R E S E A R C H
Open Access
Multiple singular integrals and Marcinkiewicz
integrals with mixed homogeneity along
surfaces
Feng Liu and Huoxiong Wu
**Correspondence: [email protected]
School of Mathematical Sciences, Xiamen University, Xiamen, 361005, China
Abstract
This paper is devoted to studying the singular integrals and Marcinkiewicz integrals with mixed homogeneity along surfaces, which contain many classical surfaces as model examples, on the product domainsRm×Rn(m,n≥2). Under rather weak size
conditions of the kernels, theLp(Rm×Rn)-boundedness for such operators is
established. These results essentially extend certain previous results. MSC: 42B20; 42B25
Keywords: singular integrals; Marcinkiewicz integrals; maximal operators; mixed homogeneity; product domains
1 Introduction
LetRd(d=morn),d≥, be thed-dimensional Euclidean space andSd–be the unit
sphere inRdequipped with the induced Lebesgue measuredσ
d. Letαd,,αd,, . . . ,αd,dbe fixed real numbers,αd,j≥ (j= , . . . ,d). Define the functionF:Rd×(,∞) –→Rby F(x,ρd) =
d j=xjρ
–αd,j
d ,x= (x,x, . . . ,xd). It is clear that for each fixedx∈Rd, the
func-tionF(x,ρd) is a decreasing function inρd> . We letρd(x) denote the unique solution of the equationF(x,ρd) = . Fabes and Riviére [] showed that (Rd,ρd) is a metric space, which is often called the mixed homogeneity space related to{αd,j}jd=. Forλ> , we letAd,λ
be the diagonald×dmatrixAd,λ=diag{λαd,, . . . ,λαd,d}. Letφ:R+–→(,∞), we denote
Ad,φ(ρd(y))ybyA
φ
d(y) fory∈Rd, wherey=Ad,ρd(y)–y∈S d–.
Letβd=max≤j≤dαd,j,γd=min≤j≤dαd,j. It is easy to check that
ρd(x)γd<|x|<ρd(x)βd, ifρd(x) > ;
ρd(x)βd<|x|<ρd(x)γd, ifρd(x) < ;
ρd(x) =|x|, ifρd(x) = .
The change of variables related to the spaces (Rd,ρ
d) is given by the transformation
x=ρ αd,
d cosθ· · ·cosθn–cosθn–,
x=ρ αd,
d cosθ· · ·cosθn–sinθn–, · · ·,
xd–=ρ αd,d–
d cosθsinθ,
xd=ρ
αd,d d sinθ.
Thus dx=ραd–
d Jd(x)dρddσd(x), whereρ
αd–
d Jd(x) is the Jacobian of the above trans-form andαd=
d
j=αd,j,Jd(x) =
d
j=αd,j(xj). Obviously,Jd(x)∈C∞(Sd–) and there exists Md> such that
≤Jdx≤Md, ∀x∈Sd–.
Let∈L(Sd–) and satisfy the following conditions:
(Ad,λx) =(x), ∀λ> andx= ,
Sd–
yJdydσd
y= .
Define the parabolic singular integral operatorT by
Tf(x) :=p.v.
Rd
(y)
ρd(y)αd
f(x–y)dy. (.)
As is well known, a singular integral operator of the type (.) originally arose from the study on the existence and regularity results of the heat equation and the more gen-eral parabolic differential operator with constant coefficients. In , Fabes and Riviére [] showed thatT is bounded onLp(Rd) for <p<∞if ∈C(Sd–). Subsequently, Nagel, Riviére and Wainger [] weakened the regularity condition on to the case
∈Llog+L(Sd–). Recently, Chen, Ding and Fan [] extended further the condition to the case∈H(Sd–).
In this paper, we will continue the research along this line. We will focus our attention on the multiple singular integrals with mixed homogeneity. Assume that∈L(Sm–×Sn–)
and satisfies the following conditions:
(Am,sx,An,ty) =(x,y), ∀s,t> , (x,y)∈Rm×Rn, (.)
Sm–
u,·Jmudσm
u=
Sn–
·,vJnvdσn
v= . (.)
We consider the multiple singular integral with mixed homogeneity defined by
T(f)(x,y) :=p.v.
Rm×Rn
(u,v)
ρm(u)αmρn(v)αn
f(x–u,y–v)du dv. (.)
In , Chen and Le [] showed that if∈L(log+L)(Sm–×Sn–), thenT
is bounded
on Lp(Rm×Rn) for <p<∞. On the other hand, in the special case α
m,i =αn,j= (i= , , . . . ,m;j= , , . . . ,n),Tis the classical multiple singular integral, which is studied
extensively by many authors (see [, , , , , , , , ] for examples). In par-ticular, Ying [] (also see [] for a more general case) proved that T is bounded on
condition:
sup
(ξ,η)∈Sm–×Sn–
Sm–×Sn–
u,vGξ,η;u,vβdσm
udσn
v<∞, (.)
where
Gξ,η;u,v=log
|ξ·u|+log
|η·v|+log
|ξ·u|·log
|η·v|.
It should be pointed out that the condition (.) for one parameter case was originally defined in Walsh’s paper [] and developed by Grafakos and Stefanov []. For the sake of simplicity, we denote that forβ> ,
Fβ
Sm–×Sn–=∈LSm–×Sn–:satisfies (.).
