# A Note on Marcinkiewicz Integrals along Submanifolds of Finite Type

Authors: Daiqing Zhang
Journal: Journal of Function Spaces
 (2018)

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## Abstract

We study the parametric Marcinkiewicz integrals along submanifolds of finite type with rough kernels. The kernels of our operators are allowed to be very rough both on the unit sphere and in the radial direction. Under the rather weakened size conditions on the integral kernels, theLp bounds will be established for such operators. As applications, the corresponding results for parametric Marcinkiewicz integrals related to area integrals and Littlewood-Paley gλ⁎ functions are also given.


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## Body

## 1. Introduction

LetRn,  n≥2, be the n-dimensional Euclidean space and Sn-1 denote the unit sphere in Rn equipped with the induced Lebesgue measure dσ. Assume that Ω∈L1(Sn-1) is a homogeneous function of degree zero that satisfies(1)∫Sn-1Ωθdσθ=0.For a suitable mapping Γ:Rn→Rd(d≥1), we define the parametric Marcinkiewicz integral operator μh,Ω,Γ,ϱ on Rd by(2)μh,Ω,Γ,ϱfx=∫0∞1tϱ∫y≤tfx-ΓyΩyhyyn-ϱdy2dtt1/2,where f∈S(Rd) (the Schwartz class), ϱ=σ+iτ (σ>0,τ∈R), and h∈Δγ(R+). Here R+≔(0,∞) and Δγ(R+)(γ>0) denotes the set of all measurable functions h:R+→R satisfying(3)hΔγ=supj∈Z∫2j2j+1htγdtt1/γ<∞.If n=d and Γ(y)=y, we denote μh,Ω,Γ,ϱ=μh,Ω,ϱ.Whenh(t)=ϱ=1, the operator μh,Ω,ϱ reduces to the classical Marcinkiewicz integral operator μΩ, which was first introduced by Stein [1] and has been studied by many authors. For example, see [2] for the case Ω∈H1(Sn-1) (the Hardy space on the unit sphere (see [3, 4])), [5] for the case Ω∈L(log+L)1/2(Sn-1), [6] for the case Ω∈Bq(0,-1/2)(Sn-1) (the Block space generated by q-block), and [7] for the case Ω∈Fβ(Sn-1) (the Grafakos-Stefanov function class (see [8])). The study of parametric Marcinkiewicz integral operator μh,Ω,ϱ has attracted the attention of many authors; we refer the readers to consult [9–12] for relevant results. For general mapping Γ, the operator μh,Ω,Γ,ϱ has also been investigated extensively by many authors. For example, see [7, 13–16] for polynomial curves, [13, 14, 17, 18] for polynomial mapping, [18, 19] for homogeneous mappings, [14, 20, 21] for surfaces of revolution, and [18, 22] for submanifolds of finite type. For other interesting works related to this topic we can consult [23–29].We notice that the following inclusion relations are valid:(4)LrSn-1⊊Llog+Lβ1Sn-1⊊Llog+Lβ2Sn-1∀r>1,0<β2<β1;Llog+LβSn-1⊊H1Sn-1∀β≥1;Llog+LβSn-1⊈H1Sn-1⊈Llog+LβSn-1∀0<β<1.⋃r>1LrSn-1⊊Bq0,vSn-1∀q>1,v>-1;Bq0,v2Sn-1⊊Bq0,v1Sn-1∀q>1,v2>v1>-1;⋃q>1Bq0,vSn-1⊈⋃r>1LrSn-1∀v>-1;Bq0,vSn-1⊂H1Sn-1+Llog+L1+vSn-1∀q>1,v>-1.The main purpose of this paper is to study theLp boundedness of parametric Marcinkiewicz integrals along submanifolds of finite type with rough kernels as well as parametric Marcinkiewicz integrals related to area integrals and Littlewood-Paley gλ∗ functions. Precisely, let B(0,1)={x∈Rn:|x|<1} and Φ:B(0,1)→Rd be a smooth function. In what follows we always assume that Φ is of finite type at the origin; that is, for any ξ∈Sd-1, there exists a multi-index α=(α1,…,αn) such that |α|≥1 and ∂xα〈Φ(x),ξ〉|x=0≠0, where ∂xα=(∂/∂x1)α1⋯(∂/∂αn)αn, |α|=α1+⋯+αn, and the notation “〈·,·〉” denotes the inner product in Rd. We define the parametric Marcinkiewicz integral operator by(5)μh,Ω,Φϱfx=∫011tϱ∫y≤tfx-ΦyΩyhyyn-ϱdy2dtt1/2,where f∈S(Rd) and h,Ω,ϱ are given as in (2). When h(t)=ϱ=1, we denote μh,Ω,Φϱ by μΩ,Φ. Ding et al. [18] proved that μΩ,Φ is bounded on Lp(Rn) for all 1<p<∞, provided that Ω∈Lq(Sn-1) for some q>1. Later on, Al-Salman and Al-Qassem [22] extended the above result to the case Ω∈L(log+L)(Sn-1). To the best of our knowledge, there is nothing to introduce some relevant results on μh,Ω,Φϱ with h(t)≠1, even in the special case ϱ=1.On the other hand, the investigation of singular integrals along submanifolds of finite type has also attracted the attention of many authors (see [30–32]). In particular, in order to study singular integral operator along submanifolds of finite type with rough kernel both on the unit sphere and in the radial direction(6)Th,Ω,Φfx=p.v.∫B0,1fx-ΦyΩyhyyn.Sato [31] introduced the radial functions class Λγη(γ,η>0), which denotes the set of all functions h:R+→R satisfying(7)hΛγη=hΔγ+hΛη<∞.Here(8)hΛη=supt∈0,1t-ηωh,t<∞,where(9)ωh,t=sups<tR/2∫R2Rhr-s-hrdrr,t∈0,1,where the supremum is taken over all s and R such that |s|<tR/2 (see [33, 34]).We now introduce the result of [31] as follows.Theorem A (see [31]).
