Let d≥2 (d=n or d=m) and Sd−1 be the unit sphere in ℝd equipped with the normalized Lebesgue measure dσ. Let Ω∈L1(Sn−1) be a homogeneous function of degree zero on ℝn satisfying

(1.1)∫Sn−1Ω(y′)dσ(y′)=0

where y′=y|y|∈Sn−1 for y≠0. The Marcinkiewicz integral operator μΩ is given by

(1.2)μΩ(f)(x)=∫−∞∞∫|y|<2tf(x−y)Ω(y′)|y|n−1dy2dt22t12.

In [18], E. M. Stein established the Lp boundedness (1<p≤2) of μΩ provided that Ω∈Lipα(0<α≤1). Subsequently, A. Benedek, A. Calderón, and R. Panzone proved the Lp boundedness of μΩ under the stronger condition that Ω∈C1(Sn−1) [10]. Since then, several authors have studied the Lp boundedness of μΩ under various conditions on the function Ω. A particular result that is of interest to us in this paper is the main result in [12]. In [12], Fan and Pan proved that μΩ is bounded on Lp for all p∈2+2ϵ1+2ϵ,2+2ϵ provided that Ω satisfies

(1.3)supξ′∈Sn−1∫Sn−1|Ω(y′)|log1ξ′ ⋅y′ 1+ϵdσ(y′)<∞

for some ϵ>0. For ϵ>0, we let F(ϵ,Sn−1) be the space of all integrable functions on Sn−1 that satisfy the condition (1.3). The set of conditions (1.3) were introduced by Grafakos and Stevanov in [17]. Grafakos and Stevanov showed that

F(ϵ,Sn−1)⊈L(log+L)(Sn−1)andL(log+L)(Sn-1)⊈F(ϵ,Sn-1).

Furthermore, it can be easily seen that

∪ q>1Lq(Sn−1)⊂F(ϵ,Sn−1),ϵ>0

For additional background information and related results on the operator μΩ, we advice readers to consult [2], [4]-[9], [12], and [15], among others.

Our aim in this paper is to study the Lp boundedness of a related class of Marcinkiewicz integral operators on product domains. For suitable functions Φ,Ψ:ℝ+→ℝ and Ω∈L1(Sn−1×Sm−1) satisfying

(1.4)∫Sn−1Ω(u′,.)dσ(u′)=∫Sm−1Ω(.,v′)dσ(v′)=0

and

(1.5)Ω(tx,sy)=Ω(x,y)

for any t,s>0, we define the associated Marcinkiewicz integral operator on ℝn×ℝm by

(1.6)MΩ,Φ,Ψf(x,y)=∫ −∞∞ ∫ −∞ ∞ F t′,s′Φ,Ψ(f)(x,y)2dt′ds′ 22( t ′ + s ′ ) 12,

where

(1.7)Ft′,s′Φ,Ψ(f)(x,y)=∫ ∫ Λ( t ′, s ′)f(x−Φ(|u|)u′,y−Ψ(|v|)v′)Ω(u′,v′)|u|n−1|v|m−1dudv

and Λ(t′,s′)={(u,v)∈ℝn×ℝm:|u|≤2t′and|v|≤2s′}.

For the sake of simplicity, we denote MΩ,Φ,Ψ by MΩ,c when Φ(t)=Ψ(t)=t. In [14], Ding proved that the operator MΩ,c is bounded on L2(ℝn×ℝm) provided that Ω∈L(log+L)2(Sn−1×Sm−1). Subsequently, Chen, Fan, and Ying extended Ding's result to Lp for all 1<p<∞ [11]. The condition Ω∈L(log+L)2(Sn−1×Sm−1) was very much relaxed by AL-Qassem, Al-Salman, Pan, and Chang in [2]. In fact, the authors of [2] proved that MΩ,c is bounded on Lp for all 1<p<∞ provided that kernel satisfies the weaker condition Ω∈L(logL)(Sn−1×Sm−1) (see [13] for the case p=2). In the same paper [2], the authors showed that condition Ω∈L(log+L)(Sn−1×Sm−1) can not be replaced by any condition in the form Ω∈L(log+L)α(Sn−1×Sm−1) for some α<1.

