Multiple singular integrals and maximal operators related to homogeneous mappings

R E S E A R C H

Open Access

Multiple singular integrals and maximal

operators related to homogeneous mappings

Feng Liu

_{, Daiqing Zhang}

_{and Jing Zhang}

*_{Correspondence:}

[email protected]

1_{College of Mathematics and}

Systems Science, Shandong University of Science and Technology, Qingdao, Shandong 266590, China

Full list of author information is available at the end of the article

Abstract

In this paper, we present theLp_{mapping properties of multiple singular integrals}

related to homogeneous mappings with rough kernels given by the radial function

h∈

γ

γ

bounds for the related maximal operators are also given. Our main results extend and improve some known ones.

MSC: 42B20; 42B25

Keywords: multiple singular integrals; rough kernels; maximal operators; homogeneous mappings

1 Introduction

LetR_{(=morn),≥, be the-dimensional Euclidean space, andS}–_{denote the}

unit sphere inR_{equipped with the induced Lebesgue measuredσ}

. For anyx∈R, we

also letx=x/|x|. Let∈L_(Rm_×Rn_{) satisfy}

Sm–

u,·dσm

u=

Sn–

·,vdσn

v= . (.)

ForM,N≥ and two suitable mappings::Rm→RMandϒ:Rn→RN, define the mul-tiple singular integral operatorsTh,,,ϒ along the surfacesS,ϒ={((u),ϒ(v)) : (u,v)∈ Rm_×Rn_}by

Th,,,ϒ(f)(x,y) := p.v.

Rm_×Rn

fx–(u),y–ϒ(v)Kh,(u,v)du dv, (.)

whereKh,(u,v) =(u,v)h(|u|,|v|)|u|–m|v|–nandh∈(R+×R+). HereR+= (,∞), and

γ(R+×R+) (γ ≥) is the set of all measurable functionsh(r,s) onR+×R+satisfying the

condition

hγ(R+×R+):=sup k,j∈Z

j+

j

k+

k

_h(r,s)γdr ds

rs

/γ

<∞.

For convenience, we denoteγ(R+×R+) byγ forγ ≥. Obviously,γγ forγ>

γ> .

For simplicity, we denoteTh,,,ϒ=T,,ϒ ifh≡ andT,,ϒ=T ifM=m,N=n,

(u) =u, andϒ(v) =v. The operatorT is the classic multiple singular integral

opera-tor, which was first introduced by Fefferman and Stein (see [, ]) and has been studied extensively by many authors (see [–], etc.). In particular, Duoandikoetxea [] proved thatTis bounded onLp(Rm×Rn) for  <p<∞, provided that∈Lq(Sm–×Sn–) with

q> . Later on, Chen [] improved the result of [] to the case∈L(log+L)_(Sm–_×Sn–_).

Subsequently, Ying and Chen [] (resp., Al-Salman et al. []) extended the result of [] to the multiple singular integrals along polynomial curves (resp., associated with poly-nomial mappings). We also refer the readers to [–], among others. Recall that ∈

L(log+L)α_(Sm–_×Sn–_{) forα>  is the set of all functionsonS}m–_×Sn–_{satisfying L}(log+L)α_(Sm–_×Sn–₎:=

Sm–_×Sn–

(u,v)logα_{ +(u,v)dσ}

m(u)dσn(v) <∞.

Note that, for anyq>  andβ>α> ,

LqSm–×Sn–Llog+LβSm–×Sn–Llog+LαSm–×Sn–.

The aim of this paper is to investigate theLp_{bounds for multiple singular integral}

op-erators associated with homogeneous mappings and the related maximal opop-erators. For

l∈N\{}andd= (d, . . . ,dl)∈Rl, define the family of dilations{δt}t>onRlby

δt(x, . . . ,xl) =

td_x

, . . . ,tdlxl

.

We say that a mapping:Rn_→Rl_{is homogeneous of degreedif}

(tx) =δt

(x)

for allx∈Rn_{andt> . When,ϒare two homogeneous mappings, Al-Qassem and Ali}

[] proved thatT,,ϒ is bounded onLp(RM ×RN) for  <p<∞if∈B(,)q (Sm–×

Sn–_{) for someq> . We note that the question with regard to the relationship between}

B(,α–)q (Sm–×Sn–) withq>  andL(log+L)α(Sm–×Sn–) (forα> ) remains open. In

, Al-Qassem and Ali [] gave the following result.

Theorem A ([]) Let == (,, . . . ,M) andϒ = = (,, . . . ,N) be two

homogeneous mappings of degrees d= (d, . . . ,dM)and v= (v, . . . ,vN),respectively,with

dι,vκ= for≤ι≤M and≤κ≤N.Assume that|Sm– andϒ|_Sn–are real-analytic.

Suppose that∈L(log+L)_(Sm–_×Sn–_{)satisfies(.).Then,for any <p<∞,there exists}

C> such that

T,,ϒ(f)_Lp_(RM_×RN₎≤CfLp_(RM_×RN₎

for all f ∈Lp_(RM_×RN_).

A natural question, which arises from the above results, is the following:

Question . Forh∈γwithγ > , isTh,,,ϒbounded onLp(RM×RN) under the same

Our investigation not only addresses this problem, but also deals with a more general class of operators. More specially, we shall establish the following:

Theorem . Let(y) =(ϕ(|y|)y)andϒ(y) =(ψ(|y|)y),where= (,, . . . ,M)

and= (,, . . . ,N)are given as in Theorem A,andϕ,ψ∈ForF,whereF(resp.,

F)is the set of all functionsφsatisfying the following condition(a) (resp., (b)):

(a) φ:R+→R+is an increasingC_{function such thatφis monotonous and}

tφ(t)≥Cφφ(t)andφ(t)≤cφφ(t)for allt> ,whereCφandcφare independent

oft.