Leth∈Λ1η for some η>0. Suppose that Ω∈L(log+L)(Sn-1) and satisfies condition (1). Then(10)Th,Ω,ΦfLpRd≤CpfLpRdfor all 1<p<∞.Based on the above, we find that it is natural to ask the following question.Question B. Is the operator μh,Ω,Φϱ bounded on Lp(Rd) under the conditions Ω∈L(log+L)(Sn-1) and h∈Λ1η for some η>0?This question can be addressed in the affirmative by the following theorem.Theorem 1.
Leth∈Λ1η for some η>0 and Ω satisfy condition (1).(i)
IfΩ∈L(log+L)(Sn-1), then(11)μh,Ω,ΦϱfLpRd≤CphΛ1η1+ΩLlog+LSn-1fLpRdfor all p∈(1,2).(ii)
IfΩ∈L(log+L)1/2(Sn-1), then(12)μh,Ω,ΦϱfLpRd≤CphΛ1η1+ΩLlog+L1/2Sn-1fLpRdfor all p∈[2,∞).Actually, Theorem1 can be obtained by extrapolation arguments (see [17] or [35]) and the following result.Theorem 2.
LetΩ∈Lq(Sn-1) for some q∈(1,2] and h∈Λ1η for some η>0. Then(13)μh,Ω,ΦϱfLpRd≤CpAp,qhΛ1ηΩLqSn-1fLpRdfor all 1<p<∞. Here the constant Cp>0 is independent of h,Ω and Ap,q=(q-1)-1χ1<p<2+(q-1)-1/2χ2≤p<∞.Applying Theorem2 and extrapolation argument, we can obtain the following result.Theorem 3.
Leth∈Λ1η for some η>0 and Ω satisfy condition (1).(i)
IfΩ∈Bq(0,0)(Sn-1) for some q>1, then(14)μh,Ω,ΦϱfLpRd≤CphΛ1η1+ΩBq0,0Sn-1fLpRdfor all p∈(1,2).(ii)
IfΩ∈Bq(0,-1/2)(Sn-1) for some q>1, then(15)μh,Ω,ΦϱfLpRd≤CphΛ1η1+ΩBq0,-1/2Sn-1fLpRdfor all p∈[2,∞).Finally, we also consider the corresponding parametric Marcinkiewicz integral operatorsMh,Ω,Φϱ,λ,∗ and Mh,Ω,Φ,Sϱ related to the Littlewood-Paley gλ∗-function and the area integral, respectively, which are defined by(16)Mh,Ω,Φϱ,λ,∗fx≔∬R+n+1tt+x-ynλ1tϱ∫y≤tΩyhyyn-ϱfx-Φydy2dydttn+11/2,where λ>0, 1<q<∞, and R+n+1=Rn×(0,∞);(17)Mh,Ω,Φ,Sϱfx≔∬Sx1tϱ∫y≤tΩyhyyn-ϱfx-Φydy2dydttn+11/2,where 1<q<∞, S(x)={(y,t)∈R+n+1:|x-y|<t}, and Ω,h,ϱ are given as in (2); Φ is of finite type.As applications of Theorems1 and 3, we can get the following result.Theorem 4.
Letλ>0, h∈Λ1η for some η>0 and Ω∈L(log+L)1/2(Sn-1)∪(⋃q>1Bq(0,-1/2)(Sn-1)), satisfying condition (1). Then, for p∈[2,∞) and f∈Lp(Rd), the following inequalities are true:(18)Mh,Ω,Φϱ,λ,∗fLpRd≤CphΛ1ηfLpRd,Mh,Ω,Φ,SϱfLpRd≤CphΛ1ηfLpRd.Remark 5.