Employing the ideas in [], one easily verifies that for β>β> ,Fβ(Sm–×Sn–)
Fβ(S
m–×Sn–) and
β> Fβ
Sm–×Sn–Llog+LSm–×Sn–Llog+LSm–×Sn–
β> Fβ
Sm–×Sn–. (.)
Based on the above, a natural question is as follows.
Question . For the general caseαm,i≥ (i= , . . . ,m) andαn,j≥ (j= , . . . ,n), isT
bounded onLp(Rm×Rn) under the condition (.) for someβ> ?
One of the main purposes of this paper is to give a positive answer to the above question. The method we use allows us to treat a family of operators broader than those given by (.). To be precise, for suitable functionsϕ,ψ :R+–→(,∞) and two real polynomial
PNionRwithPNi() = andPNi(t) > fort= , whereNiis the degree ofPNi(i= , ), we define the multiple singular integral operatorTP
along surfacesS(PN(ϕ),PN(ψ)) by
TP(f)(x,y)
=p.v.
Rm×Rn
(u,v)
ρm(u)αmρn(v)αn
fx–APN(ϕ)
m (u),y–A PN(ψ)
n (v)
du dv, (.)
where
SPN(ϕ),PN(ψ)
:=APN(ϕ) m (u),A
PN(ψ)
n (v)
: (u,v)∈Rm×Rn.
Obviously,Tis the special case ofTPforPNi(s) =ϕ(s) =ψ(s) =s(i= , ). Also, in the
special caseαm,i=αn,j= (i= , . . . ,m;j= , . . . ,n),
SPN(ϕ),PN(ψ)
=PN
ϕ|u|u,PN
Moreover, for the special caseϕ(s) =ψ(s) =sandαm,i=αn,j= (i= , . . . ,m;j= , . . . ,n),
SPN(ϕ),PN(ψ)
=PN
|u|u,PN
|v|v: (u,v)∈Rm×Rn.
Wu and Yang [] proved that if∈Fβ(Sm–×Sn–) withβ> , thenTPis bounded on
Lp(Rm×Rn) for β/(β– ) <p< β. In this paper, we will extend the result above as follows.
Theorem . Let PNand PNbe two real polynomials onRwith PNi() = and PNi(t) > for t= , where Niis the degree of PNi(i= , ), and letϕ,ψ∈F, whereFis the set of functions
φsatisfying the following properties:
(i) φ:R+–→(,∞)is continuous strictly increasing andφ∈C((,∞))satisfying that φis monotonous;
(ii) there exist constantsCφ,cφ> such thattφ(t)≥Cφφ(t)andφ(t)≤cφφ(t)for all
t> .
Suppose that satisfies (.)-(.) and∈Fβ(Sm–×Sn–)for someβ > . Then TP
defined as in (.) is bounded on Lp(Rm×Rn)forβ/(β– ) <p< β. The bound is inde-pendent of the coefficients of PNi(i= , ), but depends onϕ,ψ, N, N, m, n andβ.
Remark . For anyφ∈F, there exists a constantBφ> such thatφ(r)≥Bφφ(r) for all
r> . To see this, by the mean-valued theorem, for anyr> , there existss∈(r, r) such thatφ(r) –φ(r) =rφ(s). The properties (i) and (ii) ofφimply that
φ(r) –φ(r) =rφ(s)≥rCφ φ(s)
s ≥ Cφ
φ(r).
TakingBφ= +Cφ/, this is the desired constant.
Remark . We remark that the model examples for functions φ ∈Fare tα (α > ),
tln( +t),tln ln(e+t) and real-valued polynomialsPonRwith positive coefficients and P() = (see []). Theorem . extends the result of [], which is the multiple-parameter generalization of the result in [, ], to the mixed homogeneity setting, even in the spe-cial caseϕ(s) =ψ(s) =s. Also, by (.), Theorem . is distinct from the result of [], even in the special casePN(s) =PN(s) =ϕ(s) =ψ(s) =s.
On the other hand, we also consider the multiple Marcinkiewicz integral operatorMP
along the surfacesS(PN(ϕ),PN(ψ)) defined by
MP
(f)(x,y) =
∞
∞
FPN(ϕ),PN(ψ) s,t (x,y)
ds dt
st
/
, (.)
where
FPN(ϕ),PN(ψ) s,t (x,y) =
(s,t)
(u,v)
ρm(u)αm–ρn(v)αn–
fx–APN(ϕ)
m (u),y–A PN(ψ)
n (v)
du dv,
and(s,t) ={(u,v)∈Rm×Rn:ρm(u)≤s,ρn(v)≤t}.
WhenPN(s) =PN(s) =ϕ(s) =ψ(s) =s,αm,i=αn,j= (i= , . . . ,m;j= , . . . ,n), we denote
MP
is studied extensively by many authors (see [, , , , , , –]et al.). In particular, Al-Qassem, Al-Salman, Cheng and Pan [] showed that if∈Llog+L(Sm–×Sn–), then M is bounded onLp(Rm×Rn) for <p<∞; Hu, Lu and Yan [] (also see [, ])
proved that if∈Fβ(Sm–×Sn–) forβ> /, thenMis bounded onLp(Rm×Rn) for
+ /(β) <p< + β. For the general operatorMP
, whenPNi(t) =t(i= , ) andϕ,ψ∈F,
Al-Salman [] showed thatMP
is bounded onLp(Rm×Rn) for <p<∞provided that ∈Llog+L(Sm–×Sn–).