The main purpose of this paper is to investigate the Lp boundedness of MΩ,Φ,Ψ for mappings Φ and Ψ more general than polynomials and convex functions, provided that Ω satisfies the condition

(1.8)sup(ξ′,η′)∈(Sn−1×Sm−1)∫Sn−1×Sm−1|Ω(u′,v′)|{G(ξ′,η′)}1+ϵdσ(u′)dσ(v′)<∞,

for some ϵ>0, where

G(ξ′,η′)=log+(|ξ′⋅u′|−1)+log+(|η′⋅v′|−1)+log+(|ξ′⋅u′|−1)log+(|η′⋅v′|−1).

For ϵ>0, we let F(ϵ,Sn−1,Sm−1) be the class of all Ω∈L1(Sn−1×Sm−1) that satisfy (1.8). The class F(ϵ,Sn−1,Sm−1) is the analogy of the class F(ϵ,Sn−1) in the one parameter setting above. It is clear that, for any ϵ>0, we have

∪ q>1Lq(Sn−1×Sm−1)⊂F(ϵ,Sn−1,Sm−1).

Moreover, it was observed in [4] that

F(ϵ,Sn−1,Sm−1)⊈L(log+L)(Sn−1×Sm−1)

and

L(log+L)(Sn−1×Sm−1)⊈F(ϵ,Sn−1,Sm−1).

Historically, in [4], Al-Salman proved the Lp boundedness of MΩ,Φ,Ψ for all p∈2+2ϵ1+2ϵ,2+2ϵ provided that Ω satisfies (1.8) and the functions Φ and Ψ are either convex increasing functions or satisfy a growth condition in the form

(1.9)|φ(t)|≤C1td,  |φ′′(t)|≤C2td−2
(1.10)C3td−1≤|φ′(t)|≤C4td−1

where d≠0, t∈(0,∞) and C1,C2,C3 and C4 are positive constants independent of t.

In [3], Al-Salman introduced a class of functions generalizing the convexity property. To be more specific, a function ψ:[0,∞)→ℝ is said to belong to the class PCλ(d) (d>0) if there exist, λ∈ℝ, a polynomial P, and φ∈C0([0,∞)) such that

(1.11)(i)ψ(t)=P(t)+λφ(t)(ii)P(0)=0 and φ(j)(0)=0 for 0⩽j⩽d(iii)φ(j)is positive nondecreasing on (0,∞) for 0⩽j⩽d+1.

It was shown in [3] that the class ∪d≥0(PCλ(d)) contains properly the class of polynomials Pd as well as the class of convex increasing functions. The author of [3] pointed out that the function θ(t)=−t2+t2ln(1+t) is in PCλ(2) which is neither convex nor polynomial.

In light of the aforementioned discussion, it is natural to ask the following:

Question. Let MΩ,Φ,Ψ be given by (1.6) and assume that Ω∈F(ϵ,Sn−1,Sm−1) atisfying (1.4)-(1.5) for some ϵ>0. Suppose that Φ∈PCλ(d),Ψ∈PCα(b) for d,b>0 and λ,α∈ℝ Is MΩ,Φ,Ψ bounded on Lp for some 1<p<∞?

In the following theorem, we give an affirmative answer to the above question:

Theorem 1.1. Suppose that Ω∈F(ϵ,Sn−1,Sm−1) satisfying (1.4)-(1.5). If Φ∈PCλ(d),Ψ∈PCα(b) for d,b>0 and λ,α∈ℝ. Then MΩ,Φ,Ψ is bounded on Lp(ℝn×ℝm) for p∈2+2ϵ1+2ϵ,2+2ϵ with Lp bounds independent of λ,α∈ℝ and the coefficients of the particular polynomials involved in the standard representation (i) of Φ and Ψ in (1.11).

We remark here that Theorem 1.1 is a fundamental generalization of Theorem 1.1 in [4].

Throughout this paper, the letter C will denote of a constant that may vary at each occurrence but it is independent of the essential variables.

We start for the following result in [16]:

Lemma 2.1. ([16]) Suppose that P(y)=∑|α|=maαyα is polynomial of degree m on ℝn and ε<1m. Then there exists Aε>0 such that

∫Sn−1|P(y′)|−εdσ(y′)≤Aε‖P‖,

where

‖P‖=∑ |α|=m|aα|.

The bound Aε may depend on ε,m and n but it is independent of the coefficients of the polynomial.

Also, we shall need the following lemma in [1]:

Lemma 2.2.([1]) If φ∈Cd+1[0,∞) and satisfies the conditions i)-(ii) (1.11), then

(i) φ(αr)≤α φ(r)    for 0≤α≤1and r>0

(ii)  φ(αr)≥α φ(r)    for α≥1and r>0.