(b) φ:R+→R+is a decreasingCfunction such thatφis monotonous and

tφ(t)≤–Cφφ(t)andφ(t)≤cφφ(t)for allt> ,whereCφandcφare independent

oft.

Suppose that h∈γ for someγ > and∈L(log+L)(Sm–×Sn–)satisfies(.).Then

for|/p– /|<min{/γ, /},there exists C> such that

Th,,,ϒ(f)_Lp_(RM_×RN₎≤CL(log+_L)_(Sm–_×Sn–)f_Lp_(RM_×RN₎

for all f ∈Lp_(RM_×RN_).

Remark . Theorem . represents an improvement of the corresponding results in [, , ]. Also, Theorem . implies Theorem A whenh≡, even in the particular caseϕ(t) = ψ(t) =t. There are some model examples for the classF, such astαlnβ( +t) (α> ,β≥),

tln ln(e+t), real-valued polynomialsPonRwith positive coefficients andP() = , and so on (see []). The model examples for a functionϕ∈Faretδ(δ< ) andt–ln( + /t). We

point out that there exist a constantBϕ>  such thatϕ(t)≥Bϕϕ(t) (orϕ(t)≥Bϕϕ(t))

forϕ∈F(orF) (see []).

Remark . We remark that the one-parameter case of Theorem . was studied by many authors (see [–] for examples). It follows from Example . in [] that Theorem . is not true if there existdi=  orvj=  for somei∈ {, , . . . ,M}andj∈ {, , . . . ,N}. It should

be pointed out that the index  in∈L(log+L)_(Sm–_×Sn–_{) is best possible since it has}

been shown in [] that for any> , there is∈L(log+L)–(Sm–×Sn–) such thatTis

not bounded onLp_(Rm_×Rn_{) for anyp∈(,∞).}

Whenγ ≥, the range ofpin Theorem . is (,∞), but for  <γ < , the range ofpis shrunk to (_γγ_+ , γ

γ–). In light of the aforementioned facts, it is natural to ask the following

question.

Question . Can the range ofpin Theorem . be enlarged for the case  <γ < ? The next aim of this paper is to address this question by imposing some more restrictive conditions onh. Precisely, for ≤γ ≤ ∞, letUγ(R+×R+) be the set of all measurable

functionshonR+×R+satisfying

hUγ(R+×R+):= ∞

 ∞



h(r,s)γdr ds

rs

/γ

<∞. (.)

For simplicity, we denoteUγ(R+×R+) byUγ forγ≥. Obviously,Uγγfor  <γ <∞

Theorem . Let,ϒbe as in Theorem..Suppose that h∈Uγ for someγ ≥and that

∈L(log+L)/γ(Sm–×Sn–)satisfies(.).Then Th,,,ϒ is bounded on Lp(RM×RN)if

one of the following conditions holds:

(i) γ = ,p=∞; (ii) γ > , <p<∞.

Remark . Obviously, whenγ =∞, Theorem . coincides with Theorem .. When

 <γ<∞, the condition onin Theorem . is strictly weaker than that in Theorem .. Moreover, for  <γ < , the range ofpin Theorem . is larger than that in Theorem .. Meanwhile, we also obtain the result at the endpoint caseγ = . Therefore, it is worth to impose the above restriction onh. It is interesting, but not clear, whether the restriction onhcan be removed.

To prove Theorem ., we need to consider the related maximal operatorsM(γ_,,ϒ) de-fined by

M(γ_,,ϒ) f(x,y) = sup hUγ≤

Th,,,ϒf(x,y),

which are interesting themselves. When(u) =uandϒ(v) =v, we shall denoteM(γ_,,ϒ) by

M(γ) . Historically, Ding [] proved that the operatorM() is bounded onL_(Rm_×Rn_),

provided that∈L(log+L)_(Sm–_×Sn–_{). This result was greatly improved by Al-Salman}

[], who obtained the Lp boundedness of M() for ≤p<∞ under the weaker

con-dition that ∈L(log+_L)(Sm–_{× S}n–_{). Moreover, Al-Salman showed that the}

condi-tion ∈L(log+L)(Sm–_×Sn–_{) cannot be replaced by any condition of the form ∈}

L(log+L)–_(Sm–_×Sn–_{), > . Particularly, Al-Qassem and Pan [] proved that the}

operator M(γ) is bounded onLp(Rm ×Rn) for γ≤p<∞(for γ = , p=∞) if ∈

L(log+L)/γ_(Sm–_×Sn–_{) and ≤γ ≤ (also see [] for the nonisotropic case).}

The remaining main results can be formulated as follows.

Theorem . Let,ϒbe as in Theorem.,and let∈L(log+L)/γ(Sm–×Sn–)for≤ γ ≤satisfy(.).Then M(γ)_,,ϒis bounded on Lp_(RM_×RN_{)for <γ≤withγ≤p<∞,}

and it is bounded on L∞(RM_×RN_{)forγ = .}

Remark . Theorem . improves and generalizes the results of [, ]. Also, Theo-rem . generalizes the result of [].

