PII. S0161171204312342 http://ijmms.hindawi.com © Hindawi Publishing Corp.

**ROUGH SINGULAR INTEGRALS ON PRODUCT SPACES**

**AHMAD AL-SALMAN and HUSSAIN AL-QASSEM**

Received 26 December 2003

We study the mapping properties of singular integral operators deﬁned by mappings of ﬁnite type. We prove that such singular integral operators are bounded on the Lebesgue spaces under the condition that the singular kernels are allowed to be in certain block spaces.

2000 Mathematics Subject Classiﬁcation: 42B20, 42B15, 42B25.

**1. Introduction and results.** For*n∈*N,*n≥*2, let*K(·)*be a Calderón-Zygmund
ker-nel deﬁned onR*n*_{, that is,}

*K*Ω*(y)=*Ω*(y)|y|−n,* (1.1)

whereΩ*∈L*1_{(}_{S}*n−1 _{)}*

_{is a homogeneous function of degree zero that satisﬁes}

**S***n−*1Ω(y)dσ (y)*=*0 (1.2)

with*dσ (·)*being the normalized Lebesgue measure on the unit sphere.

Let *Bn(*0*,*1*)* be the unit ball centered at the origin in R*n*. For a Ꮿ*∞* mapping Φ :

*Bn(*0*,*1*)→*R*d*,*d≥*1, consider the singular integral operator

*T*Φ*,*Ω*f (x)=*p.v.

*Bn(*0*,*1*)*

*fx−*Φ*(y)K*Ω*(y)dy,* (1.3)

where, p.v. denotes the principal value.

It is known that ifΦis of ﬁnite type at 0 (seeDeﬁnition 2.2) andΩ*∈*Ꮿ1_{(}_{S}*n−1 _{)}*

_{, then}

*T*

_{Φ}

*,*Ωis bounded on

*Lp*for 1

*< p <∞*[15]. Moreover, it is known that

*T*Φ

*,*Ωmay fail to be

bounded on*Lp*_{for any}_{p}_{if the ﬁnite-type condition is removed. In [}_{8}_{], Fan et al. showed}
that the*Lp*_{boundedness of the operator}_{T}

Φ*,*Ωstill holds if the conditionΩ*∈*Ꮿ1*(***S***n−1)*

is replaced by the weaker conditionΩ*∈Lq _{(}*

_{S}*n−*1

_{)}_{for some}

_{q >}_{1. Subsequently, the}

*(1*

_{L}p*< p <∞*) boundedness of

*T*Φ

*,*Ωwas established under conditions much weaker than

Ω*∈Lq _{(}*

_{S}*n−1*

_{)}_{[}

_{1}

_{,}

_{6}

_{]. In particular, Al-Qassem et al. [}

_{1}

_{] established the}

_{L}p_{boundedness}of

*T*

_{Φ}

*,*Ωunder the condition that the functionΩbelongs to the block space

*Bq*0

*,*0

*(*

**S**

*n−*1

*)*introduced by Jiang and Lu in (see [14]). In fact, they proved the following theorem.

**Theorem _{1.1}.**

_{Let}_{T}_{Φ}

_{,}_{Ω}

*Ω*

_{be given by (}_{1.3}_{). Suppose that}*∈*0

_{B}_{q}*,*0

_{(}

_{S}*n−*1

_{)}_{for some}_{q >}_{1}

_{.}*If*Φ*is of ﬁnite type at*0*, then for*1*< p <∞there exists a constantCp>*0*such that*

*T*Φ*,*Ω*fLp(Rd)≤CpfLp _{(R}d)* (1.4)

It should be pointed out here that the conditionΩ*∈Bq*0*,*0*(***S***n−1)*inTheorem 1.1was
recently proved to be nearly optimal. In fact, Al-Qassem et al. [2] showed that if the
conditionΩ*∈Bq*0*,*0*(***S***n−1)*is replaced byΩ*∈B*0*q,ν(***S***n−1)*for some*ν <*0, then the
corre-sponding classical Calderón-Zygmund operator

*T*_{Ω}*f (x)=*p.v.

R*nf (x−y)K*Ω*(y)dy* (1.5)

may fail to be bounded on*Lp*_{at any 1}_{< p <}_{∞}_{.}

Feﬀerman [11] and Feﬀerman and Stein [12] studied singular integrals on product domains. Namely, they studied operators of the form

*P*Ω*f(x, y)=*p.v.

