on (0, ∞) for 1 ≤ j ≤ n + 1. Proof that

The proof is straightforward. First, by an inductive argument, it can be easily shown D1 γj = jtj−1 ,

1 ≤ j ≤ n,

Dk γj = Ck,j tj−k , where Ck,j =

j k−1

n ≥ j ≥ k ≥ 2,

(2.2)

 k−1   k−1   (j − i) (k − i)−1 . i=1

i=2

Thus, α1 = 1 and αk = k − 1

for k ≥ 2.

Therefore, an inductive argument implies that Dk γn+1 =

ϕ(k) (k − 1)!

for k ≥ 1.

Hence, γ = (t, . . . , tn , ϕ(t)) is highly monotone if and only if ϕ(j) (0) = 0 for 1 ≤ j ≤ n + 1 and that ϕ(j) is positive nondecreasing on (0, ∞) for 1 ≤ j ≤ n. This completes the proof.  The main result of this section is the following: Theorem 2.4 Suppose that γ(t) = (γ1 (t), . . . , γd (t)) : R → Rd is a polynomial mapping. Suppose also that ψ : (0, ∞) → R is a function satisfying ψ(t) = θ(t) + ϕ(t), where θ is a polynomial, ϕ(j) (0) = 0 for 0 ≤ j ≤ N and ϕ(j) is positive nondecreasing on (0, ∞) for 0 ≤ j ≤ N + 1 where N = max{deg(γ), deg(θ)}. Let Mγ,ψ be the maximal function given by (1.3). Then for 1 < p < ∞, there exists a constant Cp > 0 independent of the coefficients of the polynomial mappings γj and the polynomial θsuch that Mγ,ψ (f )p ≤ Cp f p .

(2.3)

Al-Salman A. and Jarrah A. M.

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Before presenting the proof of Theorem 2.4, we recall the following two lemma that will play useful role the proof: Lemma 2.5 ([2]) Let φ be a C 2 , convex and increasing function satisfying φ(0) = 0. Let m ∈ N and let M (m) be the maximal function on R defined by    2m(j+1)  dt  f (u − φ(t)) . M (m) f (u) = sup  t mj j∈Z 2

Then for 1 < p ≤ ∞, there exists a positive constant Cp independent of m such that M (m) f p ≤ Cp m f p for all f ∈ Lp (R). Lemma 2.6 Let {μk }k∈Z and {τk }k∈Z be sequences of non-negative Borel measures on Rn . Let L : Rn → Rm be a linear transformation. Suppose that for all k ∈ Z, ξ∈Rn , for some a ≥ 2, α, C > 0 and for some constant B > 1, we have (i) μk  ≤ B; τk  ≤ B; α (ii) |ˆ μk (ξ)| ≤ CB (a kB |L(ξ)| )− B ; α kB (iii) |ˆ μk (ξ)−ˆ τk (ξ)| ≤ CB (a |L(ξ)| ) B ; (iv) Suppose that τ ∗ (f )p ≤ B f p

for all 1 < p ≤ ∞ and f ∈ Lp (Rn ).

(2.4)

Then the inequality μ∗ (f )p ≤ B f p

(2.5)

holds for all 1 < p ≤ ∞ and f in Lp ( Rn ). The constant Cp is independent of B and the linear transformation L. Now, we are ready to prove Theorem 2.4. Proof of Theorem 2.4

If N = 0, then the mappings γ and θ are constants. Thus  2j+1 dt |f (x − c, xd+1 − ψ(t))| , Mγ,ψ (f )(x, xd+1 ) = sup t j∈Z 2j

where c is constant and ψ is as in Lemma 2.5. Thus, by the translation invariance property of Lebsgue measure and Lemma 2.5 (with m = 1), we obtain that (2.3) holds for Mγ,ψ for the case N = 0. Now, we assume that N ≥ 1. For each integer j, define the measures μj and τj by  2j+1  dt f (γ(t), ψ(t)) f dμj = t j 2 and



 f dτj =

∗

2j+1 2j

f (γ(t), θ(t))

dt . t

∗

Let μ and τ be the maximal functions μ∗ (f )(x, xn+1 ) = sup |μj ∗ f (x, xn+1 )| j

931

and τ ∗ (f )(x, xn+1 ) = sup |τj ∗ f (x, xn+1 )| . j

Let L = (L1 , . . . , LN ) : R

n+1

→R

N

be the linear transformation satisfying

(ξ, λ) · (γ(t), ψ(t)) = (L(ξ, λ), λ) · (t, t2 , . . . , tN , ϕ(t)) for (ξ, λ) ∈ Rn × R and t ∈ R. By Proposition 2.3, it follows that the mapping Γ(t) = (t, t2 , . . . , tN , ϕ(t)) is highly monontone and |DN +1 ((L(ξ, λ), λ) · Γ(t))| =

|λϕ(N +1) (2j )| |λϕ(N +1) (t)| ≥ . N! N!