The rest of this paper is organized as follows. In Section , we shall recall some nota-tion and establish some preliminary lemmas. The proofs of the main results will be given in Section . We remark that some ideas of our methods are taken from [, ], but our meth-ods and technique are more delicate and complex than those used in [, , ]. Through-out this paper, letpdenote the conjugate index ofp, that is, /p+ /p= . The letterC, sometimes with additional parameters, will stand for positive constants, not necessarily the same at each occurrence but independent of the essential variables. We also use the conventions _j∈∅aj=  and

2 Notation and auxiliary lemmas

Following the notation in []. Let∈L(log+L)α_(Sm–_×Sn–_{) satisfy (.) for someα> .}

Forμ∈N, letaμ= μ+,Eμ={(x,y)∈Sm–×Sn–:aμ/≤ |(x,y)|<aμ},bμ=χEμ, and

λμ=bμ. LetD={μ∈N:λμ≥–μ}and

μ(x,y) =λ–μ

bμ(x,y) –

Sm–bμ

u,ydσm

u–

Sn–bμ

x,vdσn

v

+

Sm–_×Sn–

bμ

u,vdσm

udσn

v (.)

forμ∈D, and=– μ∈Dλμμ. Letλ= . Then

Sm–

μ

u,·dσm

u=

Sn–

μ

·,vdσn

v= , (.)

μ_L_(Sm–_×Sn–₎≤a_μ and μ_L_(Sm–_×Sn–₎≤, (.)

(x,y) =

μ∈D∪{}

λμμ(x,y), (.)

μ∈D∪{}

λμ(μ+ )α≤CL(log+L)α_(Sm–_×Sn–₎ for anyα> . (.)

The following lemma of Van der Corput type was proved by Cheng [].

Lemma .([]) Let l∈N\{},α, . . . ,αl∈R,andν, . . . ,νlbe distinct nonzero real

num-bers.Letψ∈C([, ]).Then there exists a positive constant C,independent of{αj}lj=,such

that

_δτexpiαtν+· · ·+αltνl

ψ(t)dt≤C|α|–/lψ(τ)+

τ

δ

ψ(t)dt

for/≤δ<τ≤.

Applying Lemma ., we have the following:

Lemma . Let(t) =μtα+μtα+· · ·+μntαn,whereμ,μ, . . . ,μn∈R,andα, . . . ,αn

are distinct nonzero real numbers.Then for any r> ,there exists C> independent of r such that

_rexpi(t)dt

t

≤C +r–α/n|_μ|–/n_.

Proof We can choose an integerl≥ such that l_{≤r< }l+_{. By the change of variables}

we have

r



expi(t)dt

t

≤

l–

j= j+

j

expi(t)dt

t

+

r

l

expi(t)dt

t

≤ l–

j= 

/

expij+tdt t

+



l r

expi(rt)dt

t

Note that / < l_{/r≤. Applying Lemma . and using (.), we obtain}

_rexpi(t)dt

t

≤C

_l–

j=

|μ|–/nj+–α/n+|μ|–/nr–α/nr

l

≤C +r–α/n|_μ|–/n_,

whereC>  is independent ofr. This proves Lemma ..

Lemma .([]) Let= (, . . . ,m)be real analytic on Sn–.Suppose that{, . . . ,m}

is a linearly independent set.Then there exists a positive numberδ=δ(m,)such that

sup

z∈Sm–

Sn–_×Sn–

z·(x) –(y)–δdx dy<∞.

Let,ϒ,,,ϕ,ψbe as in Theorem .. Letλbe the number of distinctdi, andλthe

number of distinctvj. Without loss of generality, we may assume that

=,, . . . ,λ_,

=,, . . . ,λ_,

whereι_{= (}

ι,, . . . ,ι,aι) withι,i(ty) =t

dαι

ι,i(y) for any ≤ι≤λ and ≤i≤ι. and

κ_{= (}

κ,, . . . ,κ,bκ) withκ,j(ty) =tvβκκ,j(y) for any ≤κ≤λand ≤j≤κ. Obviously,

λ

ι=aι=Mand{α, . . . ,αλ} ⊂ {, . . . ,M}, λ

κ=bκ=Mand{β, . . . ,βλ} ⊂ {, . . . ,N}. We

also assume that{ι,, . . . ,ι,oι}forms a basis forspan{ι,, . . . ,ι,aι}for any ≤ι≤λand

{κ,, . . . ,κ,κ}forms a basis forspan{κ,, . . . ,κ,bκ}for any ≤κ≤λ. Thus, there exist

two sequences of numbers{aι,i,l}and{bκ,j,k}such that

ι,i(x) =aι,i,ι,(x) +· · ·+aι,i,oιι,oι(x)

for any ≤ι≤λ, ≤i≤aι, andx∈Rmand

κ,j(y) =bκ,j,κ,(y) +· · ·+bκ,j,κκ,κ(y)

for any ≤κ≤λ, ≤j≤bκ, andy∈Rn. Define two sequences of linear transformations {Rι,i}oi=ι :Raι→Rand{Hκ,j}j=κ:Rbκ →Ras follows:

Rι,i(y) =aι,,ix+· · ·+aι,aι,ixaι, ≤i≤oι;

Hκ,j(y) =bκ,,jy+· · ·+bκ,bκ,jybκ, ≤j≤κ.

Define two families of linear transformations{Rι}λ

ι=and{Hκ}λ_κ= by

Rι= (Rι,, . . . ,Rι,oι),

In what follows, let˜ι_{= (}

ι,, . . . ,ι,oι) and˜

κ_{= (}

κ,, . . . ,κ,κ). Letξ = (ξ, . . . ,ξM) =

(ξ, . . . ,ξλ_{) withξ}ι_{= (ξ}

ι,,ξι,, . . . ,ξι,aι) for any ≤ι≤λ. Letη= (η, . . . ,ηN) = (η, . . . ,η

λ₎

withηκ_{= (η}

κ,,ηκ,, . . . ,ηκ,bκ) for any ≤κ≤λ. Thus,

ξι·ι=Rι

ξι· ˜ι, ≤ι≤λ; (.)

ηκ·κ=Hκ

ηκ· ˜κ, ≤κ≤λ. (.)

For ≤ι≤λ and ≤κ≤λ, we define two linear transformationsLι:RM→Roι and

κ:RN→Rκ by

Lι(ξ) =Rι

ξι, κ(η) =Hκ

ηκ.