It should be point out that theLp bounds for Marcinkiewicz integrals along submanifolds of finite type for Ω∈H1(Sn-1) or Ω∈Fβ(Sn-1) are still unknown. Another inviting possibility is the investigation of the validity of Theorems 1-4 if h∈Λ1η for some η>0 is replaced by h∈Δγ(R+) for some γ≥1.The paper is organized as follows. Section2 is devoted to presenting some preliminary notations and lemmas, which play key roles in the proofs of our main results. The proofs of main results will be given in Section 3. We remark that the methods employed in this paper follow from a combination of ideas and arguments in [17, 30, 31, 35], among others. Throughout this note, we let p′ denote the conjugate index of p which satisfies 1/p+1/p′=1. The letter C will stand for positive constants not necessarily the same one at each occurrence but is independent of the essential variables.


## 2. Notations and Lemmas

Let us begin with introducing two lemmas, which follow from [30, 31].Lemma 6 (see [30] or [31]).
LetΦ:B(0,1)→Rn be smooth of finite type at the origin. Define Gm:B(0,1)×Sd-1→R by(19)Gmx,ξ=∑α=mξ,∂xαΦxxαfor m≥1 and ξ∈Sd-1. Then, there exist constants R,δ∈(0,1/4) and a mapping l from Sn-1 to a finite set of positive integers such that(20)CΦ≔supξ∈Sd-1∫x≤RGlξx,ξ-δdx<∞.Lemma 7 (see [30] or [31]).
Letψ,φ∈C∞(R) be real-valued. Let s∈(0,1] and a,b∈R with a<b. Suppose that φ is compactly supported and that(21)ddxkψx≤s∀x∈a,b,ddxk+1ψx≤1∀x∈a-s,b-s,where k is a positive integer. Then, there exists a positive constant C depending only on k and φ such that(22)∫abeiλψxφxdx≤Cλ-ϵ/k∫a-sb-sddxkψx-ϵ1+1/kdxfor all λ∈R∖{0} and ϵ∈(0,1].It was known that, for a functionF defined on an appropriate subinterval of R+ by F(t)=〈ξ,Φ(tx)〉 for fixed Sd-1 and x∈B(0,1), (d/dt)mF(t)=t-mGm(tx,ξ), where Gm is as in Lemma 6 (see [31]).Define a family of measures{σh,t}0<t<1 on Rd by(23)σh,t^ξ=1tϱ∫t/2<y≤te-2πiξ,ΦyΩyhyyn-ϱdy.Lemma 8.
LetΩ∈Lq(Sn-1) for some q∈(1,2] and h∈Λ1η for some η>0. Then there exist a positive integer N, a positive number ϵ0∈(0,1/4), and a positive constant C>0, such that(24)σh,t^ξ≤ChΛ1ηΩLqSn-1tNξ-ϵ0/q′.The constants N,ϵ0 and C are independent of h,Ω,q.Proof.
Lett∈(0,1) and Φ,δ,R and l(ξ) be as in Lemma 6. Without loss of generality, we assume that R≥2. Set l0=maxξ∈Sd-1l(ξ). Set(25)ddrlξ′,Φsrθ<14for 1≤l≤l0+1,  s∈[1,2],  r∈(0,5),  ξ′∈Sd-1, and θ∈Sn-1. By the changes of variables, one finds(26)σh,t^ξ=1tϱ∫t/2t∫Sn-1e-2πiξ,ΦrθhrΩθdσθrϱdrr=∫1/21∫Sn-1e-2πiξ,ΦtrθhtrΩθdσθrϱdrrfor any ξ∈Rd∖{0}. Choose a function ϕ∈C∞(R) satisfying supp(ϕ)⊂(0,10-9),ϕ≥1,∫Rϕ(s)ds=1. Define g(r)=∫s<r/2h(t(r-s))ϕu(s)ds,r>0, where ϕu(s)=1/uϕ(s/u),u>0. Then, if u<1, the following holds: (27)∫1/21htr-grdrr=∫1/21∫s<r/2htr-s-htrϕusdsdrr≤∫s<1/4∫1/21htr-s-htrdrrϕusds≤∫s<1/4∫t/2thr-ts-hrdrrϕusds≤Cωh,u.We take u=(tNξ)-ζ/q′ for a suitable N with N≥l0 and ζ>0, which will be