(iii)  φd+1(r)⩾r−d−1φ(r) for r>0.

The following well known theorem on maximal functions is significant:

Theorem 2.3. ([3]) Suppose that Υ:ℝn→ℝd is a non-constant mapping and assume that ψ∈PCλ(d) for some d⩾0 and λ∈ℝ. Suppose also that ρ>0. If Ω∈L1(Sn−1) is homogeneous of degree zero in ℝn, then the maximal function MΨ,Ω given by

(2.1)MΨ,Ω(f)(x)=supj∈ℤ∫ρj<|y|<ρj+1f(x−ψ(|y|)Υ(y′))Ω(y′)|y|n dy

satisfies

‖MΨ,Ω(f)‖p≤Cp‖Ω‖1‖f‖p.

for 1<p<∞. Here, the constant Cp is independent of λ,Υ(y′) and the coefficients of the particular polynomials involved in the representation (1.11) of ψ.

Now, we move to obtain the needed oscillatory estimates. For Ω∈L1(Sn−1×Sm−1) and suitable mappings Φ,Ψ:ℝ+→ℝ, we define the family of measures {σΦ, Ψ,Ω,t′,s′:t′,s′∈ℝ} by

(2.2)∫ℝn×ℝnfdσΦ, Ψ,Ω,t′,s′ =∫Γ(2t′,2s′)f(x−Φ(|u|)u′,y−Ψ(|v|)v′)Ω(u′,v′)|u|n−1|v|m−1 dudv

where

Γ(2t′,2s′)={(u,v)∈ℝn×ℝm:2t′−1<|u|≤2t′and 2s′−1<|v|≤2s′}.

The corresponding maximal function is defined by

(2.3)(σΦ, Ψ,Ω)∗f(x,y)=supt′,s′∈ℝ|σΦ, Ψ,Ω,t′,s′|∗(f)(x,y).

For simplicity, we shall let

σt′,s′=σΦ, Ψ,Ω,t′,s′.

Now, for Φ∈PCλ1(d) and Ψ∈PCλ2(b) for some b,d>0, let

Φ(t)=P(t)+λ1φ1(t)Ψ(r)=Q(r)+λ2φ2(r),

where λ1,λ2∈ℝ, P∈Pd, Q∈Pb, φ1∈Cd+1[0,∞), and φ2∈Cb+1[0,∞). Let

(2.4)P(t)=∑ k=0dck,1tk    and    Q(t)=∑ k=0bck,2tk.

For 0≤l≤d and 0≤s≤b, let

(2.5)Pl(t)=∑ k=0 lck,1tk    and    Qs(t)=∑ k=0 sck,2tk,

where we use the convention that ∑ j∈Θ=0. Now, by (2.2), we defined the family of measure {σt′,s′(d+1,b+1):t′,s′∈ℝ} via the Fourier transform by

(2.6)σ^t′,s′(d+1,b+1)(ξ,η)=2−(t′+s′)∫ ∫ Γ( 2 t ′, 2 s ′)e−iΦ(|u|) ξ.u′+Ψ(|v|) η.v′ Ω(u′,v′)|u|n−1|v|m−1dudv.

For 0≤l≤d,0≤s≤b, we defined the family of measures {σt′,s′(l,s):t′,s′∈ℝ} by

(2.7)σ^t′,s′(l,s)(ξ,η)=2−(t′+s′)∫ ∫ Γ( 2 t ′, 2 s ′)e−i Pl (|u|)ξ.u′+Qs (|v|)η.v′ Ω(u′,v′)|u|n−1|v|m−1dudv.

It is clear that

σ^t′,s′(0,0)=σ^t′,s′(0,b+1)=σ^t′,s′(d+1,0)=0.

By (2.6)-(2.7), we have

(2.8)MΩ,Φ,Ψ(f)(x,y)=∫−∞∞ ∫ −∞ ∞ σt′,s′(d+1,b+1)∗(f)(x,y)2dt′ds′12.