Define two families of mappings{ι}λ

ι=and{ϒκ} λ ι=by

= (, . . . , ); ϒ= (, . . . , );

ι=

, . . . ,ι, , . . . , , ≤ι<λ and λ=

, . . . ,λ_;

ϒκ=

, . . . ,κ, , . . . , , ≤κ<λ and ϒλ=

, . . . ,λ_.

Letμbe as in (.). For anyr,s> , define two families of measures{σ_μ;ι,κ_r,s}and{|σ_μ;ι,κ_r,s|}

for ≤ι≤λand ≤κ≤λas follows:

σμ;ι,κr,s(ξ,η) =

Sm–_×Sn–

exp–πiξ·ι

ϕ(r)u

+η·ϒκ

ψ(s)vμ

u,vdσm

udσn

v,

and{|σι,κ

μ;r,s|}is defined in the same way as{σμ;ι,κr,s}, but withμreplaced by|μ|. Clearly,

σ_μ;,κ_r,s=σ_μ;ι,_r,s=  for ≤ι≤λ, ≤κ≤λ.

Lemma . Letϕ,ψ ∈F orF.Forμ∈D∪ {}, k,j∈Z, ≤ι≤λ, ≤κ ≤λ,and

(ξ,η)∈RM_×RN_{,there exist positive constants}

,,and C such that

sup

r,s>

σμ;ι,κr,s(ξ,η)≤C; (.) ajμ+

ajμ

akμ+

akμ

σμ;ι,κr,s(ξ,η) –σμ;ι–,κr,s(ξ,η) dr ds

rs

≤C(μ+ )ϕakμ dαι_L

ι(ξ) 

μ+_min,ψaj μ

vβκ

κ(η)

–_μ+_{; (.)} ajμ+

ajμ

akμ+

akμ

σμ;ι,κr,s(ξ,η) –σμ;ι,κ–r,s(ξ,η) dr ds

rs

≤C(μ+ )ψajμ vβκ

κ(η) 

μ+_min,ϕak μ

dαι_L

ι(ξ)

–_μ+ _{; (.)} ajμ+

ajμ

akμ+

akμ

σμ;ι,κr,s(ξ,η) dr ds

≤C(μ+ )min,ϕakμ dαι_L

ι(ξ) –

μ+_, ψajμ

v_βκ

κ(η) –

μ+_,ϕak μ

dαι_L

ι(ξ) –

μ+_ψaj μ

v_βκ

κ(η) –

μ+_{; (.)} ajμ+

ajμ

ak+

μ

ak

μ

σμ;ι,κr,s(ξ,η) –σμ;ι–,κr,s(ξ,η) –σμ;ι,κ–r,s(ξ,η) +σμ;ι–,κ–r,s (ξ,η)

dr ds rs

≤C(μ+ )min,ϕakμ dαι_L

ι(ξ) 

μ+_,ψaj μ

v_βκ

κ(η) 

μ+_, ϕakμ

dαι_L

ι(ξ) 

μ+_ψaj μ

v_βκ

κ(η) 

μ+_{. (.)}

Proof Note that for anyϕ∈ForF, there existC,C>  such thatC<ϕ(_ϕ(rr₎)<C. Thus,

we only prove the caseϕ,ψ∈F, and other cases are analogous. Estimate (.) is obvious.

By the change of variables,

σμ;ι,κr,s(ξ,η)

=

Sm–_×Sn–

exp–πiξ·ι

ϕ(r)u

+η·ϒκ

ψ(s)vμ

u,vdσm

udσn

v



=

(Sm–_×Sn–₎

exp–πiξ·ι

ϕ(r)u–ι

ϕ(r)θ

×exp–πiη·ϒκ

ψ(s)v–ϒκ

ψ(s)ω

×μ

u,vμ(θ,ω)dσm

udσn

vdσm(θ)dσn(ω).

It follows that

ajμ+

ajμ

ak+

μ

akμ

σμ;ι,κr,s(ξ,η)

dr ds rs

≤

(Sm–_×Sn–₎ Hk,μ

u,θ,ξJj,μ

v,ω,η

×μ

u,vμ(θ,ω)dσm

udσn

vdσm(θ)dσn(ω), (.)

where

Hk,μ

u,θ,ξ:=

akμ+

akμ

exp–πiξ·ι

ϕ(r)u–ι

ϕ(r)θdr

r,

Jj,μ

v,ω,η:=

ajμ+

ajμ

exp–πiξ·ϒκ

ψ(s)v–ϒκ

ψ(s)ωds

s .

By the change of variables and (.) we have

Hk,μ

u,θ,ξ

= μ l= l+akμ

l_ak

μ

exp–πiξ·ι

ϕ(r)u–ι

ϕ(r)θdr

r

=

μ

l=

ϕ(l+akμ)

ϕ(l_ak

μ)

exp–πiξ·ι

ru–ι(rθ) dr r = μ l=

ϕ(l+ak_μ)

ϕ(l akμ)



exp–πiξ·ι

ϕlak_μru–ι

ϕlak_μrθdr

r = μ l=

ϕ(l+ak_μ)

ϕ(l akμ) 

exp

–πi

ι

j=

ϕlakμ dαj

rdαj_R j

ξj·˜ju–˜j(θ)

dr r

. (.)