R*n _{×R}mf (x−u, y−v)K*Ω

*(u, v)du dv,*(1.6)

where*n, m≥*2,

*K*_{Ω}*(u, v)=*Ω*|u|−1 _{u,}_{|}_{v}_{|}−1_{v}_{|}_{u}_{|}−n_{|}_{v}_{|}−m_{,}*

Ω*∈L*1_{S}*n−1 _{×}*

_{S}*m−1*

_{,}_{Ω}

_{(tx, sy)}_{=}_{Ω}

_{(x, y)}_{for any}

_{t, s >}_{0}

_{,}**S***n−*1Ω*(u,·)dσ (u)=*0*,*

**S***m−*1Ω*(·, v)dσ (v)=*0*.*

(1.7)

In [12], it was shown that*P*Ωis bounded on*Lp(*R*n+m)*for 1*< p <∞*ifΩsatisﬁes

some regularity conditions. Subsequently, the*Lp*_{(1}_{< p <}_{∞}_{) boundedness of}* _{P}*
Ωwas

established under weaker conditions onΩ, ﬁrst in [7] forΩ*∈Lq _{(}*

_{S}*n−*1

_{×}

_{S}*m−*1

_{)}_{with}

_{q >}_{1}and then in [9] forΩ

*∈q>*1

*Bq*0

*,*1

*(*

**S**

*n−1×*

**S**

*m−1)*which contains

*q>*1

*Lq(*

**S**

*n−1×*

**S**

*m−1)*as a proper subspace, where

*Bq*0

*,*1represents a special class of block space on

**S**

*n−1×*

**S**

*m−1*; for

*p=*2, it was proved by Jiang and Lu in [13]). The deﬁnition of block spaces will be recalled inSection 2(see Deﬁnitions2.2and2.3).

The analogue of the operators*T*Φ*,*Ωin (1.3) on product domains is deﬁned as follows.

For*N, M∈*N, letΦ:*Bn(*0*, r )→*R*N* and Ψ:*Bm(*0*, r )→*R*M* beᏯ*∞*mappings. Deﬁne
the singular integral operator*P*Ω*,*Φ*,*Ψby

*P*Ω*,*Φ*,*Ψ*f(x, y)=*p.v.

*Bn(*0*,*1*)×Bm(*0*,*1*)*

*fx−*Φ(u), y*−*Ψ(v)*K*Ω*(u, v)du dv.* (1.8)

Using the ideas developed in [4,8], we can easily show that*P*_{Ω}*,*Φ*,*Ψ is bounded on*Lp*

*(*1*< p <∞)*provided thatΦandΨare of ﬁnite type at 0 andΩ*∈Lq _{(}*

_{S}*n−*1

_{×}

_{S}*m−*1

_{)}_{for}some

*q >*1. However, the natural question that arises here is as follows.

**Question _{1.2}.**

_{Suppose that}Ω

*∈*

_{q>}_{1}

*0*

_{B}_{q}*,*1

_{(}

_{S}*n−1×*

_{S}*m−1*

_{)}_{and}Φ

_{and}Ψ

_{are of ﬁnite}type at 0. Is the operator

*P*

_{Ω}

*,*Φ

*,*Ψ bounded on

*Lp(*1

*< p <∞)*?

**Theorem1.3.** *LetP*_{Ω}_{,}_{Φ}_{,}_{Ψ} *be given by (1.8). Suppose that* Ω*∈B _{q}*0

*,*1

*(*

**S**

*n−1×*

**S**

*m−1)for*

*someq >*1*. If*Φ*and*Ψ *are of ﬁnite type at*0*, then for*1*< p <∞there exists a constant*

*Cp>*0*such that*

_{P}_{Ω}_{,}_{Φ}_{,}_{Ψ}_{(f )}_{L}p_{(}_{R}*N _{×R}M)≤CpfLp(*R

*N*(1.9)

_{×R}M)*for anyf∈Lp _{(}*

_{R}

*N*

_{×}_{R}

*M*

_{)}_{.}Regarding the conditionΩ*∈Bq*0*,*1*(***S***n−*1*×***S***m−*1*)* inTheorem 1.3, we should remark
here that in a recent paper [5], Al-Salman was able to obtain a similar result to that in
[2]. More precisely, Al-Salman showed that the size conditionΩ*∈Bq*0*,*1*(***S***n−*1*×***S***m−*1*)*is
sharp in the sense that ifΩ*∈Bq*0*,*1*(***S***n−*1*×***S***m−*1*)*is replaced byΩ*∈B*0*q,*1−*ε(***S***n−*1*×***S***m−*1*)*
for some*ε >*0, then the operator*P*Ωmay fail to be bounded on*Lp*for any*p*.

Also, in this paper we will give a similar result for the truncated singular integral operator

*P*_{Ω}*∗ _{,}*

_{Φ}

_{,}_{Ψ}

*f (x, y)=*sup

*ε*1

*,ε*2

*>*0

_{E(ε}

1*,ε*2*,r )*

*fx−*Φ*(u), y−*Ψ*(v)K*_{Ω}*(u, v)du dv,* (1.10)

where*E(ε1, ε2, r )= {(u, v)∈*R*n _{×}*

_{R}

*m*

_{:}

_{ε1}_{≤ |}_{u}_{|}_{< r , ε2}_{≤ |}_{v}_{|}_{< r}_{}}_{,}

_{x}_{∈}_{R}

*N*

_{and}

_{y}_{∈}_{R}

*M*

_{.}In fact, we have the following.