specified below. We assume tN|ξ|≥1 and define(28)sh,tξ=∫1/21∫Sn-1e-2πiξ,ΦtrθgrΩθdσθrϱdrr.Then by (27) we get(29)σh,t^ξ-sh,tξ≤CΩL1Sn-1ωh,u≤CΩL1Sn-1hΛηtNξ-ηζ/q′.Let(30)Fς=∫1/2ςe-2πiξ,Φtrθrϱ-1dr,ς∈12,1.By Lemma 7, we have(31)Fς≤Cξ-ϵ/lξ′∫1/45/4Glξ′trθ,ξ′-ϵ/1+lξ′drfor ς∈(1/2,1], where ξ′=ξ/|ξ|. It is easy to see that |g(1)|≤Cu-1hΔ1 and ∫1/21g′rdr≤Cu-1hΔ1. Therefore, applying integration by parts and Hölder’s inequality, we see that(32)∫1/21e-2πiξ,Φtrθgrrϱ-1dr=g1F1+∫1/21Fug′udu≤Cu-1hΔ1ξ-ϵ/lξ′∫1/45/4Glξ′trθ,ξ′-ϵ/1+lξ′dr,∫Sn-1∫1/45/4Glξ′trθ,ξ′-ϵ/1+lξ′drrΩθdσθ≤Ct-n/q′∫x≤3Glξ′x,ξ′-2ϵΩx′x-n-1/qdx≤Ct-n/q′ΩLqSn-1∫x≤RGlξ′x,ξ′-2ϵq′dx1/q′,where ϵ∈(0,1]. Therefore, we have(33)∫1/21∫Sn-1e-2πiξ,ΦtrθgrΩθdσθrϱdrr≤Cu-1hΔ1ξ-ϵ/lξ′ΩLqSn-1t-n/q′∫x≤RGlξ′x,ξ′-2ϵq′dx1/q′,where C is independent of ϵ,q,h, and Ω. We put ϵ=δ/(2q′), then by Lemma 6 we have(34)sh,tξ≤CCΦ1/q′hΔ1ΩLqSn-1tNξ-ζ/q′t2nlξ′/δξ-δ/2q′lξ′.Therefore, if N is a positive integer such that N-1<2nl0/δ≤N and ζ<δ/(2l0),(35)sh,tξ≤CCΦ1/q′hΔ1ΩLqSn-1tNξ-δ/2q′l0-ζ/q′.This combining with (29) implies(36)σh,t^ξ≤ChΛ1ηΩLqSn-1tNξ-ϵ0/q′where ϵ0=min(ηζ,δ/(2q′l0)-ζ). If tN|ξ|≤1, the conclusion of Lemma 8 follows from the estimate |σh,t^(ξ)|≤ChΔ1ΩL1(Sn-1). This completes the proof of Lemma 8.Applying similar arguments in Lemma8 and [31, Lemma 2.5], we can obtain the following lemma.Lemma 9.
Leth∈Λ1η for some η>0, m∈N and let R be a real-valued polynomial on Rn with deg(R)≤m-1. Suppose that P(y)=∑|α|=mcαyα+R(y), Ω is a homogeneous function of degree zero, and Ω∈Lq(Sn-1) for 1<q≤2. Then there exists a constant C>0 independent of q,h,Ω, and the coefficients of the polynomial P such that(37)1tϱ∫t<y≤2te-iPyΩyhyyn-1dy≤ChΛ1ηΩLqSn-1tm∑α=mcα-τ/mq′,where τ=4-1min(1,η).ForΦ=(Φ1,…,Φd), let N be as in Lemma 8 and P=(P1,…,Pd) at the origin be defined by(38)Pιx=∑α≤N-11α!∂αΦι∂xα0xαfor 1≤ι≤d. For 1≤s≤N-1, define Ps=(P1,s,…,Pd,s) by(39)Pι,sx=∑α≤s1α!∂αΦι∂xα0xα,forι=1,…,d,0≤s≤N-1.Set PN=Φ. For each 0≤s≤N, let o(s) denote the number of multi-indices α∈(N∪{0})n satisfying |α|=s. Label the coordinate of Ro(s) by the of multi-indices α satisfying |α|=s. That is, Ro(s)={(xα)}|α|=s. For 0≤s≤N we define the linear transformations Ls:Rn→Ro(s) by(40)Lsξ=∑β=1s1α!∂αΦl∂xα0ξβα=s,where o(N)=N.Letλs=rank(Ls). By [36, Lemma 6.1], there exist two nonsingular linear transformations Rs:Rλs→Rλs and Qs:Rn→Rn such that(41)RsπλsnQsξ≤Lsξ≤CRsπλsnQsξ,where C depends only on n and πλsn is a projection operator from Rn to Rλs.For1≤s≤N,t∈(0,1) and ρ≥2, we define the family of measures {σh,ts} and the related maximal operators σh,s∗ and Mh,ρ,s on Rd by(42)σh,ts^ξ=1tϱ∫t/2<y≤te-2πiξ,PsyhyΩyyn-ϱdy;σh,s∗fx=supt∈0,1σh,ts∗fx;Mh,ρ,sfx=supk∈Z-∫ρkρk+1σh,ts∗fxdtt,where |σh,ts| is defined in the same way as σh,ts, but with Ω replaced by |Ω| and h replaced by |h|, Z-={-1,-2,…,}. By the definition of σh,ts, it is easy to see that σh,tN=σh,t.Lemma 10.