Thus, by Lemma 2.2, we get

   N +1  λϕ(N +1) (t) D ≥ CN |2−(N +1)j λϕ(2j )|; ((L(ξ, λ), λ) · Γ(t)) = N! which combined with Lemma 2.1 implies that 1

1

|ˆ μj (ξ, λ)| ≤ C|2−(N +1)j λϕ(2j )|− N +1 ≤ C|2λϕ(2j )|− N +1 .

(2.6)

On the other hand, we have the estimate |ˆ μj (ξ, λ) − τˆj (ξ, λ)| ≤ C|λϕ(2j+1 )|.

(2.7)

Also, it can be easily seen that μj  ≤ 1 and

τj  ≤ 1.

(2.8)

By [15, Proposition 1, p. 477], it follows that τ ∗ (f )p ≤ Cp f p

(2.9)

for 1 < p < ∞ with constant Cp independent of the coefficients of the polynomial mappings γj and the polynomial θ. ˆ ˆ Now, choose ψ ∈ S(R) such that ψ(λ) = 1 for |λ| ≤ 1/2 and ψ(λ) = 0 for |λ| ≥ 1, and let −1 ψt (x) = t ψ(x/t) for t > 0. For each j ∈ Z, let νj be the measure defined by j ˆ νˆj (ξ, λ) = μ ˆj (ξ, λ) − τˆj (ξ, λ)ψ(ϕ(2 )λ).

(2.10)

Thus, by (2.6)–(2.8), and the choice of ψ, we have 1

|ˆ νj (ξ, λ)| ≤ C min{|2λϕ(2j )|− N +1 , |λϕ(2j+1 )|}. Let Sf (x, xn+1 ) =



2

|νj ∗ f |

(2.11)

 12 (2.12)

j∈Z

and (νj )∗ (f )(x, xn+1 ) = sup |(|νj | ∗ f )(x, xn+1 )| .

(2.13)

μ∗ (f )(x, xn+1 ) ≤ Sf (x, xn+1 ) + Cτ ∗ (M ⊗ IRn )(f )(x, xn+1 )

(2.14)

j∈Z

Then it follows that

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and (νj )∗ (f )(x, xn+1 ) ≤ Sf (x, xn+1 ) + 2Cτ ∗ (M ⊗ IRn )(f )(x, xn+1 ).

(2.15)

Here, M denotes the Hardy–Littlewood maximal function on R. By (2.8)–(2.11), (2.14)–(2.15), and the bootstrapping argument in [2], we obtain (2.3). This completes the proof. 3

The Higher Dimensional Maximal Function

The main aim of this section is to prove Theorems 1.1 and 1.2. Moreover, we highlight one of the main consequences of Theorem 1.1 that will play an important role in subsequent sections. We start by the proof of Theorem 1.1. Proof of Theorem 1.1 we obtain

If the mapping Ψ is radial, then by Theorem 2.4 and H¨ older’s inequality, MΨ,ψ,q (f )p ≤ δq (Ω)Cp Ω1 f p

for 1 < p < ∞. Thus, assume that Ψ is not radial and let dΨ = max{deg(Ψj ) : 1 ≤ j ≤ d}. We write dΨ  Ψ(y) = Ψ(l) (y), l=1 (l)

where Ψ

is homogeneous of degree l. We let AΨ = {1 ≤ l ≤ dΨ : Ψ(l) is no radial}.

Since Ψ is not radial, the set AΨ is not empty and we may assume that AΨ = {l1 < l2 < · · · < lM }. Now, for lj ∈ AΨ , we let Vlj (n) be the space of real-valued homogenous polynomials of l / Ulj (n). Thus, for each degree lj on Rn . Also, we let Ulj (n) be a subspace of Vlj (n) with |x| j ∈ lj ∈ AΨ , we can write ˜ (lj ) (y) + R(lj ) (|y|) = (Ψ ˜ (lj ) (y), . . . , Ψ ˜ (lj ) (y)) + R(lj ) (|y|), Ψ(lj ) (y) = Ψ 1 d (l )

(l )

where {Ψ1 j , . . . , Ψd j } ⊂ Ulj (n) and R(lj ) is a radial polynomial with degree less than or equal lj . Thus, M  ˜ (lj ) (y) + R(|y|), Ψ Ψ(y) = (3.1) j=1

where R(|y|) =

M 

R(lj ) (|y|).

j=1

For 1 ≤ j ≤ M and 1 ≤ s ≤ d, we let ˜ s(lj ) (y) = Ψ

 |α|=lj

For 1 ≤ u ≤ M , let Ψu (y) =

u  j=1

(s)

aα,lj y α .