A natural question which arises from the above is the following:
Question . Under the condition (.) withβ> /, isMP
also bounded onLp(Rm×Rn)
for + /(β) <p< + β?
This question will be addressed by our next theorem.
Theorem . Let PNi(i= , ),ϕ,ψbe as in Theorem .. Suppose∈Fβ(Sm–×Sn–)for
someβ> /and satisfies (.)-(.). ThenMP
defined as (.) is bounded on Lp(Rm×Rn)
for + /(β) <p< + β. The bound is independent of the coefficients of PNi(i= , ) but depends onϕ,ψ, N, N, m, n andβ.
Remark . Theorem . extends the result of [] to the mixed homogeneity setting, even for the special casePN(s) =PN(s) =ϕ(s) =ψ(s) =s. And by (.), Theorem . is
distinct from the result of [], even in the special casePN(s) =PN(s) =s.
The rest of this paper is organized as follows. After recalling some notation and estab-lishing some preliminary lemmas, we will prove Theorem . in Section . And the proof of Theorem . will be given in Section . We remark that our some ideas in the proofs of our main results are taken from [, , , ], but our methods and technique are more delicate and complex than those used in [, , , ].
Throughout this paper, the letterCor c, sometimes with additional parameters, will stand for positive constants, not necessarily the same at each occurrence but independent of the essential variables.
2 On multiple singular integrals
Let us begin with some notations and lemmas. For given positive polynomialsPN(t) =
N
i=βiti,PN(t) =
N
i=γitiand two smooth functionsϕ,ψ∈F, we set
Pl(t) =PN(t)
αm,l :=
Nαm,l i=
ai,lti forl∈ {, , . . . ,m};
Pk(t) =PN(t)
αn,k
:= Nαn,k
j=
bj,ktj fork∈ {, , . . . ,n}.
Then forx,ξ∈Rm;y,η∈Rn,
APN(ϕ) m (x)·ξ=
m
l=
PN
ϕρm(x)
αm,l xl·ξl=
m
=
Nαm,l i=
ai,lϕ
ρm(x)
APN(ψ) n (y)·η=
n
k=
PN
ψρn(y)
αn,k yk·ηk=
n
k=
Nαn,k j=
bj,kψ
ρn(y)
j yk·ηk.
We denoteN:=max{Nαm,l: ≤l≤m},N:=max{Nαn,k: ≤k≤n}and setai,l= wheneveri>Nαm,l;bj,k= wheneverj>Nαn,k. So we can write
APN(ϕ) m (x)·ξ=
m
=
Nαm,l i=
ai,lϕ
ρm(x)
i xl·ξl=
N
i=
Li(ξ)·xϕρm(x)
i ,
whereLi(ξ) = (ai,ξ,ai,ξ, . . . ,ai,mξm). Similarly,
APN(ψ) n (y)·η=
N
j=
Ij(η)·y
ψρn(y)
j ,
whereIj(η) = (bj,η,bj,η, . . . ,bj,nηn). Forμ∈ {, , . . . ,N},ν∈ {, , . . . ,N}, we set
Qμ(x) =
μ
i=
ai,xϕ
ρm(x)
i , . . . ,
μ
i=
ai,mxmϕ
ρm(x)
i
,
Rν(y) =
ν
j=
bj,yψ
ρn(y)
j , . . . ,
ν
j=
bj,nynψ
ρn(y)
j
.
Here we use the conventioni∈∅ai= . Hence,
Qμ(x)·ξ= μ
i=
Li(ξ)·x
ϕρm(x)
i
, ≤μ≤N;
Rν(y)·η= ν
j=
Ij(η)·yψρn(y)
j
, ≤ν≤N.
For anyκ,∈Zandμ∈ {, , . . . ,N},ν∈ {, , . . . ,N}, we define the measures{σκ,;μ,ν}
and{|σκ,;μ,ν|}as follows.
σκ,;μ,ν(ξ,η) =
κ,
(x,y)
ρm(x)αmρn(y)αn
exp–iQμ(x)·ξ+Rν(y)·η
dx dy, (.)
|σκ,;μ,ν|(ξ,η) =
κ,
|(x,y)|
ρm(x)αmρn(y)αn
exp–iQμ(x)·ξ+Rν(y)·η
dx dy, (.)
whereκ,={(x,y)∈Rm×Rn: κ–≤ρm(x) < κ, –≤ρn(y) < }. By (.) andQ(x) =
(, , . . . , )∈Rm,R
(y) = (, , . . . , )∈Rn, forμ∈ {, , . . . ,N}andν∈ {, , . . . ,N}we
have
σκ,;,ν(ξ,η) =σκ,;μ,(ξ,η) = . (.)
Then it is easy to see that
TP(f)(x,y) =
κ,∈Z
Lemma . (cf. [, pp.-]) Let P be a polynomial mapping R+ –→Rd, where
P(t) = (P(t),P(t), . . . ,Pd(t))and Piis a real polynomial defined onR+(i= , . . . ,d). Then
the maximal function MP(f)(x)defined by
MP(f)(x) =sup r>
r
|t|≤rfx–P(t)dt
is bounded on Lp(Rd)for <p<∞. The bound is independent of the coefficients of Pi(i= , . . . ,d) and f .