We have the following two lemmas:

Lemma 2.4. Let {σt′,s′(d+1,b+1):t′,s′∈ℝ} be the measures given in (2.6). Suppose that Ω∈F(ϵ,Sn−1,Sm−1) for some ϵ>0. Then

(i) ‖σt′,s′(d+1,b+1)‖≤C;

(ii) σ^t′,s′(d+1,b+1)(ξ,η)≤Clog+|λ1φ1(2t′−1)ξ|−1−ϵlog+|λ2φ2(2s′−1)η|−1−ϵ;

(iii) σ^t′,s′(d+1,b+1)(ξ,η)−σ^t′,s′(d,b+1)(ξ,η)≤Cλ1φ1(2t′)ξlog+|λ2φ2(2s′−1)η|−1−ϵ;

(iv) σ^t′,s′(d+1,b+1)(ξ,η)−σ^t′,s′(d+1,b)(ξ,η)≤Cλ2φ2(2s′)ηlog+|λ1φ1(2t′−1)ξ|−1−ϵ;

(v) σ^t′,s′(d+1,b+1)(ξ,η)−σ^t′,s′(d,b+1)(ξ,η)−σ^t′,s′(d+1,b)(ξ,η)+σ^t′,s′(d,b)(ξ,η)≤C|λ1φ1(2t′)ξ||λ2φ2(2s′)η|;

(vi) σ^t′,s′(d+1,b)(ξ,η)−σ^t′,s′(d,b)(ξ,η)≤C|λ1φ1(2t′)ξ|;

(vii) σ^t′,s′(d,b+1)(ξ,η)−σ^t′,s′(d,b)(ξ,η)≤Cλ2φ2(2s′)η.

Here C is independent of t′,s′∈ℝ and (ξ,η)∈ℝn×ℝm.

Lemma 2.5. Let {σt′,s′(l,s):0≤l≤d,0≤s≤b} be as in (2.7. Suppose that Ω∈F(ϵ,Sn−1,Sm−1) for some ϵ>0.Then

(i) ‖σt′,s′(l,s)‖≤C;

(ii) σ^t′,s′(l,s)(ξ,η)≤Clog+|cl,1(2 t′ −1)ll!ξ|−1−ϵlog+|cs,2(2 s′ −1)ss!η|−1−ϵ;

(iii) σ^t′,s′(l,s)(ξ,η)−σ^t′,s′(l−1,s)(ξ,η)≤C|cl,1(2t′)lξ|log+|cs,2(2 s′ −1)ss!η|−1−ϵ;

(iv) σ^ω,t′,s′(l,s)(ξ,η)−σ^ω,t′,s′(l,s−1)(ξ,η)≤Ccs,2(2s′)sηlog+|cl,1(2t′ −1)ll!ξ|−1−ϵ;

(v) σ^t′,s′(l,s)(ξ,η)−σ^t′,s′(l−1,s)(ξ,η)−σ^t′,s′(l,s−1)(ξ,η)+σ^t′,s′(l−1,s−1)(ξ,η) ≤C|cl,1(2t′)lξ||cs,2(2s′)sη|;

(vi) σ^t′,s′(l,s−1)(ξ,η)−σ^t′,s′(l−1,s−1)(ξ,η)≤C|cl,1(2t′)lξ|;

(vii) σ^t′,s′(l−1,s)(ξ,η)−σ^t′,s′(l−1,s−1)(ξ,η)≤Ccs,2(2s′)sη.

Here C is independent of t′,s′∈ℝ and (ξ,η)∈(ℝn,ℝm).

Now, we shall start by presenting the proof of Lemma 2.4.

Proof of Lemma 2.4. To prove (i), we have

‖σt′,s′(d+1,b+1)‖⩽2−(t′+s′)∫ 2 t ′−1 2 t ′ ∫ 2 s ′ −1 2 s ′ ∫ S n−1 ×S m−1 | Ω(u′,v′)|dtdrdσ(u′)dσ(v′)    ⩽C‖Ω‖L1 ⩽C.

To get (ii), by polar coordinates, we have

(2.9)σ^ j,k (d+1,b+1)(ξ,η)= ∫Sn−1×Sm−1 ∫ 2 t′ −1 2 t′ ∫ 2 s ′ −1 2 s ′ 2 −( t ′+s ′) e −i Φ(t)ξ.u′ +Ψ(r)η.v′ Ω(u′,v′)dtdrσ(u′)dσ(v′)≤2−(t′+s′)∫S n−1 ×S m−1 |Ω(u′,v′)|AΦ,Ψ,t′ ,s′ dσ(u′)dσ(v′)

where

(2.10)A Φ,Ψ,t′,s′= ∫ 2 t ′−1 2 t ′ ∫ 2 s ′ −1 2 s ′ e−i Φ(t)ξ.u ′ +Ψ(r)η.v ′ dtdr.