Letq∈(1,2], Ω∈Lq(Sn-1), and h∈Λ1η for some η>0. Then, for 1≤s≤N and t>0,ξ∈Rn, there exists a constant C>0 such that(43)maxσh,ts,σh,ts^ξ,σh,ts^ξ≤ChΛ1ηΩLqSn-1;(44)maxσh,ts^ξ-σh,ts-1^ξ,σh,ts^ξ-σh,ts-1^ξ≤ChΛ1ηΩLqSn-1tsLsξβs;(45)maxσh,ts^ξ,σh,ts^ξ≤ChΛ1ηΩLqSn-1tsLsξ-βs,where βs=τ/(q′s) for 1≤s≤N-1 and βN=ϵ0/(q′s).Proof.
By the straightforward calculations (43) holds. By Lemmas 8-9 we have(46)σh,ts^ξ≤ChΛ1ηΩLqSn-1tsLsξ-βs.Similarly,(47)σh,ts^ξ≤ChΛ1ηΩLqSn-1tsLsξ-βs,where βs=τ/(q′s) for 1≤s≤N-1 and βN=ϵ0/(q′s). Thus (45) holds. On the other hand, by a change of variable, we have(48)σh,ts^ξ-σh,ts-1^ξ≤1tϱ∫t<y≤2te-2πiLsξ,yαα=s-1Ωyhyyn-ϱdy≤ChΛ1ηΩLqSn-1tsLsξ,which combining with (43) implies that(49)σh,ts^ξ-σh,ts-1^ξ≤ChΛ1ηΩLqSn-1tsLsξβs.Similarly, we can obtain(50)σh,ts^ξ-σh,ts-1^ξ≤ChΛ1ηΩLqSn-1tsLsξβs.This completes the proof of Lemma 10.Motivated by the idea in [37], we have the following result, which will play a key role in the estimates on some vector-valued norm inequalities.Lemma 11.
LetΩ,h be as in Lemma 8 and ρ≥2. Then for 0≤s≤N and any 1<p<∞, there exists a constant C>0 such that(51)σh,s∗fLpRd≤Clog⁡ρhΛ1ηΩLqSn-1fLpRd;(52)Mh,ρ,sfLpRd≤Clog⁡ρhΛ1ηΩLqSn-1fLpRd.The constant C is independent of ρ,q,h, and Ω.Proof.
For convenience, we setA=(log⁡ρ)hΛ1ηΩLq(Sn-1). It is easy to verify that(53)σh,s∗fx≤supk∈Z-1ρk+1∫ρk<y≤ρk+1fx-PsyΩyhyyn-1dy.For 0≤s≤N, we define the family of measures {μk,s}k∈Z- and maximal operators μs∗ on Rd by(54)∫Rnfxdμk,sx=supk∈Z-1ρk+1∫ρk<y≤ρk+1Ωyhyyn-1fPsydu,μs∗fx=supk∈Z-μk,s∗fx,where |μk,s| is defined in the same way as μk,s, but with Ω replaced by |Ω| and h replaced by |h|. Thus, we have(55)σh,s∗fx≤μs∗fx.Therefore, to prove (51), it suffices to prove that(56)μs∗fLpRd≤CpAfLpRd,1<p<∞.Then by the proof of Lemma 10 and a straightforward calculation we get that, for 1≤s≤N,(57)μk,s^ξ≤CAmin⁡1,ρksLsξ-11/βs,(58)μk,s^ξ-μk,s-1^ξ≤CAmin1,ρksLsξ1/βs.In what follows, we prove (56) by induction on s.
Case  1. Without loss of generality, we may assume that f is nonnegative and fLp(Rd)<∞. It is easy to check that μ0∗(f)(x)≤CAf(x), which implies (56) for s=0.