Lemma . Let P be a polynomial mapping R+ –→ Rd, where P(t) = (P
(t),P(t),
. . . ,Pd(t))and Piis a real polynomial defined onR+(i= , . . . ,d). Suppose thatφ∈F. Then
the operator Mφdefined by
MP(φ)(f)(x) =sup
r>
r r
fx–Pφ(t)dt t
is bounded on Lp(Rd)for <p<∞. The bound is independent of the coefficients of Pi(i= , . . . ,d) and f , but depends onφ.
Proof For anyr> , by the change of variable, it can be easily seen that
r r
fx–Pφ(t)dt t
=
φ(r) φ(r)
fx–P(s) ds
φ(φ–(s))φ–(s)
≤
Cφ
φ(r) φ(r)
fx–P(s)ds s ≤
Cφφ(r)
φ(r) φ(r)
fx–P(s)ds
≤ φ(r)
Cφφ(r)
φ(r)
φ(r)
fx–P(s)ds≤ cφ Cφ
MP|f|(x).
This implies that MP(φ)(f)(x)≤C(φ)MP(|f|)(x). Then Lemma . follows from
Lem-ma ..
Lemma . Letϕ,ψ∈F. Suppose that∈L(Sm–×Sn–)and satisfies (.)-(.). Then,
forμ∈ {, , . . . ,N},ν∈ {, , . . . ,N}, the maximal operator defined by
σμ*,ν(f)(x,y) = sup
κ,∈Z
|σκ,;μ,ν| ∗f(x,y)
is bounded on Lp(Rm×Rn)for <p<∞. The bound is independent of the coefficients of PNi(i= , ) and f , but depends onϕ,ψ, N, N, m, n.
Proof By the definition of|σκ,;μ,ν|, we have
|σκ,;μ,ν| ∗f(ξ,η)
=
κ,
|(u,v)|
ρm(u)αmρn(v)αn
fx–Qμ(u),y–Rν(v)
≤C
Sm–×Sn–
κ κ–
–
|f(x–Qμ(Am,ρmu),y–Rν(An,ρnv))|
ρmρn
dρmdρn
×u,vdσm
udσn
v
≤C
Sm–×Sn–
u,vMQμ,Rν;u,v(f)(x,y)dσm
udσn
v,
where
MQμ,Rν;u,v(f)(x,y) :=sup s,t>
st
s s
t t
fx–Qμ
Am,ru
,y–Rν
An,hvdr dh.
By Lemma ., using iterated integration, it is easy to see that
MQμ,Rν;u,v(f)p≤Cfp for <p<∞,
whereCis independent ofu,v. Thus
σμ*,ν(f)p≤C
Sm–×Sn–
u,vMQμ,Rν;u,v(f)pdσm
udσn
v≤Cfp,
which completes the proof of Lemma ..
Lemma . (cf. [, p., Corollary]) Let (t) =tα +μ
tα +· · ·+μntαn and ∈
C([a,b]), whereμ, . . . ,μ
n are real parameters, andα, . . . ,αn are distinct positive (not necessarily integer) exponents. Then
abexpiλ(t)(t)dt≤Cλ–
sup a≤t≤b
(t)+
b a
(t)dt
,
with=min{/α, /n}and C does not depend onμ, . . . ,μmas long as≤a<b≤.
Lemma . Suppose thatϕ,ψ∈F. Then for anyμ∈ {, , . . . ,N}andν∈ {, , . . . ,N}, there exist= /μand= /νsuch that for any r>
r r/
exp–iQμ
Am,ρmx
·ξdρm ρm ≤
C(ϕ)ϕ(r)μL μ(ξ)·x
–
;
r r/
exp–iRν
An,ρny
·ηdρn ρn ≤
C(ψ)ψ(r)νI ν(η)·y
–
.
The constant C(ϕ)is independent of the coefficients of PN but depends onϕ; and C(ψ)is independent of the coefficients of PNbut depends onψ.
Proof We only prove the first inequality, since a similar argument can get the second in-equality. By the change of variables, we have
r r/
exp–iQμ
Am,ρmx
·ξdρm
ρm
=
r r/
exp
–i
μ
j=
Lj(ξ)·xϕ(ρm)j
dρm
=
ϕ(r) ϕ(r/)
exp
–i
μ
j=
Lj(ξ)·xtj
dt
ϕ–(t)ϕ(ϕ–(t))
=ϕ(r)
ς
exp
–i
μ
j=
Lj(ξ)·xϕ(r)jtj
φ(t)gr,ϕ(t)dt,
whereς=ϕ(r/)/ϕ(r),φ(t) = (ϕ–(ϕ(r)t))–,gr,ϕ(t) = (ϕ(ϕ–(ϕ(r)t)))–. Let
I(t) =
t
ς
exp
–i
μ
j=
Lj(ξ)·x
ϕ(r)jsj
φ(s)ds, ς≤t≤.
By Lemma ., there exists= /μsuch that
I(t)≤Cϕ(r)μLμ(ξ)·x –
sup s∈[ς,t]
φ(s)+
t
ς
φ(s)ds
≤Cϕ(r)μLμ(ξ)·x –
(/r+ /r)
≤C
rϕ(r)
μ
Lμ(ξ)·x –
.