˜ (lj ) (y) + R(|y|). Ψ

(3.2)

(3.3)

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Notice that ΨM (y) = Ψ(y). Define the sequence of measures {σu,k,ψ : 1 ≤ u ≤ M, k ∈ Z} by   |Ω(y  )| f (Ψu (y), ψ(|y|)) dy. f dσu,k,ψ = |y|n 2δq (Ω)k <|y|≤2δq (Ω)(k+1) We define σ0,k,ψ by

 σ ˆ0,k,ψ (ξ, η) =

2δq (Ω)k <|y|≤2δq (Ω)(k+1)

eiηψ(|y|)

(3.4)

|Ω(y  )| dy. |y|n

It is clear that MΨ,ψ,q (f )(x, xd+1 ) = sup |σM,k,ψ ∗ f (x, xd+1 )| . k∈Z

Notice that for ξ ∈ R , we have d

ξ · Ψu (y) =

u 

˜ (lj ) (y) + ξ · R(|y|) ξ·Ψ

j=1

=

u  d 

ξs

 

j=1 s=1

=

u 

j=1 |α|=lj

=

u 

|α|=lj

d  



(s)

aα,lj y α ) + ξ · R(|y|

 (s) ξs aα,lj y α + ξ · R(|y|)

s=1

Fj (y) + ξ · R(|y|),

j=1

where Fj (y) =

d   |α|=lj

 (s) ξs aα,lj y α .

s=1

Clearly, Fj ∈ Ulj (n). For 1 ≤ j ≤ l, let Nj be the number of multi-indices β = (β1 , . . . , βn ) ∈ (N ∪ {0})n with |β| = β1 + · · · + βn = lj and define the linear transformation Ls : Rd → RNj by   d (s) ξj aα,lj . Lj (ξ) = Lj (ξ1 , . . . , ξd ) = s=1

|α|=lj

Now, |ˆ σu,k,ψ (ξ, η)|2 2     u  |Ω(y  )|   = exp i Fj (y) + ξ · R(|y|) + ηψ(|y|) |y|n  Eq,k

j=1

2    u    dr  |Ω(y )| exp i Fj (ry ) + ξ · R(r) + ηψ(r) dσ(y )  r n−1 Eq,k S j=1   2     u       dr  ≤ |Ω(y )| exp i F (ry ) + ξ · R(r) + ηψ(r) dσ(y ) , j  n−1 r Eq,k S j=1   ≤ 





where Eq,k = [2δq (Ω)k , 2δq (Ω)(k+1) ]. Thus, by Proposition 5.1 in [7], we get |ˆ σu,k,ψ (ξ, η)| ≤ Aδq (Ω) ΩLq (2δq (Ω)klu |Lu (ξ)|)

− 4s1 l

u

,

(3.5)

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where s = min{2, q} and A is a constant independent of the essential variables. By combining the estimate (3.5) and the estimate |ˆ σu,k,ψ (ξ, η)| ≤ Aδq (Ω)ΩL1 , we get |ˆ σu,k,ψ (ξ, η)| ≤ Aδq (Ω) ΩL1 (2δq (Ω)klu |Lu (ξ)|)

(3.6) − 4s l

1 u δq (Ω)

.

(3.7)

On the other hand, we have |ˆ σu,k,ψ (ξ, η) − σ ˆu−1,k,ψ (ξ, η)| ≤ ΩL1 (2δq (Ω)klu |Lu (ξ)|),

(3.8)

which combined with (3.8) implies 1

|ˆ σu,k,ψ (ξ, η) − σ ˆu−1,k,ψ (ξ, η)| ≤ δq (Ω) ΩL1 (2δq (Ω)klu |Lu (ξ)|) δq (Ω) .

(3.9)

Now, the result follows by Lemma 2.6 and an induction argument. In order to carry out the induction argument, we only need to observe that the maximal function (σ0,ψ )∗ (f )(x, xd+1 ) = sup ||σ0,k,ψ | ∗ f (x, xd+1 )|

(3.10)

(σ0,ψ )∗ (f )p ≤ δq (Ω)Cp ΩL1 f p

(3.11)

k∈Z

satisfies for all 1 < p < ∞. The inequality (3.11) follows by Lemma 2.6. This completes the proof. Now, we move to the proof of Theorem 1.2. Proof of Theorem 1.2 Let Ω ∈ L(log L)(Sn−1 ) and satisfy (1.1). By the argument in [2], we can find a subset D of N0 = N ∪ {0}, a sequence {λm : m ∈ N0 } of non-negative real numbers, a constant B ≥ 1, and a sequence of functions {Am : m ∈ D} in L1 (Sn−1 ) such that  Am (y  )dσ(y  ) = 0, Sn−1

Am L1 (Sn−1 ) ≤ 1, Am L2 (Sn−1 ) ≤ 22(m+1) ,  λm δ2 (Am ) ≤ C ΩL(log L)(Sn−1 ) , m∈D