Case  2. Suppose that (56) holds for s=m-1, m∈{1,…,N}. We shall prove (56) for s=m. Let ψ∈C0∞(R) be supported in {|x|≤1} and ψ(x)≡1 for |x|≤1/2. Define the Borel measures {ωk,m}k∈Z- on Rd by(59)ωk,m^ξ=μk,m^ξ-ψρkmRmπλmnQmξμk,m-1^ξfor ξ∈Rd, where Rm,πλmn,Qm are as in (41). It follows from (41) and (57)-(58) that(60)ωk,m^ξ≤CAmin1,ρkmLmξ,ρkmLmξ-11/βm.In addition, by (59) and a well-known result on maximal functions (see [36]), we have(61)σh,m∗fLpRd≤GmfLpRd+Cσh,m-1∗fLpRd,1<p<∞,ωm∗fLpRd≤GmfLpRd+Cσh,m-1⋆fLpRd,1<p<∞,where(62)ωm∗f=supk∈Z-ωk,m∗f,Gmf=∑k∈Z-ωk,m∗f21/2.By our assumption we have(63)μm-1∗fLpRd≤CpAfLpRd,1<p<∞,where Cp is independent of ρ,q,h, and Ω. It remains to prove that(64)GmfLpRd≤CpAfLpRd,1<p<∞,where Cp is as above. By a well-known property of Rademacher’s functions, (64) follows from(65)VϵmfLpRd≤CpAfLpRd,1<p<∞,where Vϵm(f)=∑k∈Z-ϵkωk,m∗f with ϵ=ϵk, ϵk=1 or -1. Below we prove (65). Choose a sequence of nonnegative functions {Ψk}k∈Z in C0∞(R) such that(66)suppΨk⊂ρ-mk+1,ρ-mk-1,∑k∈ZΨk2ξ=1,ddxjΨkx≤Cjx-jj=1,2,⋯,∀x>0,j∈N,where Cj are independent of ρ,t, and k. Define the Fourier multiplier operator Sj by(67)Sjf^ξ=ΨjRmπλmnQmξf^ξforj∈Z.Then(68)Vϵmf=∑k∈Z-ϵkωk,m∗∑j∈ZSj+kSj+kf=∑j∈Z∑k∈Z-ϵkSj+kωk,s∗Sj+kf≔∑j∈ZVjmf.By the Littlewood-Paley theory, we have(69)VjmfLpRd≤Cp∑k∈Z-ωk,m∗Sj+kf21/2LpRd,1<p<∞.This combining with Plancherel's theorem yields(70)VjmfL2Rd2≤C∑k∈Z-ωk,m∗Sj+kf21/2L2Rd2≤C∑k∈Z-∫Dj+kωk,m^ξ2f^ξ2dξ,where(71)Dk=ξ∈Rd:ρ-mk+1≤RmπλmnQmξ≤ρ-mk-1.We get from (41) and (60) that(72)VjmfL2Rd≤CAρ1-jm/βmχj≥1+ρ1+jm/βmχj<1fL2Rd,where βm is as in Lemma 10. (72) together with (68) yields that(73)VϵmfL2Rd≤CAfL2Rd.Thus,(74)GmfL2Rd≤CAfL2Rd,which combining the Littlewood-Paley theory and (60)-(63) with the proof of Lemma in [37, p.544] implies that(75)VjmfLpRd≤CpAfLpRd,p=4orp=43.Interpolating between (72) and (75) and combining with (68), we get(76)VϵmfLpRd≤CAfLpRd,43<p<4,which leads to(77)GmfLpRd≤CAfLpRd,43<p<4.Reasoning as above, (60)-(63), (77), the proof of Lemma in [37, p.544], the Littlewood-Paley theory, and interpolation imply(78)VϵmfLpRd≤CAfLpRd,87<p<8.By using this argument repeatedly, we can obtain ultimately that(79)VϵmfLpRd≤CAfLpRd,1<p<∞.This proves (65) and completes the proof of (51).
Now, we prove (52). It suffices to prove that(80)Mh,ρ,sfLpRd≤CpAfLpRdfor all 0≤s≤N and 1<p<∞, Cp is independent of Ω,h,q,γ, and the coefficients of Pj for all 1≤j≤n. We shall prove (80) by induction on s. When s=0, it is easy to see that(81)Mh,ρ,0fx≤CAfx,which implies (80) for s=0. Assuming that (80) holds for s=m-1,m∈{1,…,N}, we will prove that (80) holds for s=m. Let ψ∈C0∞(R) be as in (57). Define the family of measures {λk,m}k∈Z- by(82)λk,m^ξ=∫ρkρk+1σh,tm^ξdtt-ψρkmRmπλmnQmξ∫ρkρk+1σh,tm-1^ξdtt.By Lemma 10 and (41), one can easily check that(83)λk,m^ξ≤CAmin1,ρkmLmξ,ρkmLmξ-1βm.By the definition of λk,m and a well-known result on maximal function (see [36]), we have(84)Mh,ρ,mfLpRd≤gmfLpRd+CMh,ρ,m-1fLpRd,1<p<∞,(85)λm∗fLpRd≤gmfLpRd+CMh,ρ,m-1fLpRd,1<p<∞,where(86)gmf=∑k∈Z-λk,m∗f21/2,λm∗f=supk∈Z-λk,m∗f.From our assumption, (83)-(85), and similar arguments to those in getting (41), we get (80) for s=m. Thus (80) holds. Lemma 11 is proved.Applying Lemma11, we can obtain the following result.Lemma 12.