Note that ϕ(l+akμ)

ϕ(l_ak

μ) ≤cϕ. Combining (.) with Lemma . yields that

Hk,μ

u,θ,ξ≤

μ

l=

min,ϕlak_μdαj

Rι

ξι·˜ιu–˜ι(θ)–/ι

≤C(μ+ )min,maxϕakμ dαj

,ϕμakμ dαj–

×Rι

ξι·˜ιu–˜ι(θ)–

(.)

for any  <≤/λ. Similarly, we have

Jj,μ

v,ω,η≤C(μ+ )min,maxψaj_μvβκ_,ψμ_aj μ

vβκ–

×Hκ

ηκ·˜κv–˜κ(ω)– (.)

for any  <≤/λ. For any ≤ι≤λ andx∈Soι–, since{ι,, . . . ,ι,oι}is linear

in-dependent,x· ˜ι(·) is a nonzero real-analytic function. Invoking Lemma ., there exists δ>  such that

sup

x∈Soι–

(Sm–₎

x·˜ιu–˜ι(θ)–δdσm

udσm(θ) <∞. (.)

Similarly, for any ≤κ≤λ, there existsδ>  such that

sup

y∈Sκ–

(Sn–₎

y·˜κu–˜κ(θ)–δ

dσn

udσn(θ) <∞. (.)

From (.) and (.)-(.) we have

ajμ+

ajμ

ak+

μ

akμ

σμ;ι,κr,s(ξ,η)

dr ds rs

≤C(μ+ )min,maxϕakμ dαj

,ϕμakμ dαj–

Rι

ξι–

×min,maxψajμ

vβκ_,ψμ

ajμ

vβκ–

Hκ

ηκ–

×

(Sm–_×Sn–₎ Rι(ξι)

|Rι(ξι)|

·˜ιu–˜ι(θ)

– Hκ(ηκ)

|Hκ(ηκ)|

·˜κv–˜κ(ω)

–

×μ

u,vμ(θ,ω)dσm

udσn

for any  <≤/λand  <≤/λ. Take=min{/λ,δ/}and=min{/λ,δ/}.

Using (.), (.)-(.) and Hölder’s inequality along with (.), we obtain

ajμ+

ajμ

akμ+

akμ

σμ;ι,κr,s(ξ,η) dr ds

rs

≤Caμ(μ+ )min

,maxϕakμ dαj

,ϕμakμ dαj–

Rι

ξι–

×min,maxψajμ

vβκ_,ψμ

ajμ

vβκ–

Hκ

ηκ–

. (.)

We can easily check that

ajμ+

ajμ

akμ+

akμ

σμ;ι,κr,s(ξ,η)

dr ds

rs ≤C(μ+ )

_{, (.)}

ajμ+

ajμ

akμ+

akμ

σμ;ι,κr,s(ξ,η) –σμ;ι–,κr,s(ξ,η) dr ds

rs ≤C(μ+ )

_{. (.)}

Interpolating between (.) and (.) yields (.). We now prove (.) and (.). By

the change of variables,

σμ;ι,κr,s(ξ,η) –σμ;ι–,κr,s(ξ,η)

=

Sm–_×Sn–

exp–πiξ·ι

ϕ(r)u–exp–πiξ·ι–

ϕ(r)u

×exp–πiη·ϒκ

ψ(s)vμ

u,vdσm

udσn

v



=

(Sm–_×Sn–₎

exp–πiξ·ι

ϕ(r)u–exp–πiξ·ι–

ϕ(r)u

×expπiξ·ι

ϕ(r)θ–exp–πiξ·ι–

ϕ(r)θ

×exp–πiη·ϒκ

ψ(s)v–ϒκ

ψ(s)ω

×μ

u,vμ(θ,ω)dσm

udσn

vdσm(θ)dσn(ω),

which, together with (.), implies that

ajμ+

ajμ

akμ+

akμ

σμ;ι,κr,s(ξ,η) –σμ;ι–,κr,s(ξ,η)

dr ds rs

≤C(μ+ )min,ϕakμ dαι_R

ι

ξι

(Sm–_×Sn–₎ Jj,μ

v,ω,η

×μ

u,vμ(θ,ω)dσm

udσn

Using (.), (.), (.), (.), and Hölder’s inequality, we have

ajμ+

ajμ

ak+

μ

ak

μ

σμ;ι,κr,s(ξ,η) –σμ;ι–,κr,s(ξ,η)

dr ds rs

≤Caμ(μ+ )min

,ϕakμ dαι

Rι

ξι

×min,maxψa_μjvβκ_,ψμ_aj μ

vβκ–

Hκ

ηκ–. (.)

Estimate (.) follows form (.) and (.). Similarly, we can get (.). Estimate (.) follows from the inequality

σμ;ι,κr,s(ξ,η) –σμ;ι–,κr,s(ξ,η) –σμ;ι,κ–r,s(ξ,η) +σμ;ι–,κ–r,s (ξ,η)

=

Sm–_×Sn–

exp–πiξ·ι–

ϕ(r)u+η·ϒκ–

ψ(s)v

×exp–πiξ·ι

ϕ(r)u–ι–

ϕ(r)u– 

×exp–πiη·ϒκ

ψ(s)v–ϒκ–

ψ(s)v– 

×u,vJm

uJn

vdσm

udσn

v

≤Cmin,ϕ(r)dαι_Rιξι_min,ψ(s)vβκ_Hκηκ_.

This proves Lemma ..

For anyμ∈D∪ {}andk,j∈Z, we define two families of measures{τ_μ;ι,κ_k,j}and{|τ_μ;ι,κ_k,j|} for ≤ι≤λand ≤κ≤λas follows:

τ_μ;ι,κ_k,j(ξ,η) =

μ;k,j

μ(u,v)h(|u|,|v|) |u|m_|v|n exp

–πiξ·ι

ϕ|u|u

+η·ϒκ

ψ|v|vdu dv,

τ_μ;ι,κ_k,j(ξ,η) =

μ;k,j

|μ(u,v)h(|u|,|v|)| |u|m_|v|n exp

–πiξ·ι

ϕ|u|u

+η·ϒκ

ψ|v|vdu dv, where (ξ,η)∈RM_×RN _and

μ;k,j=

(u,v)∈Rm×Rn:|u|,|v|akμ,akμ+

×ajμ,ajμ+

. (.)