Ω(y) = B



λm Am (y  )

(3.12) (3.13) (3.14)

m∈D

with constant C independent of m. Thus,  (m) λm MΨ,ψ,2 (f )(x, xd+1 ) MΨ,ψ (f )(x, xd+1 ) ≤ m∈D (m)

where MΨ,ψ,2 has the same definition as MΨ,ψ,q in Theorem 1.1 with q = 2 and Ω is replaced by Am . By Theorem 1.1, we immediately obtain  (m) MΨ,ψ (f )p ≤ λm MΨ,ψ,2 (f )p m∈D

≤

  m∈D

 λm δ2 (Am ) Cp f p ≤ Cp ΩL(log L)(Sn−1 ) f p

935

for all 1 < p < ∞. This completes the proof. As a consequence of Theorem 1.2, we obtain the following result that will play an important role in the proof of Theorem 1.3: Corollary 3.1 Let Ψ = (Ψ1 , . . . , Ψd ) be a polynomial mapping from Rn into Rd . Suppose m ∈ N, Ω ∈ LLogL(Sn−1 ), and Φ is a homogeneous function of degree β = 0 and Φ ∈ L∞ (Sn−1 ). Let ψ : (0, ∞) → R be a function satisfying ψ(t) = θ(t) + ϕ(t), where θ is a polynomial and that ϕ(j) (0) = 0 for 0 ≤ j ≤ N and that ϕ(j) is positive nondecreasing on (0, ∞) for 0 ≤ j ≤ N + 1 where N = max{deg(Ψ), deg(θ)}. For m ∈ N and ε > 0, (0) (1) (2) let Eε,m , Eε,m , and Eε,m be the sets given by (0) Eε,m = (|y| > ε) ∩ (|y| < 22m ) ∩ (|y| ≥ 1), (1) Eε,m = (|y| > ε) ∩ (|y| > 22m ), (2) Eε,m = (|y| > ε) ∩ (|y| < 22m ). (0,m)

(1,m)

(2,m)

For m ∈ N, let MΨ,Φ,ψ,Ω , MΨ,Φ,ψ,Ω , and MΨ,Φ,ψ,Ω be given by    Ω(y  )  (0,m) iΦ(y)  e f (x − Ψ(y), xd+1 − ψ(|y|)) dy , MΨ,Φ,ψ,Ω (f ) = sup  (0) |y|n ε>0 Eε,m    Ω(y  )  (1,m) iΦ(y)  MΨ,Φ,ψ,Ω (f ) = sup  (e − 1)f (x − Ψ(y), xd+1 − ψ(|y|)) dy , (1) |y|n ε>0 E   ε,m   Ω(y  )  (2,m) iΦ(y)  MΨ,Φ,ψ,Ω (f ) = sup  (e − 1)f (x − Ψ(y), xd+1 − ψ(|y|)) dy . (2) |y|n ε>0 Eε,m

Then for all 1 < p < ∞, there exists a constant Cp > 0 independent of Ω and m such that (0,m)

(i) MΨ,Φ,ψ,Ω (f )p ≤ mΩLLog(Sn−1 ) Cp f p . (1,m)

(ii) MΨ,Φ,ψ,Ω (f )p ≤ mΩLLog(Sn−1 ) Cp f p provided that β < 0. (2,m)

(iii) MΨ,Φ,ψ,Ω (f )p ≤ mΩLLog(Sn−1 ) Cp f p provided that β > 0. The proof of Corollary 2.7 follows by an application of Theorem 1.2 and arguing in a similar way as in the proof of Lemma 2.2 in [1]. We omit details. 4

L2 Estimates of Truncated Maximal Oscillatory Operators

This section is devoted to prove the main estimates needed to present a proof of Theorem 1.3. We start by establishing the following lemma which can be considered as an analogue of Lemma 2.4 in [1]: Lemma 4.1 Suppose that Ω ∈ Lq (Sn−1 ), q > 1 is a homogeneous function of degree zero on Rn that satisfies ΩL1 ≤ 1. Let δq (Ω) = log(e + Ωq ). Suppose that Ψ = (Ψ1 , . . . , Ψd ) is a polynomial mapping from Rn into Rd and that Φ is a homogeneous function that satisfies (1.9)–(1.10) with either the index β = 0 is not a positive integer or β is a positive integer larger than max{deg(Ψj ) : 1 ≤ j ≤ d}. Suppose that ψ : (0, ∞) → R is a function satisfying ψ(t) = θ(t) + ϕ(t),

Al-Salman A. and Jarrah A. M.