Thus by integration by parts and the fact thatϕis monotonous, we have
r/r exp–iQμ
Am,ρmx
·ξdρm ρm
=ϕ(r)
ς
gr,ϕ(t)dI(t)
≤ϕ(r)I()gr,ϕ()+
ς
I(t)gr,ϕ(t)dt
≤ϕ(r)/rϕ(r)μLμ(ξ)·x –
ϕ(r)–+ϕ(r/)–.
Usingtϕ(t)≥Cϕϕ(t), we get
r r/
exp–iQμ
Am,ρmx
·ξdρm ρm ≤
c Cϕ
( + cϕ)ϕ(r)μLμ(ξ)·x –
≤C(ϕ)ϕ(r)μLμ(ξ)·x –
.
This proves Lemma ..
Lemma . Letϕ,ψ∈F. Suppose that∈Fβ(Sm–×Sn–)for someβ> and satisfies
(.)-(.). Then forμ∈ {, , . . . ,N}andν∈ {, , . . . ,N}, there exists a constant C> such that
(i) if|ψ()νI
ν(η)|> , then
σκ,;μ,ν(ξ,η) –σκ,;μ–,ν(ξ,η) ≤CϕκμLμ(ξ)min
,logψνIν(η) –β
(ii) if|ϕ(κ)μL
μ(ξ)|> , then
σκ,;μ,ν(ξ,η) –σκ,;μ,ν–(ξ,η) ≤CψνIν(η)min
,logϕκμLμ(ξ) –β
; (.)
(iii) if|ϕ(κ)μL
μ(ξ)|> and|ψ()νIν(η)|> , then
σκ,;μ,ν(ξ,η)≤Cmin
,logϕκμLμ(ξ) –β
,logψνIν(η) –β
; (.)
σκ,;μ,ν(ξ,η)≤Cmin
,logϕκμLμ(ξ) –β
logψνIν(η) –β
; (.)
(iv)
σκ,;μ,ν(ξ,η) –σκ,;μ–,ν(ξ,η) –σκ,;μ,ν–(ξ,η) +σκ,;μ–,ν–(ξ,η) ≤Cmin,ϕκμLμ(ξ),ψ
νIν(η),
ϕκμLμ(ξ)ψ
νIν(η). (.)
The constant C is independent of the coefficients of PNand PN.
Proof Let
Hκ,μ
x,ξ=
κ κ–
exp–iQμ
Am,ρmx
·ξdρm ρm
;
J,ν
y,η=
–
exp–iRν
An,ρny
·ηdρn
ρn .
By Lemma ., there exist,∈(, ] such that
Hκ,μ
x,ξ≤Cmin,ϕκμLμ(ξ)·x –
;
J,ν
y,η≤Cmin,ψνI ν(η)·y
–
.
When|ϕ(κ)μL
μ(ξ)|> , sincet/(logt)βis increasing in (eβ,∞), we have
Hκ,μ
x,ξ≤C(loge
β|(ϕ(κ)μL
μ(ξ))·x|–)β
(log|ϕ(κ)μL μ(ξ)|)β
.
Then
Hκ,μ
x,ξ≤Cmin
,(loge
β|(ϕ(κ)μL
μ(ξ))·x|–)β
(log|ϕ(κ)μL μ(ξ)|)β
. (.)
Similarly, when|ψ()νI
ν(η)|> ,
J,ν
y,η≤Cmin
,(loge
β|(ψ()νI
ν(η))·y|–)β
(log|ψ()νI ν(η)|)β
By the definition ofσκ,;μ,ν, we have
σκ,;μ,ν(ξ,η) –σκ,;μ–,ν(ξ,η)
≤C
Sm–×Sn–
/
exp–iQμ
Am,κρmx·ξ–exp–iQμ–Am,κρmx·ξd
ρm
ρm
×J,ν
y,ηx,ydσm
xdσn
y
≤C
Sm–×Sn–
J,ν
y,ηϕκμL μ(ξ)
x,ydσm
xdσn
y.
Combining (.) with the fact∈Fβ(Sm–×Sn–), we obtain (.). Similarly, we can
conclude (.). To prove (.) and (.), we write
σκ,;μ,ν(ξ,η)≤C
Sm–×Sn–
x,yHκ,μ
x,ξJ,ν
y,ηdσm
xdσn
y.
Then (.) and (.) follow from (.)-(.) with the fact∈Fβ(Sm–×Sn–). Finally,
(.) follows from the inequality
σκ,;μ,ν(ξ,η) –σκ,;μ–,ν(ξ,η) –σκ,;μ,ν–(ξ,η) +σκ,;μ–,ν–(ξ,η)
≤C
Sm–×Sn–
x,y
×
/
/
exp–iQμ–
Am,κρmx·ξ+Rν–An,ρ
ny
·η ×exp–iϕκρm
μ
Lμ(ξ)·x
–
×exp–iψρn
ν
Iν(η)·y
– dρm
ρm dρn
ρn
dσm
xdσn
y.
This completes the proof of Lemma ..
Now we take two radial Schwartz functionsφ∈S(Rm) andφ∈S(Rn) such thatφ i(t)≡ for|t| ≤ andφi(t)≡ for|t|>min{Bϕ,Bψ}(i= , ), whereBϕ,Bψare as in Remark ..