By change of variables, we get

(2.11)AΦ,Ψ,t′,s′≤2t′+s′−2AΦ,t′AΨ,s′

where

AΦ,t′=∫12e−i(Φ(2t′−1t)ξ.u′dt

and

AΨ,s′=∫12e−i(Ψ(2s′−1r)η.v′dr.

By Lemma 2.2, we have

(2.12)Φ(d+1)(2t′−1t)≥C|λ1φ1(2t′−1)|

and

(2.13)Ψ(b+1)(2s′−1r)≥C|λ2φ2(2s′−1)|

where 2t′−1<t<2t′ and 2s′−1<r<2s′. Thus, by Van der Corput lemma in [19], we obtain

(2.14)AΦ,t′⩽|λ1φ1(2t′−1)ξ⋅u′|−1d+1

and

(2.15)AΨ,s′⩽|λ2φ2(2s′−1)η⋅v′|−1b+1.

Thus, by combining the trivial estimates AΦ,t′≤C and AΨ,s′≤C with the estimates (2.14)-(2.15), we have

AΦ,t′≤min{C,|λ1φ1(2t′−1)ξ⋅u′|−1d+1}

and

AΨ,s′≤min{C,|λ2φ2(2s′−1)η⋅v′|−1b+1}.

Therefore,

(2.16)AΦ,t′≤ C log+ |ξ ′ ⋅u ′ | −1d+1 log+ |λ1 φ1 (2 t ′−1 )ξ| 1d+1 1+ϵ  ≤C log+ |λ1 φ1 (2 t ′ −1 )ξ|−1−ϵ log+ 1 |ξ ′ ⋅u ′ | 1+ϵ,

where ξ′=ξ|ξ|. Similarly, we get

(2.17)AΨ,s′≤Clog+|λ2φ2(2s′ −1)η|−1−ϵlog+1|η′ ⋅v′ |1+ϵ.

Thus, we arrive at the following estimate

(2.18)AΦ,Ψ,t′,s′≤C2(t′+s′) log+ |λ1 φ1 ( 2 t ′ −1 )ξ|−1−ϵ log+ 1 |ξ ′ ⋅u ′ | 1+ϵ   log+ |λ 2 φ 2 ( 2 s ′ −1 )η|−1−ϵ log+ 1 |η ′ ⋅v ′ | 1+ϵ.

Finally, by (2.9), (2.18) and (1.8), we obtain

(2.19)σ^t,′s′(d+1,b+1)(ξ,η)≤Clog+|λ1φ1(2t′−1)ξ|−1−ϵlog+|λ2φ2(2s′−1)η|−1−ϵ

For the proof of (iii), we have

(2.20)σ^ t,′s′ (d+1,b+1)(ξ,η)−σ^ t,′s′ (d,b+1)(ξ,η)= ∫ Sn−1 ×Sm−1 ∫ 2 t ′ −1 2 t ′ ∫ 2 s ′ −1 2 s ′ Ω (u′,v′)e −iΨ(r)η. v ′ e −iΦ(t)ξ. u ′ − e −iP(t)ξ. u ′ dtdrdσ(u′)dσ(v′).

Thus, by Fubini's Theorem, we get

(2.21)σ^ t,′s′ (d+1,b+1)(ξ,η)−σ^ t,′s′ (d,b+1)(ξ,η)≤2−(t′+s′)∫ Sn−1 ∫ 2 t′ −1 2 t′ ∫ Sm−1 ∫ 2 s ′ −1 2 s ′ Ω (u ′ ,v ′ )e−iΨ(r)η. v′ drdσ(v ′ )⋅            e−iP(t)ξ.u′ e −iλ1φ1(t)ξ. u′−1dtdσ(u′)≤Cλ1φ1(2 t′)ξ∫ S n−1×S m−1|Ω(u′,v′)|dσ(u′)dσ(v′)∫12 ∫ 1 2 e−iΨ(2 s′ −1 r)η.v ′ drdt≤C‖Ω‖L1 λ1 φ1 (2 t ′ )ξ∫12A Ψ,s ′ dt

where φ1 is increasing, 2t′−1<t<2t′ and As′,Ψ as in (2.17). Thus,

(2.22)σ^t,′s′(d+1,b+1)(ξ,η)−σ^t,′s′(d,b+1)(ξ,η)≤C‖Ω‖L1λ1φ1(2t′)ξ log+ | λ 2 φ 2 ( 2 s ′ −1 )η| −1−ϵ log+ 1 |η ′ ⋅v ′ | 1+ϵ.