936

where θ is a polynomial, ϕ(j) (0) = 0 for 0 ≤ j ≤ N and ϕ(j) is positive nondecreasing on (0, ∞) for 0 ≤ j ≤ N + 1 where N = max{deg(Ψ), deg(θ)}. Let ψk,q be a smooth function on R that dψ satisfies 0 ≤ ψk,q ≤ 1, supp(ψk,q ) ⊆ [2−δq (Ω)(k+1) , 2−δq (Ω)(k−1) ], and | duk,q (u)| ≤ Cu−1 with constant C independent of δq (Ω) and k. Then    ∞ β    i{t Φ(y  )−P(ty  )·ξ} ψk,q (t)dtdσ  βα(k+1)  Jk (Φ, ξ, η, Ω) =  Ω(y ) e  ≤ δq (Ω)C2 t Sn−1 0 for some constants 0 < α < 1 and C > 0 which are independent of δq (Ω), k, and the coefficients of Ψ1 , . . . , Ψd . Proof

By the properties of ψk,q and the fact that ΩL1 ≤ 1, we have Jk (Φ, ξ, Ω) ≤ 2δq (Ω) ln 2.

On the other hand, q

(Jk (Φ, ξ, Ω)) ≤ Next, let

q Ωq

 S

    n−1

  Ik (Φ, ξ) = 

∞

∞

β

e

i{t Φ(y  )−P(ty  )·ξ} ψk,q (t)

t

0

β

ei{t

Φ(y  )−P(ty  )·ξ} ψk,q (t)

t

0

q   dt dσ(y  ).

  dt .

Then by the support property of ψk,q , (4.1) reduces to    22δq (Ω) β   i{t Φ(y  )−P(ty  )·ξ} ψk,m (ak,q t)   dt, Ik (Φ, ξ) =  e t 1 where we set ak,q = 2−δq (Ω)β(k+1) . Now, by Van der Corput lemma (see [15]), we have   u β   1 i{t ak,m Φ(y  )−P(ak,q ty  )·ξ}   −l  e dt  ≤ C |ak,q Φ(y )| 

(4.1)

(4.2)

(4.3)

1

for all u > 1, where l is the degree of P and C is a constant independent of δq (Ω). Therefore, by (4.2), (4.3) and integration by parts, we obtain − 1l

Ik (Φ, ξ) ≤ C |ak,q Φ(y  )| where ψk,q (ak,q 22δq (Ω) ) C(k, q) = + 22δq (Ω)



22δq (Ω)

1

C(k, q),

(4.4)

    (ak,q tψk,q (ak,q t) − ψk,q (ak,q t)   .   t2

By the properties of ψk,q , we immediately obtain C(k, q) ≤

1 22δq (Ω)

−

2 22δq (Ω)

+2=1−

1 22δq (Ω)

≤ 1.

(4.5)

Thus, by (4.4) and (4.5), we get − 1l

Ik (Φ, ξ) ≤ C |ak,q Φ(y  )|

;

(4.6)

which interpolated with the trivial estimate Ik (Φ, ξ) ≤ 2δq (Ω) ln 2, implies that − δl

Ik (Φ, ξ) ≤ δq (Ω)C |ak,q Φ(y  )|

.

(4.7)

937

By (4.7) and (1.10), we obtain δ

Jk (Φ, ξ, Ω) ≤ δq (Ω)C2δq (Ω) |ak,q |− l .

(4.8)

Now, by an interpolation between the estimate σk (Φ, q, Ω) ≤ 2δq (Ω) ln 2 and (4.8), we get the desired result. This completes the proof.  We end this section by the following theorem: Theorem 4.2 Suppose that Ω is a homogeneous function of degree zero on Rd and that Ω ∈ L2 (Sn−1 ) with ΩL1 ≤ 1. Suppose also that Ψ = (Ψ1 , . . . , Ψd ) is a polynomial mapping and that ψ : (0, ∞) → R is a function satisfying ψ(t) = θ(t) + ϕ(t), where θ is a polynomial, ϕ(j) (0) = 0 for 0 ≤ j ≤ N and ϕ(j) is positive nondecreasing on (0, ∞) for 0 ≤ j ≤ N + 1 where N = max{deg(Ψ), deg(θ)}. Then the operator     −n ∗   f (x − Ψ(y), xd+1 − ψ(|y|))Ω(y )h(|y|) |y| dy  SΨ,Ω,h (f )(x, xd+1 ) = sup  ε>0

satisfies

|y|>ε

  ∗ SP,Ω,h (f ) ≤ δ2 (Ω) h Cp f  ∞ p p

for 1 < p < ∞. Here, Cp is a positive constant independent of δ2 (Ω) and the coefficients of the polynomial mappings {Ψj : 1 ≤ j ≤ d}. ˜ (lj ) , R(lj ) , and R be as in the proof of Theorem 1.1. For Proof Let dΨ , Ψj , Ψ(l) , M, AΨ , lj , Ψ 1 ≤ u ≤ M, k ∈ Z, let λu,k,ψ be the measure given by (3.4) with |Ω(y  )| is replaced by Ω and δq (Ω) is replaced by δ2 (Ω). By assumption and similar argument as that led to (3.9) and (3.7), we get λu,k,ψ  ≤ δq (Ω)C h∞ , ˆ u,k,ψ (ξ, η)| ≤ δq (Ω)A h (2δq (Ω)klu |Lu (ξ)|) |λ ∞