Define the measures{ωκ,;μ,ν}by
ωκ,;μ,ν(ξ,η) =σκ,;μ,ν(ξ,η) N
i=μ+ φ
ϕκi
Li(ξ)
N
j=ν+ φ
ψj
Ij(η)
–σκ,;μ–,ν(ξ,η) N
i=μ
φϕκiL
i(ξ)
N j=ν+
φψjI
j(η)
–σκ,;μ,ν–(ξ,η) N
i=μ+ φ
ϕκiLi(ξ)
N
j=ν φ
ψjIj(η)
+σκ,;μ–,ν–(ξ,η) N
i=μ φ
ϕκiLi(ξ)
N
j=ν φ
for κ, ∈ Z, μ ∈ {, , . . . ,N} and ν ∈ {, , . . . ,N}, where we use the convention
j∈∅aj= . By (.), it is easy to see that
σκ,;N,N=
N
μ= N
ν=
ωκ,;μ,ν. (.)
Lemma . Let,ϕ,ψ be as in Lemma .. Forμ∈ {, , . . . ,N}andν∈ {, , . . . ,N},
κ,∈Z, we have
(i)
ωκ,;μ,ν(ξ,η)≤Cϕ
κμLμ(ξ)ψ
νIν(η); (.)
(ii) if|ϕ(κ)μL
μ(ξ)|>Bϕ, then
ωκ,;μ,ν(ξ,η)≤C
logϕκμLμ(ξ) –β
ψνIν(η); (.)
(iii) if|ψ()νI
ν(η)|>Bψ, then
ωκ,;μ,ν(ξ,η)≤Cϕ
κμLμ(ξ)logψ
νIν(η)– β
; (.)
(iv) if|ϕ(κ)μL
μ(ξ)|>Bϕand|ψ()νIν(η)|>Bψ, then
ωκ,;μ,ν(ξ,η)≤C
logϕκμL μ(ξ)
–β
logψνI ν(η)
–β
. (.)
Here and below, Bφ(φ=ϕorψ) is as in Remark ., the constant C is independent of the
coefficients of PNi(i= , ).
Proof We write(μ) =
N
i=μ+φ(ϕ(κ)iLi(ξ)),(ν) =
N
j=ν+φ(ψ()jIj(η)). Then
ωκ,;μ,ν(ξ,η) =σκ,;μ,ν(ξ,η)(μ)(ν) –σκ,;μ–,ν(ξ,η)(μ– )(ν)
–σκ,;μ,ν–(ξ,η)(μ)(ν– )
+σκ,;μ–,ν–(ξ,η)(μ– )(ν– ). (.)
Thus, it is easy to see that
ωκ,;μ,ν(ξ,η)=(μ)(ν)σκ,;μ,ν(ξ,η) –σκ,;μ–,ν(ξ,η)φ
ϕκμLμ(ξ)
–σκ,;μ,ν–(ξ,η)φ
ψνI ν(η)
+σκ,;μ–,ν–(ξ,η)φ
ϕκμLμ(ξ)
φ
ψνIν(η)
≤Cσκ,;μ,ν(ξ,η) –σκ,;μ–,ν(ξ,η) –σκ,;μ,ν–(ξ,η) +σκ,;μ–,ν–(ξ,η)
+Cσκ,;μ–,ν(ξ,η) –σκ,;μ–,ν–(ξ,η) –φ
ϕκμLμ(ξ)
+Cσκ,;μ,ν–(ξ,η) –σκ,;μ–,ν–(ξ,η) –φ
ψνI ν(η)
+Cσκ,;μ–,ν–(ξ,η) –φ
ϕκμLμ(ξ) –φ
Notice that
–φ
ϕκμL
μ(ξ)≤Cϕ
κμL
μ(ξ), (.)
–φ
ψνIν(η)≤Cψ
νIν(η). (.)
Invoking Lemma ., we get (.). On the other hand, since
(μ– ) = , ifϕ
κμLμ(ξ)>Bϕ, (.)
(ν– ) = , ifψ
νIν(η)>Bψ, (.)
by (.) and (.), we have
ωκ,;μ,ν(ξ,η)=σκ,;μ,ν(ξ,η)(μ)(ν) –σκ,;μ,ν–(ξ,η)(μ)(ν– ) ≤σκ,;μ,ν(ξ,η) –σκ,;μ,ν–(ξ,η)φ
ψνIν(η) ≤σκ,;μ,ν(ξ,η) –σκ,;μ,ν–(ξ,η)+σκ,;μ,ν–(ξ,η) –φ
ψνIν(η).
Then (.) follows from (.)-(.) with (.). Similarly, we get (.). Finally, (.) fol-lows from (.), (.), (.) and (.). This completes the proof of Lemma ..
By Lemma . and the definition of{μκ,;μ,ν}, it is easy to verify the following lemma.
Lemma . Let ,ϕ,ψ be as in Lemma .. Then for μ∈ {, , . . . ,N} andν∈ {, , . . . ,N}, we have
sup
κ,∈Z
|ωκ,;μ,ν| ∗f(·,·)
Lp(Rm×Rn)≤CfLp(Rm×Rn)
for <p<∞. The constant C is independent of the coefficients of PNand PN.