Finally, by combining (1.8) and (2.22), we establish the estimate (iii). Similarly, we can obtain the estimate (iv). We omit details.

Now, to get the estimate (v), we have

σ^ t,′s′ (d+1,b+1)(ξ,η)−σ^ t,′s′ (d,b+1)(ξ,η)−σ^ t,′s′ (d+1,b)(ξ,η)+σ^ t,′s′ (d,b)(ξ,η)≤2−(t′+s′)∫ Sn−1×Sm−1 ∫ 2t ′ −1 2t ′ ∫2 s′ −1 2 s′ | Ω(u′ ,v′ )||e −iλ1φ1(t)ξ −1||e −iλ2φ2(r)η −1|dtdrdσ(u′ )dσ(v′ )≤2−(t′+s′)|λ1φ1(2t′ )ξ||λ2φ2(2s′ )η|∫ S n−1×S m−1 ∫ 2 t ′−1 2 t ′ ∫ 2 s ′ −1 2 s ′ | Ω(u′ ,v′ )|dtdrdσ(u′ )dσ(v′ )≤C‖Ω‖L 1 |λ1φ1(2t′ )ξ||λ2φ2(2s′ )η|≤C|λφ1(2t′ )ξ||λ2φ2(2s′ )η|.

For the estimate (vi), we have

σ^ t,′s′ (d+1,b)(ξ,η)−σ^ t,′s′ (d,b)(ξ,η)≤2−(t′+s′)∫ S n−1×S m−1 ∫ 2t′−12t′ ∫2s′−12s′ |Ω(u′ ,v′ )| e −iλ1φ1(t)ξ⋅u ′ −1dtdrdσ(u′ )dσ(v′ ).

Now, since φ1 is increasing and 2t′−1<t<2t′, then by change of variables, we obtain

(2.23)σ^t,′s′(d+1,b)(ξ,η)−σ^t,′s′(d,b)(ξ,η)≤C|λ1φ1(2t′)ξ|.

Similarly, we can prove (vii). We omit details. This completes the proof.

Proof of Lemma 2.5. The proof of Lemma 2.5 follows the same procedure as in the proof of Lemma 2.4. We only need to notice here that

dldtPl(2t′−1t)=cl,1(2t′−1)ll!   and   dsdrQs(2s′−1r)=cs,2(2s′−1)ss!.

We omit details.

Now, we have the following lemma on the concerned maximal functions:

Lemma 2.6. Suppose that Ω∈L1(Sn−1×Sm−1) and Φ,Ψ:ℝ+→ℝ. Let MΦ,Ψ,Ω be the maximal function defined by

MΦ,Ψ,Ω(f)(x,y)=supt′,s′∈ℝ12t′+s′∫ ∫ Γ( 2 t ′ , 2 s ′ ) f(x−Φ(|u|)u′,y−Ψ(|v|)v′)Ω(u′,v′)|u|n−1|v|m−1dudv.

If Φ∈PCλ(d) and Ψ∈PCα(b) for some d,b>0 ,then

‖MΦ,Ψ,Ωf‖Lp≤C‖Ω‖L1‖f‖Lp.

Proof. For Φ∈PCλ(d) and Ψ∈PCα(b), let

MΨ,Ω(f)(x,y)=sups′∈ℝ12s′∫2s′ −1<|v|<2s′ f(⋅,y−Ψ(|v|)v′)Ω(⋅,v′)|v|m−1d(v)

and

MΦ,Ω(f)(x,y)=supt′∈ℝ12t′∫2t′ −1<|u|<2t′ f(x−Φ(|u|)u′),⋅)Ω(u′,⋅)|u|n−1d(u)

By using the observation MΦ,Ψ,Ω(f)≤MΦ,Ω∘MΨ,Ω(f) and Theorem 2.3, we get

(2.24)‖MΦ,Ψ,Ω(f)‖p≤‖MΦ,Ω∘MΨ,Ω(f)‖p    ≤Cp‖Ω‖1‖MΨ,Ω(f)‖p    ≤C‖Ω‖1‖f‖p,

where ∘ denotes the composition of operators. This ends the proof.