− 4s l

1 u δq (Ω)

,

ˆ u,k,ψ (ξ, η) − λ ˆ u−1,k,ψ (ξ, η)| ≤ δq (Ω) h (2δq (Ω)klu |Lu (ξ)|) |λ ∞

(4.9) 1 δq (Ω)

.

(4.10)

Here, we should observe that λ0,k,ψ = 0. On the other hand, by Theorem 2.4, we get   ∗ λu,ψ (f ) ≤ δq (Ω)C h f  ∞ p p for 1 ≤ u ≤ M and 1 < p < ∞. Thus, by a similar argument as in [2, Theorem 2.2], the result follows. This completes the proof.  5

Lp Boundedness of Truncated Maximal Oscillatory Operators

This section is devoted to the proof of Theorem 1.3. Proof of Theorem 1.3 Assume that Ω ∈ L log L(Sn−1 ) and satisfies (1.1). We let D, {λm : ∗ be m ∈ N}, and {Am : m ∈ D} be as in the proof of Theorem 1.2. For m ∈ D, let TΦ,Ψ,ψ,A m

Al-Salman A. and Jarrah A. M.

938

given by (1.8) with Ω is replaced by Am . By (3.14), we get  ∗ ∗ f (x, xd+1 ) ≤ B λm TΦ,Ψ,ψ,A (x, xd+1 ). TΦ,Ψ,ψ,Ω m

(5.1)

m∈D

By the property (3.13), we only need to prove that   ∗ TΦ,Ψ,ψ,A f  ≤ δ2 (Am )Cp f  p m p for all 1 < p < ∞ with constant Cp independent of m. For each m, we let δm = δ2 (Am ) = log(2 + Am L2 ). Choose a collection of C ∞ functions {ψk,m }k∈Z on (0, ∞) with the properties:  ψk,m (u) = 1, supp(ψk,m ) ⊆ [2−δm (k+1) , 2−δm (k−1) ], 0 ≤ ψk,m ≤ 1, k∈Z

  s   d ψk,m −s    dus (u) ≤ Cs u

with constants Cs independent of m (see [2] for more details). We shall present the proof when β > 0. The case where β < 0 follows by minor modifications. As in [1], we set η(y) =

∞ 

ψk,m (|y|),

θ(y) =

k=0

−1 

ψk,m (|y|);

k=−∞

Km,∞ (y) = Am (y  )θ(y); Km,0 (y) = Am (y  )η(y). Then, it is clear that supp(Km,∞ ) ⊂ {y ∈ Rn : |y| ≥ 1};

(5.2)

Km,∞ (y) = Am (y  ) for all |y| > 22δ2 (Am )+2 ;

(5.3)

2δ2 (Am )+2

supp(Km,0 ) ⊂ {y ∈ R : |y| ≤ 2 n

};

θ(y) + η(y) = 1.

(5.4) (5.5)

By the identity (5.5), we get ∗ ∗ ∗ f (x, xd+1 ) ≤ TΦ,Ψ,ψ,K f (x, xd+1 ) + TΦ,Ψ,ψ,K f (x, xd+1 ). TΦ,Ψ,ψ,A m m,∞ m,0

(5.6)

Notice that by (5.2)–(5.3), it follows that ∗ f (x, xd+1 ) TΦ,Ψ,ψ,K m,∞ (0,m)

(1,m)

∗ (f )(x, xd+1 ) + MΨ,Φ,ψ,Am (f )(x, xd+1 ) + MΨ,Φ,ψ,Am (f )(x, xd+1 ), ≤ SΦ,A m ,hm+1 (0,m)

(1,m)

where hm = χ{|y|>22δ2 (Am ) } , MΨ,Φ,ψ,Am and MΨ,Φ,ψ,Am are as in Corollary 2.7, and    Am (y  )  ∗  SΦ,Am ,hm+1 f (x, xd+1 ) = sup  f (x − Ψ(y), xd+1 − ψ(|y|)) n dy . |y| ε>0 (|y|>ε)∩(|y|>22δ2 (Am )+2 ) Thus, by Corollary 2.7 and Theorem 3.2, we obtain ∗ (f )p ≤ δ2 (Am )Cf p TP,Φ,K m,∞

for all 1 < p < ∞.