Applying Lemma . and [, p., Lemma], we can obtain
Lemma . Let , ϕ,ψ be as in Lemma .. Then for μ∈ {, , . . . ,N} andν∈ {, , . . . ,N}, we have
κ,∈Z
ωκ,;μ,ν∗gκ,(·,·) /
Lp(Rm×Rn)≤ C
κ,∈Z
gκ,(·,·) /
Lp(Rm×Rn)
for <p<∞and any arbitrary functions{gκ,}. The constant C is independent of the
coef-ficients of PNand PN.
Now we are in the position of proving Theorem ..
Proof of Theorem . Combining (.) with (.), we write
TP
(f) = N
μ= N
ν=
κ,∈Z
ωκ,;μ,ν∗f := N
μ= N
ν=
TP
It suffices to show that forμ∈ {, , . . . ,N}andν∈ {, , . . . ,N},
TP,μ,ν(f)p≤Cfp for β/(β– ) <p< β. (.)
For fixedμ∈ {, , . . . ,N}andν∈ {, , . . . ,N}, choose two collections ofC∞functions
{λi}i∈Zand{ηj}j∈Zon (,∞) with the following properties:
(i) suppλi⊂[ϕ(i+)–μ,ϕ(i–)–μ],suppηj⊂[ψ(j+)–ν,ψ(j–)–ν]; (ii) ≤λi,ηj≤,
i∈Zλi(t)=
j∈Zηj(t)= ; (iii) |λi(t)|,|ηj(t)| ≤C/t, whereCis a constant.
Define the multiplier operatorSi,jonRm×Rnby
Si,jf(x,y) =λiLμ(x)ηjIν(y)fˆ(x,y). (.)
Then
TP,μ,ν(f)(x,y) =
κ,∈Z
ωκ,;μ,ν∗f(x,y)
=
κ,∈Z
ωκ,;μ,ν∗
i,j∈Z
Si+κ,j+Si+κ,j+f
(x,y)
=
i,j∈Z
κ,∈Z
Si+κ,j+(ωκ,;μ,ν∗Si+κ,j+f)(x,y)
:=
i,j∈Z
Ti,jf(x,y). (.)
Now we consider theLp-boundedness ofTi
,j. By the Littlewood-Paley theory and Lem-ma ., we have
Ti,jfp≤C
κ,∈Z
Si+κ,j+(ωκ,;μ,ν∗Si+κ,j+f) /
p
≤C
κ,∈Z
|ωκ,;μ,ν∗Si+κ,j+f|
/
p
≤C
κ,∈Z
|Si+κ,j+f|
/
p
≤Cfp, <p<∞,i,j∈Z. (.)
On the other hand, by the Littlewood-Paley theory and Plancherel’s theorem, we have
Ti,jf≤C
κ,∈Z
|ωκ,;μ,ν∗Si+κ,j+f|
/
=C
κ,
Rm×Rn
ωκ,;μ,ν(ξ,η)
λi+κLμ(ξ)ηj+Iν(η) ˆ
f(ξ,η)dξdη
≤C
κ,
Ei+κ,j+
ωκ,;μ,ν(ξ,η) ˆ
where Ei+κ,j+ ={(ξ,η)∈ Rm × Rn : ϕ(i+κ+)–μ ≤ |Lμ(ξ)| ≤ϕ(i+κ–)–μ,ψ(j++)–ν ≤ |Iν(η)| ≤ψ(j+–)–ν}. Using Lemma . and Remark ., we have
Ti,jf≤C(ϕ,ψ,μ,ν)Bi,jf, (.)
where
Bi,j=
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
B–iμ ϕ B
–jν
ψ , i,j> –;
B–iμ
ϕ |j|–β, i> –,j≤–; |i|–βB–jν
ψ , i≤–,j> –; |ij|–β, i,j≤–.
(.)
Interpolating (.) and (.), for anyp∈(β/(β– ), β), we can obtainδ∈(, ) such thatδβ> and
Ti,jfp≤C(ϕ,ψ,μ,ν)–δBδi,jfp, β/(β– ) <p< β.
Then we have
i,j∈Z
Ti,jfp≤C(ϕ,ψ,μ,ν) i,j>–
B–ϕiμδB –jνδ ψ +
i>–,j≤–
B–ϕiμδ|j|– δβ
+
i≤–,j>–
|i|–δβB–jνδ ψ +
i,j≤–
|ij|–δβ
fp
≤C(ϕ,ψ,μ,ν)fqp, for β/(β– ) <p< β.
This together with (.) and (.) completes the proof of Theorem ..
3 On the multiple Marcinkiewicz integrals
This section is devoted to the proof of Theorem .. We first introduce some notations and lemmas. Forμ∈ {, , . . . ,N},ν∈ {, , . . . ,N} andi,j∈Z,s,t∈R+, we define the
measures{σiμ,j;,sν,t}and{|σiμ,j;,sν,t|}by
σiμ,j;,sν,t(ξ,η) = i+jst
si,,jt
(x,y)
ρm(x)αm–ρn(y)αn–
×exp–iQμ(x)·ξ+Rν(y)·η
dx dy, (.)