Leth,Ω be as in Lemma 8. Then, for 1≤s≤N and any 1<p<∞, there exists a constant C>0 such that(87)∑k∈Z-∫ρkρk+1σh,ts∗gk2dtt1/2LpRd≤Clog⁡ρ1/2×hΔ1ΩLqSn-1∑k∈Zgk21/2LpRd,2≤p<∞;(88)∑k∈Z-∫ρkρk+1σh,ts∗gk2dtt1/2LpRd≤Clog⁡ρ×hΔ1ΩLqSn-1∑k∈Zgk21/2LpRd,1<p<2The constant C is independent of ρ,q,h, and Ω.Proof.
We shall prove Lemma12 by using the method in [17]. First we prove (87). For fixed 2≤p<∞, by duality, there exists a nonnegative function f in Lp/2′(Rd) with fLp/2′(Rd)≤1 such that(89)∑k∈Z-∫ρkρk+1σh,ts∗gk2dtt1/2LpRd2=∫Rd∑k∈Z-∫ρkρk+1σh,ts∗gk2dttfxdx.By a change of variable and Hölder’s inequality, we obtain(90)σh,ts∗gkx2≤1tϱ∫t<y≤2tgkx-PsyhyΩyyn-1dy2≤ChΔ1ΩLqSn-1∫t2t∫Sn-1gkx-Psy2Ωθdσθhrdrr.Thus by (89)-(90) and Hölder’s inequality, one can check that(91)∑k∈Z-∫ρkρk+1σh,ts∗gk2dtt1/2LpRd2≤ChΔ1ΩLqSn-1∫Rd∑k∈Z-gkx2Mh,ρ,sf~-xdx≤ChΔ1ΩLqSn-1∑k∈Zgk21/2LpRd2Mh,ρ,sf~∥Lp/2′Rd,where f~(x)=f(-x). We get from (52) that(92)Mh,ρ,sf~Lp/2′Rd≤CAfLp/2′Rd≤CA,which combining with (91) implies (87). Next, we shall prove (88) for 1<p<2. By duality, there exist functions {fk(x,t)} defined on Rd×R+ with fk·,·Lp′(Rd,l2(L2([ρk,ρk+1],dt/t)))≤1 such that(93)∑k∈Z-∫ρkρk+1σh,ts∗gk2dtt1/2LpRd≤∫Rd∑k∈Z-∫ρkρk+1σh,ts∗gkxfkx,tdttdx≤Clog⁡ρ1/2∑k∈Zgk21/2LpRdHf1/2Lp′Rd,where(94)Hfx=∑k∈Z-∫ρkρk+1σh,ts∗f~kx,t2dtt,f~kx,t=fk-x,t.Since p′>2, there exists a nonnegative function u∈Lp′/2′Rd such that(95)HfLp′/2Rd=∑k∈Z-∫Rd∫ρkρk+1σh,ts∗f~kx,t2dttuxdx.By a similar argument to in (90) and (51), we have(96)HfLp′/2Rd≤ChΔ1ΩLqSn-1∫Rdσh,s∗u~-x∑k∈Z-∫ρkρk+1f~kx,t2dttdx≤ChΔ1ΩLqSn-1∑k∈Z∫ρkρk+1fkx,t2dttLp′/2Rdσh,s∗u~Lp′/2′Rd≤Clog⁡ρhΔ12ΩLqSn-12,where u~(x)=u(-x). Equation (88) follows from (93) and (96). This completes the proof of Lemma 12.


## 3. Proof of Theorem2

This section is devoted to the proofs of our main results.Proof of Theorem2.