(5.7)

939

Next, by (5.4), we have ∗ TΦ,Ψ,ψ,K f (x, xd+1 ) = m,0

   ∞ sup  0<ε<2m

|y|>ε

k=0

where Gk,m (y) =

  eiΦ(y) f (x − Ψ(y), xd+1 − ψ(|y|))Gk,m (y)dy , (5.8) Am (y  )ψk,m (|y|) . n |y|

As in [1], it can be easily verified that (m)

∗ f (x, xd+1 ) ≤ MΨ,ψ,2 (f )(x, xd+1 ) + I(f )(x, xd+1 ), TΦ,Ψ,ψ,K m,0

where

   ∞  I(f )(x, xd+1 ) =  k=0

e

iΦ(y)

2mj ≤|y|<2m

(5.9)

  f (x − Ψ(y), xd+1 − ψ(|y|))Gk,m (y)dy 

(m)

and MΨ,ψ,2 is the operator given in (1.4) with Ω is replaced by Am and δq = δ2 (Am ) ≈ m. By Theorem 1.1, we have (m)

MΨ,ψ,2 (f )p ≤ δ2 (Am )Cf p

(5.10)

for 1 < p < ∞. Next, using the support properties of ψk,m , it can be easily seen that (m)

I(f )(x, xd+1 ) ≤ R(f )(x, xd+1 ) + 2MΨ,ψ,2 f (x, xd+1 ), where

   1−j  R(f )(x, xd+1 ) = sup  j<1 k=0

e

iΦ(y)

Rn

  f (x − Ψ(y), xd+1 − ψ(|y|))Gk,m (y).

Let σm,k be the measure defined by   −n f dσm,k = eiΦ(y) |y| Am (y  )ψk,m (|y|)f (Ψ(y), ψ(|y|))dy. Then

(5.11)

   ∞  1−j    σm,k ∗ f (x, xd+1 ) ≤ Rm,k (f )(x, xd+1 ), R(f )(x, xd+1 ) = sup  j<1 k=0

(5.12)

(5.13)

k=0

where Rm,k (f )(x, xd+1 ) = sup |σm,k−j ∗ f (x, xd+1 )| . j<1

Notice that (m)

Rm,k (f )(x, xd+1 ) ≤ MΨ,ψ,2 (f )(x, xd+1 ).

(5.14)

Thus by Theorem 1.1, we get Rm,k (f )p ≤ δ2 (Am )C f p for 1 < p < ∞. On the other hand, by Plancherel’s theorem and the observation that Rm,k (f )(x, xd+1 ) ≤

1  j=−∞

|σm,k−j ∗ f (x, xd+1 )| ,

(5.15)

Al-Salman A. and Jarrah A. M.

940

we have Rm,k (f )2 ≤

1 

σm,k−j ∗ f 2 ≤ f 2

j=−∞

1 

Jk−j (Φ, ξ, η, Am ),

(5.16)

j=−∞

where Jk−j (Φ, ξ, η, Am ) is given by Lemma 3.1. Thus, by (5.16) and Lemma 3.1, we have 1 

Rm,k (f )2 ≤ δ2 (Am )2−|β|αk f 2

2|β|αj ≤ δ2 (Am )2−|β|αk C f 2 .

(5.17)

j=−∞

By interpolation between (5.15) and (5.17), we get Rm,k (f )p ≤ δ2 (Am )2−|β|αk C f p

(5.18)

for 1 < p < ∞. Thus, by (5.13) and (5.18), we obtain Rm (f )p ≤ δ2 (Am )C f p for 1 < p < ∞. By (5.7), (5.9), (5.10), (5.11), and (5.19), we get    ∗  TΦ,Ψ,ψ,Km,0 (f ) ≤ δ2 (Am )C f p

(5.19)

(5.20)

p

for all 1 < p < ∞. Hence, the result follows by (5.6), (5.7) and (5.20). This completes the proof. 6

Additional Results

As pointed out in Introduction, we shall present in this section some additional results that can be obtained using Theorem 1.1. Our aim in this section is operators with kernels in block spaces. In their study of singular integral operators, Jiang and Lu [10] introduced a special class

of block spaces. In the following, we recall the block spaces Bq0,0 Sn−1 , q > 1. A function Ω

is in Bq0,0 Sn−1 if ∞  Ω= cμ bμ , μ=1

where for each μ, cμ is a complex number and bμ is a function defined on I = B(x0 , θ0 ) = {x ∈ Sn−1 : |x − x0 | < θ0 } − q1

and satisfies bLq ≤ |I|

and Mq ({cμ }) =

∞ 

|cμ | (1 + φ (|Iμ |)) < ∞,

μ=1



where φ (t) ∼ log t−1 as t → 0. Here, x0 ∈ Sn−1 and 0 < θ0 ≤ 2. It is known that



C 1 (S n−1 ) ⊂ Lq (Sn−1 ) ⊂ Bq0,0 Sn−1 .