σiμ,j;,sν,t(ξ,η) = i+jst
si,,tj
|(x,y)|
ρm(x)αm–ρn(y)αn–
×exp–iQμ(x)·ξ+Rν(y)·η
dx dy, (.)
wheresi,,jt={(x,y)∈Rm×Rn: i–s≤ρ
m(x)≤is, j–t≤ρn(y)≤jt}andQμ,Rν were
defined as in Section . It is obvious that forμ∈ {, , . . . ,N},ν∈ {, , . . . ,N}
and
FPN(ϕ),PN(ψ)
s,t (x,y) =st
i,j=–∞
i+jσN,N
i,j;s,t ∗f(x,y). (.)
Lemma . Let s,t> , i,j∈Zandϕ,ψ∈F. Suppose that∈Fβ(Sm–×Sn–)for some β> /and satisfies (.)-(.). Then for each pairμandν, there exists a constant C> such that
(i) if|ψ(jt)νI
ν(η)|> , then
σiμ,j;,sν,t(ξ,η) –σiμ,j;–,s,tν(ξ,η)
≤CϕisμL
μ(ξ)min
,logψjtνI
ν(η) –β
; (.)
(ii) if|ϕ(is)μL
μ(ξ)|> , then
σiμ,j;,sν,t(ξ,η) –σiμ,j;,sν,–t (ξ,η)
≤CψjtνIν(η)min
,logϕisμLμ(ξ) –β
; (.)
(iii) if|ϕ(is)μL
μ(ξ)|> and|ψ(jt)νIν(η)|> , then
σiμ,j;,sν,t(ξ,η)≤Cmin,logϕisμLμ(ξ)– β
,logψjtνIν(η)– β
; (.)
σiμ,j;,sν,t(ξ,η)≤Cmin,logϕisμLμ(ξ) –β
logψjtνIν(η) –β
; (.)
(iv)
σiμ,j;,sν,t(ξ,η) –σiμ,j;–,s,tν(ξ,η) –σiμ,j;,sν,t–(ξ,η) +σiμ,j;–,s,tν–(ξ,η)
≤Cmin,ϕisμLμ(ξ),ψ
jtνIν(η),
ϕisμLμ(ξ)ψ
jtνIν(η). (.)
The constant C is independent of the coefficients of PNand PN.
Proof Set
Uiμ,sx,ξ= is
is
i–s
exp–iQμ
Am,ρmx
·ξdρm;
Vjν,ty,η= jt
jt
j–t
exp–iRν
An,ρny
·ηdρ
n.
By Lemma ., there exist,∈(, ] such that
Uiμ,sx,ξ≤Cmin,ϕisμLμ(ξ)·x –
, (.)
Vjν,ty,η≤Cmin,ψjtνIν(η)·y –
When|ϕ(is)μL
μ(ξ)|> , sincet/(logt)βis increasing in (eβ,∞), we have
Uiμ,sx,ξ≤Cmin
,(loge
β|(ϕ(is)μL
μ(ξ))·x|–)β
(log|ϕ(is)μL μ(ξ)|)β
. (.)
Similarly, when|ψ(jt)νI ν(η)|>
Vjν,ty,η≤Cmin
,(loge
β|(ψ(jt)νI
ν(η))·y|–)β
(log|ψ(jt)νI ν(η)|)β
. (.)
By the definition ofσiμ,j,;sν,t, we have
σiμ,j;,sν,t(ξ,η) –σiμ,j;–,s,tν(ξ,η)
≤C
Sm–×Sn–
/
exp–iQμ
Am,isρ mx
·ξ–exp–iQ μ–
Am,isρ mx
·ξdρ
m
×Vν
j,t
y,ηx,ydσm
xdσn
y
≤CϕisμLμ(ξ)
Sm–×Sn–
Vjν,ty,ηx,ydσm
xdσn
y.
Combining (.) with the fact∈Fβ(Sm–×Sn–), we obtain (.). Similarly, we can
conclude (.). To prove (.) and (.), we write
σiμ,j;,sν,t(ξ,η)≤C
Sm–×Sn–
x,yUiμ,sx,ξVjν,ty,ηdσm
xdσn
y.
Combining (.)-(.) with the fact that ∈Fβ(Sm–×Sn–), we get (.) and (.).
Finally, (.) follows from the inequality
σiμ,j;,sν,t(ξ,η) –σiμ,j;–,s,tν(ξ,η) –σiμ,j;,sν,t–(ξ,η) +σiμ,j;–,s,tν–(ξ,η)
≤C
Sm–×Sn–
x,y
×
/
/
exp–iQμ–
Am,isρmx
·ξ+Rν–
An,jtρ ny
·η ×exp–iϕisρm
μ
Lμ(ξ)·x
–
×exp–iψjtρn
ν
Iν(η)·y
– dρmdρn
dσm
xdσn
y.
This completes the proof of Lemma ..
We now take two radial Schwartz functionsφ∈S(Rm) andφ∈S(Rn) such thatφ i(t)≡ for|t| ≤ andφi(t)≡ for|t|>min{Bϕ,Bψ}(i= , ), whereBϕ,Bψare as in Remark ..
Define the measures{ωiμ,j,;νs,t}by
ωμi,j,;νs,t(ξ,η) =σiμ,j;,sν,t(ξ,η)(μ)(ν) –σiμ,j;–,s,tν(ξ,η)(μ– )(ν)