Assume thath∈Δ1 and Ω∈Lq(Sn-1) for some q∈(1,2] satisfying (1). By Minkowski’s inequality, we can write(97)μh,Ω,Φϱfx=∫01∑k=-∞01tϱ∫2k-1t<y≤2ktfx-ΦyΩyhyyn-ϱdy2dtt1/2≤∑k=-∞0∫011tϱ∫2k-1t<y≤2ktfx-ΦyΩyhyyn-ϱdy2dtt1/2≤1-2-σ-1∫01σh,tN∗fx2dtt1/2.Let ψ be as in (57). For 1≤s≤N,  t∈(0,1), and ξ∈Rd, we define the family of measures {νt,s} by(98)νt,s^ξ=σh,ts^ξ∏s<j≤NψtjRjπλjnQjξ-σh,ts-1^ξ∏s-1<j≤NψtjRjπλjnQjξ.It is clear that(99)σh,tN=∑s=1Nνt,s.Here we use the convention Πj∈∅aj=1. By Lemma 10 and (41), we get(100)νt,s^ξ≤ChΛ1ηΩLqSn-1min⁡1,tsLsξ,tsLsξ-1βsfor 1≤s≤N. This together with a straightforward calculation yields that(101)∫2q′k2q′k+1νt,s^ξ2dtt1/2≤Cq-1-1/2hΛ1ηΩLqSn-1×min1,2q′ksLsξ,2q′ksLsξ-1βs.It follows from (97), (99), and Minkowski’s inequality that(102)μh,Ω,Φϱfx≤C∑s=1N∫01νt,s∗fx2dtt1/2≔C∑s=1Nμsfx.Choose a sequence of nonnegative functions {Ψk}k∈Z in C0∞(R) such that(103)suppΨk⊂2-sq′k+1,2-sq′k-1,∑k∈ZΨkξ=1,ddxjΨkx≤Cjx-jj=1,2,⋯for  allx>0,j∈N,where Cj are independent of ρ,k. Define the Fourier multiplier operator Γj by(104)Γjf^ξ=ΨjRsπλsnQsξf^ξforj∈Z.Then we have(105)μsfx≤∑i∈Zμs,ifx,where(106)μs,ifx=∑k∈Z-∫2q′k2q′k+1νt,s∗Γi+kfx2dtt1/2.Below we estimate the Lp estimates for μs,i. By Lemma 12 and the definition of νt,s, we have that, for 1≤s≤N, there exists a positive constant C, which is independent of ρ,q,h, and Ω such that(107)∑k∈Z-∫2q′k2q′k+1νt,s∗gk2dtt1/2LpRd≤Cq-1-1/2×hΔ1ΩLqSn-1∑k∈Zgk21/2LpRd,2≤p<∞;(108)∑k∈Z-∫2q′k2q′k+1νt,s∗gk2dtt1/2LpRd≤Cq-1-1×hΔ1ΩLqSn-1∑k∈Zgk21/2LpRd,1<p<2.By (107)-(108) and the Littlewood-Paley theory, we have(109)μs,ifLpRd=∑k∈Z-∫2q′k2q′k+1νt,s∗Γi+kfx2dtt1/2LpRd≤Cq-1-1/2hΔ1ΩLqSn-1∑k∈ZΓi+kf21/2LpRd≤Cq-1-1/2hΔ1ΩLqSn-1fLpRd,2≤p<∞.Similarly, we can obtain(110)μs,ifLpRd≤Cq-1-1hΔ1ΩLqSn-1fLpRd,1<p<2.On the other hand, by (41),(101), and Plancherel’s theorem, we have(111)μs,ifL2Rd2=∑k∈Z-∫RdΨi+kRsπλsnQsξ2f^ξ2∫2q′k2q′k+1νt,s^ξ2dttdξ≤∑k∈Z-∫Ei+k∫2q′k2q′k+1νt,s^ξ2dttf^ξ2dξ≤Cq-1-1hΔ12ΩLqSn-12Di2fL2Rd2,where Di=ρ(1-i)s/βsχ{i≥1}+ρ(1+i)s/βsχ{i<1},βs is as in Lemma 10 and(112)Ei+k=ξ∈Rd:ρ-si+k+1≤Lsξ≤ρ-si+k-1.Thus(113)μs,ifL2Rd≤Cq-1-1/2hΔ1ΩLqSn-1DifL2Rd.The interpolation between (109) and (113) leads to(114)μs,ifLpRd≤Cq-1-1/2hΔ1ΩLqSn-1DiαpfLpRd,2≤p<∞,where 0<αp<1. It follows from (105) and (114) that(115)μsfLpR2≤Cq-1-1/2hΔ1ΩLqSn-1fLpR2,2≤p<∞.Similarly, by (105) and interpolation between (110) and (113), we have(116)μsfLpRd≤Cq-1-1hΔ1ΩLqSn-1fLpRd,1<p<2.Using (102), (115)-(116), we get(117)μh,Ω,ΦϱfLpRd≤Cq-1-1/2hΔ1ΩLqSn-1fLpRd,2≤p<∞,μh,Ω,ΦϱfLpRd≤Cq-1-1hΔ1ΩLqSn-1fLpRd,1<p<2.This finishes the proof of Theorem 2.Proof of Theorem3.
By the same arguments as in the proof of [17, Theorem 2.3 (b)], we can obtain Theorem 3. We omit the details here.Proof of Theorem4.
By the argument similar to those used in deriving Lemma 4.2 in [38], we can obtain Theorem 4. We omit the details here.



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