Moreover,



L log+ L(Sn−1 )  Bq0,0 Sn−1

and



Bq0,0 Sn−1  L log+ L(Sn−1 ).

941

Using the Theorem 1.1 and the same technique as in previous sections, we can prove the following results. Theorem 6.1 Suppose that Ω is a homogeneous function of degree zero on Rn and that

Ω ∈ Bq0,0 Sn−1 , q > 1. Suppose also that Ψ = (Ψ1 , . . . , Ψd ) is a polynomial mapping and that ψ : (0, ∞) → R is a function satisfying ψ(t) = θ(t) + ϕ(t), where θ is a polynomial, ϕ(j) (0) = 0 for 0 ≤ j ≤ N and ϕ(j) is positive nondecreasing on (0, ∞) for 0 ≤ j ≤ N + 1 where N = max{deg(Ψ), deg(θ)}. Then MΨ,ψ,Ω (f )p ≤ Cp f p

(6.1)

for 1 < p < ∞. Here, Cp is a positive constant independent of the coefficients of the polynomial mappings {Ψj : 1 ≤ j ≤ d} and the polynomial θ. Theorem 6.2 Suppose that Ω is a homogeneous function of degree zero on Rn that satisfies

(1.1) and that Ω ∈ Bq0,0 Sn−1 , q > 1. Suppose also that Ψ = (Ψ1 , . . . , Ψd ) is a polynomial mapping. Let Φ be a homogeneous function of degree β = 0 such that Φ |Sn−1 is real analytic where β is not a positive integer or β is a positive integer larger than max{deg(Ψj ) : 1 ≤ j ≤ d}. Suppose that ψ : (0, ∞) → R is a function satisfying ψ(t) = θ(t) + ϕ(t), where θ is a polynomial, ϕ(j) (0) = 0 for 0 ≤ j ≤ N and ϕ(j) is positive nondecreasing on (0, ∞) ∗ is bounded on for 0 ≤ j ≤ N + 1 where N = max{deg(Ψ), deg(θ)}. Then the operator TΦ,Ψ,ψ,Ω p L for all 1 < p < ∞. Moreover, the operator norm is independent of the coefficients of the polynomial mappings {Ψj : 1 ≤ j ≤ d}. References [1] Al-Salman, A.: Maximal oscillatory singular integrals with kernels in L log L(Sn−1 ). Turkish J. Math., 29, 259–274 (2005) [2] Al-Salman, A., Pan, Y.: Singular integrals with rough kernels in Llog+ L(Sn−1 ). J. London. Math. Soc. (2), 66, 153–174 (2002) [3] Al-Salman, A., Al-Jarrah, A.: Rough oscillatory singular integral operators-II. Turkish J. Math., 27(4) (2003) [4] Calder´ on, A. P., Zygmund, A.: On the existence of certain singular integrals. Acta Math., 88, 85–139 (1952) [5] Calder´ on, A. P., Zygmund, A.: On singular integrals. Amer. J. Math., 78, 289–309 (1956) [6] Duoandikoetxea, J., Rubio de Francia, J. L.: Maximal and singular integral operators via Fourier transform estimates. Invent. Math., 84, 541–561 (1986) [7] Fan, D., Pan, Y.: Singular integral operators with rough kernels supported by subvarieties. Amer. J. Math., 119, 799–839 (1997) [8] Fan, D., Yang, D.: Certain maximal oscillatory singular integrals. Hiroshima Math. J., 28, 169–182 (1998) [9] Fefferman, R.: A note on singular integrals. Proc. Amer. Math. Soc., 74, 266–270 (1979) [10] Jiang, Y., Lu, S. Z.: Oscillatory singular integrals with rough kernels, in “Harmonic Analysis in China, Mathematics and Its Applications”, Vol. 327, Kluwer Academic Publishers, 1995, 135–145 [11] Kim, W., Wainger, S., Wright, J., et al.: Singular integrals and maximal functions associated to surfaces of revolution. Bull. London Math. Soc., 28, 291–296 (1996) [12] Nestlerode, W. C.: Singular integrals and maximal functions associated with highly monotone curves. Trans. Amer. Math. Soc., 267, 435–444 (1981)

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[13] Ricci, F., Stein, E. M.: Harmonic analysis on nilpotent groups and singular integrals I: Oscillatory integrals. J. Funct. Anal., 73, 179–194 (1987) [14] Stein, E. M.: Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, NJ, 1970 [15] Stein, E. M.: Harmonic Analysis: Real-Variable Mathods, Orthogonality and Oscillatory Integrals, Princeton University Press, Princeton, NJ, 1993 [16] Stein, E. M., Wainger, S.: Problems in harmonic analysis related to curvature. Bull. Amer. Math. Soc., 26 1239–1295 (1978)