Observe that

τ_μ;,κ_k,j=τ_μ;ι,_k,j=  for ≤ι≤λ, ≤κ≤λ.

For convenience, forμ∈D∪ {}andλ> , we set

hμ,γ :=sup

j,k∈Z ajμ+

ajμ

akμ+

akμ

h(r,s)γdr ds

rs

/γ

Lemma . Letμ∈D∪{},hμ,γ<∞for someγ > andγ˜=max{,γ}.Supposeϕ,ψ∈ ForF.Then for k,j∈Z,there exist positive constants,,C such that,for any≤ι≤λ

and≤κ≤λ,

τ_μ;ι,κ_k,j(ξ,η)

≤Chμ,γ(μ+ )/γmin,ϕak_μdαι_L

ι(ξ) – 

˜

γ(μ+)_,ψaj μ

vβκ

κ(η) – 

˜

γ(μ+)_, ϕak_μdαι_L

ι(ξ) – 

˜

γ(μ+)_ψaj μ

vβκ

κ(η) – 

˜

γ(μ+)_{; (.)}

τ_μ;ι,κ_k,j(ξ,η) –τ_μ;ι,κ–_k,j(ξ,η)

≤Chμ,γ(μ+ )/γψajμ vβκ

κ(η) 

˜

γ(μ+)_min,ϕak μ

dαι_L

ι(ξ) –_γ˜(μ_+)

; (.)

τ_μ;ι,κ_k,j(ξ,η) –τ_μ;ι–,κ_k,j(ξ,η)

≤Chμ,γ(μ+ )/γϕakμ dαι_L

ι(ξ) 

˜

γ(μ+)_min,ψaj μ

vβκ

κ(η) –_γ˜(μ_+)

; (.)

τ_μ;ι,κ_k,j(ξ,η) –τ_μ;ι–,κ_k,j(ξ,η) –τ_μ;ι,κ–_k,j(ξ,η) +τ_μ;ι–,κ–_k,j (ξ,η)

≤Chμ,γ(μ+ )/γmin,ϕakμ dαι_L

ι(ξ) 

˜

γ(μ+)_, ψajμ

vβκ

κ(η) 

˜

γ(μ+)_,ϕak μ

dαι_L

ι(ξ) 

˜

γ(μ+)_ψaj μ

vβκ

κ(η) 

˜

γ(μ+)_{. (.)}

Proof By a change of variables, (.), and Hölder’s inequality we have

τ_μ;ι,κ_k,j(ξ,η)=

ajμ+

ajμ

akμ+

akμ

Sm–_×Sn–

exp–πiξ·ι

ϕ(r)u+η·ϒκ

ψ(s)v

×μ

u,vdσm

udσn

vh(r,s)dr ds

rs

≤Chμ,γ

ajμ+

ajμ

ak+

μ

ak

μ

σμ;ι,κr,s(ξ,η) γdr ds

rs

/γ

≤Chμ,γ(μ+ )max{/γ–/,}

ajμ+

ajμ

ak+

μ

akμ

σμ;ι,κr,s(ξ,η)

dr ds rs

/γ˜ .

This inequality, together with (.), yields (.). By a change of variables again, (.) and Hölder’s inequality we obtain

τ_μ;ι,κ_k,j(ξ,η) –τ_μ;ι,κ–_k,j(ξ,η)

=

ajμ+

ajμ

ak+

μ

akμ

Sm–_×Sn–

exp–πiξ·ι

ϕ(r)u

×exp–πiη·ϒκ

ψ(s)v–exp–πiη·ϒκ–

ψ(s)v

×μ

u,vdσm

udσn

vh(r,s)dr ds

rs

=

ajμ+

ajμ

akμ+

akμ

σμ;ι,κr,s(ξ,η) –σμ;ι,κ–r,s(ξ,η)

h(r,s)dr ds

rs

≤Chμ,γ

ajμ+

ajμ

akμ+

akμ

σμ;ι,κr,s(ξ,η) –σμ;ι,κ–r,s(ξ,η) γdr ds

rs

/γ

≤Chμ,γ(μ+ )max{/γ–/,} ajμ+

ajμ

akμ+

akμ

σμ;ι,κr,s(ξ,η) –σμ;ι,κ–r,s(ξ,η) dr ds

rs

/γ˜ .

Combining this inequality with (.) implies (.). Applying similar arguments as in get-ting (.) and using (.), we get (.). By a change of variables and Hölder’s inequality,

τ_μ;ι,κ_k,j(ξ,η) –τ_μ;ι–,κ_k,j(ξ,η) –τ_μ;ι,κ–_k,j(ξ,η) +τ_μ;ι–,κ–_k,j (ξ,η)

=

ajμ+

ajμ

akμ+

akμ

σμ;ι,κr,s(ξ,η) –σμ;ι–,κr,s(ξ,η) –σμ;ι,κ–r,s(ξ,η) +σμ;ι–,κ–r,s (ξ,η)

h(r,s)dr ds

rs

≤Chμ,γ

ajμ+

ajμ

ak+

μ

akμ

σμ;ι,κr,s(ξ,η) –σμ;ι–,κr,s(ξ,η)

–σμ;ι,κ–r,s(ξ,η) +σμ;ι–,κ–r,s (ξ,η) γdr ds

rs

/γ

≤Chμ,γ(μ+ )max{/γ–/,}

× a

j+

μ

ajμ

akμ+

akμ

σμ;ι,κr,s(ξ,η) –σμ;ι–,κr,s(ξ,η) –σμ;ι,κ–r,s(ξ,η) +σμ;ι–,κ–r,s (ξ,η)

dr ds rs

/γ˜ .

This inequality, together with (.), yields (.) and finishes the proof.

Lemma .([]) LetP be a polynomial mapping:R+−→Rn_{,whereP(t) = (P} (t), . . . ,

Pn(t))and Pi(i= , , . . . ,n)are real polynomials defined onR+.Supposeγ > andφ∈F

orF.Then the operator Mφdefined by

Mφ(f)(x) =sup κ∈Z

γ(κ+)

γ κ

_f

x–Pφ(t)dt

t satisfies

Mφ(f)_p≤Cγfp

for <p<∞.The constant C is independent ofγ and the coefficients of Pi(i= , , . . . ,n)

but depends onφ.

Applying Lemma ., we have the following:

Lemma . Letμ∈D∪{}andhμ,γ <∞for someγ > .Supposeϕ,ψ∈ForF.Then,

forι∈ {, , . . . ,λ}andκ∈ {, , . . . ,λ},the operatorτ_μ;ι,κ∗ satisfies

τ_μ;ι,κ∗ (f)_p≤Chμ,γ(μ+ )/γfp forγ<p≤ ∞, (.)

where

τ_μ;ι,κ∗ (f)(x,y) =sup

k,j∈Z

Proof We define the measures{|ι,κ_μ;k,j|}and the maximal operator∗_μ;ι,κby

ι,κ_μ;k,j(ξ,η) =

μ;k,j

exp–πiξ·ι

ϕ|u|u+η·ϒκ

ψ|v|v|μ(u

_,v)|

|u|m|_v|n du dv

and

∗_μ;ι,κ(f)(x,y) =sup

k,j∈Z

ι,κ_μ;k,j∗f(x,y),

whereμ;k,jis defied as in (.). By a change of variables we have

∗_μ;ι,κ(f)(x,y) =sup

k,j∈Z

μ;k,j

fx–ι

ϕ|u|u,y–ϒκ

ψ|v|v|μ(u

_,v)|

|u|m_|v|n du dv

≤

Sm–_×Sn–

sup

k,j∈Z ajμ+

ajμ

akμ+

akμ

fx–ι

ϕ(r)u,y–ϒκ

ψ(s)vdr ds rs

×μ

u,vdσm

udσn

v.

By the definition ofιandκ, using the iterated integration, Lemma ., and Minkowski’s

inequality, we obtain

∗_μ;ι,κ(f)_p≤C(μ+ )fp,  <p<∞. (.)

By a change of variables and Hölder’s inequality we have

τ_μ;ι,κ_k,j∗f(x,y)

=

μ;k,j

fx–ι

ϕ(r)u,y–ϒκ

ψ(s)v|μ(u

_{,v)h(|u|,|v|)|}

|u|m|_v|n du dv

≤ ajμ+

ajμ

ak+

μ

akμ

Sm–_×Sn–

fx–ι

ϕ(r)u,y–ϒκ

ψ(s)v

×μ

u,vdσm

udσn

vh(r,s)dr ds

rs

≤Chμ,γ

ajμ+

ajμ

akμ+

akμ

Sm–_×Sn–

fx–ι

ϕ(r)u,y–ϒκ

ψ(s)v

×μ

u,vdσm

udσn v γ dr ds rs /γ

≤Chμ,γ

ajμ+

ajμ

akμ+

akμ

Sm–_×Sn– _f

x–ι

ϕ(r)u,y–ϒκ

ψ(s)vγ

×μ

u,vdσm

udσn

vdr ds rs

/γ

≤Chμ,γ∗_μ;ι,κ|f|γ(x,y)/γ.

Forι∈ {, . . . ,λ}andκ∈ {, . . . ,λ}, letr,ι=rank(Lι) andr,ι=rank(κ). By a change of

basic matrices (see also [], Lemma .), there are four nonsingular linear transformations

H,ι:Rr,ι→Rr,ι,H,ι:Rr,κ →Rr,κ,G,ι:RM→RM, andG,κ:RN→RN such that

_H,ιπ_rM_,

ιG,ιξ≤Lι(ξ)≤CMH,ιπ

M

r,ιG,ιξ forξ∈R

M_{, (.)}

_H,κπ_rN_,

κG,κη≤κ(η)≤CNH,κπ

N

r,κG,κη forξ∈R

N_{, (.)}

whereCM,CN > ,πrM,ιis a projection operator fromR

M_toRr,ι_{, andπ}N

r,ιis the projection

operator fromRN_toRr,κ_.

Now we take two radial functions φ,φ ∈C∞(R) such that φ(t) =φ(s)≡  for

max{|t|,|s|} ≤ and φ(t) =φ(s)≡ for min{|t|,|s|}>aμ. For ι∈ {, . . . ,λ} andκ ∈ {, . . . ,λ}, we define the measures{ωι,κ

μ;r,s}and{λ ι,κ μ;k,j}by

ωμ;ι,κr,s(ξ,η) =σμ;ι,κr,s(ξ,η)(ι)(κ) –σμ;ι–,κr,s(ξ,η)(ι– )(κ)

–σμ;ι,κ–r,s(ξ,η)(ι)(κ– ) +σμ;ι–,κ–r,s (ξ,η)(ι– )(κ– )

and

λ_μ;ι,κ_k,j(ξ,η) =τ_μ;ι,κ_k,j(ξ,η)(ι)(κ) –τμ;ι–,κk,j(ξ,η)(ι– )(κ)

–τ_μ;ι,κ–_k,j(ξ,η)(ι)(κ– ) +τμ;ι–,κ–k,j (ξ,η)(ι– )(κ– ),

where

(ι) = λ

i=ι+

φϕ

akμ dαiH

,iπrM,iG,iξ,

(κ) = λ

ς=κ+

φψ

ajμ v_βς

H,κπrN,κG,κη.

Let{|ωι,κ_μ;r,s|}be defined in the same way as{ωι,κ_μ;r,s}, but with{σ_μ;ι,κ_r,s}replaced by{|σ_μ;ι,κ_r,s|}. It is easy to see that

σλ,λ μ;r,s =

λ

ι= λ

κ=

ωι,κ_μ;r,s (.)

and

τλ,λ μ;k,j =

λ

ι= λ

κ=

λι,κ_μ;k,j. (.)

Lemma . Letϕ,ψ ∈F orF.Forμ∈D∪ {}, k,j∈Z, ≤ι≤λ, ≤κ ≤λ,and

(ξ,η)∈RM_×RN_{,there exist positive constants}

,,and C such that

sup

r,s>

ωι,κ_μ;r,s≤C;

ajμ+

ajμ

ak+

μ

akμ

ωι,κμ;r,s(ξ,η)

dr ds rs

≤C(μ+ )min,ϕakμ dαι_L

ι(ξ) 

μ+_ψaj μ

v_βκ

κ(η) 

μ+_, ϕakμ

dαι_L

ι(ξ) 

μ+_ψaj μ

vβκ

κ(η) –

μ+_, ψajμ

vβκ

κ(η) 

μ+_ϕak μ

dαι_L

ι(ξ) –_μ+ _,

ϕakμ dαι_L

ι(ξ) – 

μ+_ψaj μ

v_βκ

κ(η) –

μ+_.

Lemma . Letμ∈D∪ {}andhμ,γ <∞for someγ > andγ˜=max{,γ}.Suppose thatϕ,ψ∈ForF.Then for k,j∈Z,there exist positive constants,,C such that,for

any≤ι≤λand≤κ≤λ,

sup

k,j,∈Z

λι,κ_μ;k,j≤Chμ,γ(μ+ )/γ;

λι,κ_μ;k,j(ξ,η)≤Chμ,γ(μ+ )/γmin,ϕakμ dαι_L

ι(ξ) 

˜

γ(μ+)_ψaj μ

v_βκ

κ(η) 

˜

γ(μ+)_, ϕak_μdαι_L

ι(ξ) 

˜

γ(μ+)_ψaj μ

vβκ

κ(η) – 

˜

γ(μ+)_, ψajμ

vβκ

κ(η) 

˜

γ(μ+)_ϕak μ

dαι_L

ι(ξ) – 

˜

γ(μ+)_, ϕak_μdαι_L

ι(ξ) – 

˜

γ(μ+)_ψaj μ

vβκ

κ(η) – 

˜

γ(μ+)_.

Applying Lemma . and the definition ofλι,κ_μ;k,j, we can establish the following:

Lemma . Letμ∈D∪ {}andhμ,γ <∞for someγ > .Suppose thatϕ,ψ∈For

F.Then,forι∈ {, , . . . ,λ}andκ∈ {, , . . . ,λ},there exists a constant C> such that sup

k,j∈Z

λι,κ_μ;k,j∗f_p≤Chμ,γ(μ+ )/γfp, γ<p≤ ∞.

Applying Lemma ., by arguments similar to those used in the proof of [], Theo-rem ., we have the following:

Lemma . Letμ∈D∪ {}andhμ,γ <∞for someγ> .Suppose thatϕ,ψ∈ForF.

Then,forι∈ {, , . . . ,λ}andκ∈ {, , . . . ,λ},there exists a constant C> such that

k,j∈Z

λι,κ_μ;k,j∗gk,j /

p

≤Chμ,γ(μ+ )/γ

k,j∈Z |gk,j|

/ p

Lemma . Letμbe as in(.)forμ∈D∪ {}.Suppose thatϕ,ψ ∈ForF.Forι∈ {, . . . ,λ}andκ∈ {, . . . ,λ},define the operatorUby

U(f)(x,y) =sup

k,j∈Z ajμ+

ajμ

akμ+

akμ

ωι,κ_μ;r,s∗f(x,y)dr ds

rs . Then there exists a constant C> such that

_U(f)_p≤C(μ+ )fp

for all f ∈Lp_(RM_×RN_{)and <p<∞.}

Proof Define the operatorHby

H(f)(x,y) =sup

k,j∈Z ajμ+

ajμ

akμ+

akμ

σ_μ;ι,κ_r,s∗f(x,y)dr ds

rs .

Then

H(f)(x,y) = sup

k,j∈Z ajμ+

ajμ

akμ+

akμ

_Sm–_×Sn–f

x–ι

u,y–κ

v

×μ

u,vdσudσvdr ds rs

≤C

Sm–_×Sn–

sup

k,j∈Z ajμ+

ajμ

akμ+

akμ

fx–ι

u,y–κ

vdr ds rs

×μ

u,vdσudσv.

Invoking Lemma ., using the iterated integration and Minkowski’s inequality, we can obtain

_H(f)_p≤C(μ+ )fp,  <p<∞.

This, together with the definition ofω_μ;ι,κ_r,s, implies Lemma ..

3 Proofs of main results

Proof of Theorem. We only prove the caseϕ,ψ∈F, and the other cases are analogous.

By Remark . there existBϕ,Bψ >  such thatϕ(t)≥Bϕϕ(t) andψ(t)≥Bψψ(t) for all

t> . It follows form (.) and (.) that

Th,,,ϒ(f)≤

μ∈D∪{}

λμTh,μ,,ϒ(f), (.)

Th,μ,,ϒ(f) =

k,j∈Z

τλ,λ μ;k,j ∗f =

λ

ι= λ

κ=

k,j∈Z

λι,κ_μ;k,j∗f =:

λ

ι= λ

κ=