**Theorem _{1.4}.**

_{Let}_{P}∗Ω*,*Φ*,*Ψ *be given by (1.1) withr=*1*. Suppose that*Ω*∈Bq*0*,*1*(***S***n−*1*×*
**S***m−1 _{)}_{for some}_{q >}*

_{1}

_{. If}_{Φ}

_{and}_{Ψ}

_{are of ﬁnite type at}_{0}

_{, then for}_{1}

_{< p <}_{∞}_{there exists a}*constantCp>*0*such that*

*P*_{Ω}*∗,*Φ*,*Ψ*(f )Lp _{(}*

_{R}

*N*

_{×R}M_{)}≤CpfLp_{(}_{R}

*N*(1.11)

_{×R}M_{)}*for anyf∈Lp _{(}*

_{R}

*N*

_{×}_{R}

*M*

_{)}_{.}It is worth pointing out that, as in the one-parameter setting, we can show that the
*Lp*_{boundedness of the operators}_{P}

Ω*,*Φ*,*Ψ and*P*Ω*∗,*Φ*,*Ψ may fail for any*p*if at least one of

the mappingsΦandΨis not of ﬁnite type at 0.

**2. Some definitions and lemmas.** We start by the following deﬁnition.

**Definition2.1.** Let*U*be an open set inR*n*andΘ:*U→*R*d*a smooth mapping. For
*x0∈U*, it is said thatΘis of ﬁnite type at*x0*if, for each unit vector*η*inR*d*_{, there is a}
multi-index*α*so that

*∂αx*

Θ*(x)·η _{x}_{=}_{x}*

0≠0*.* (2.1)

**Definition _{2.2}.**

_{For 1}

_{< q}≤ ∞_{, it is said that a measurable function}

_{b(x, y)}_{on}

**S**

*n−1*

_{×}

_{S}*m−1*

_{is a}

_{q}_{-block if it satisﬁes the following:}

(i) supp*(b)⊆I*, where*I*is a cap on**S***n−*1_{×}_{S}*m−*1_{, that is,}

*I=* *x* *∈***S***n−1*_{:}_{x}* _{−}_{x}*
0

*< α*

*×* *y* *∈***S***m−1*_{:}_{y}* _{−}_{y}*
0

*< β*

(2.2)

for some*α, β >*0,*x*_{0}*∈***S***n−1*_{, and}_{y}

**Definition2.3.** The class *B*0* _{q},*1

*(*

**S**

*n−1×*

**S**

*m−1)*, 1

*< q≤ ∞*, consists of all functions Ω

*∈L*1

_{(}

_{S}*n−*1

_{×}

_{S}*m−*1

_{)}_{of the form}

_{Ω}

_{=}∞*µ=*1*cµbµ*, where each*cµ* is a complex number;
each*bµ*is a*q*-block supported on a cap*Iµ*on**S***n−1×***S***m−1*; and

*M*0*,*1

*q* *cµ*

*=*

*∞*

*µ=1*
*cµ*

1*+*

log1

*Iµ*

2

*<∞.* (2.3)

In dealing with singular integrals along subvarieties with rough kernels, an approach
well-established by now is to decompose the operator into an inﬁnite sum of Borel
measures then to seek certain Fourier transform estimates and certain*Lp*_{estimates of}
Littlewood-Paley type. For more details, we advise the readers to consult [1,3,4,6,7,

8,10], among others. A particular result that we will need to prove our results is the following result in [4] which is an extension of a result of Duoandikoetxea in [7].

**Theorem** _{2.4}._{Let}_{M, N}*∈*N_{and let}{_{σ}_{k,j}(l,s)_{:}_{k, j}*∈*Z_{,}_{0}*≤ _{l}≤_{N,}*

_{0}

*≤*

_{s}*≤*

_{M}}_{be a}*family of Borel measures on*R*n _{×}*

_{R}

*m*0

_{with}_{σ}(l,*)*

*k,j* *=*0*andσ*
*(*0*,s)*

*k,j* *=*0*for everyk, j∈*Z*.*

*Let{al, bs*: 1*≤l≤N,* 1*≤s≤M} ⊆*R*+\(*0*,*2*),{B(l), D(s)*: 1*≤l≤N,* 1*≤s≤M} ⊆*N*,*

*{αl, βs*: 1*≤l≤N,*1*≤s≤M} ⊆*R*+, and letLl*:R*n→*R*B(l)andQs*:R*m→*R*D(s)be linear*

*transformations for*1*≤l≤N,*1*≤s≤M. Suppose that for someB >*1*andp0∈(*2*,∞)*

*the following hold fork, j∈*Z*,*1*≤l≤N,*1*≤s≤M, and(ξ, η)∈*R*n _{×}*

_{R}

*m*(i)

_{:}*σ*2

_{k,j}(l,s) ≤B_{;}

(ii) *|σ*ˆ* _{k,j}(l,s)(ξ, η)| ≤B*2

_{|}_{a}kB*l* *Ll(ξ)|−αl/B|bjBs* *Qs(η)|−βs/B;*
(iii) *|σ*ˆ* _{k,j}(l,s)(ξ, η)−σ*ˆ

*1*

_{k,j}(l−*,s)(ξ, η)| ≤B*2

_{|}_{a}kB*l* *Ll(ξ)|αl/B|bjBs* *Qs(η)|−βs/B;*
(iv) *|σ*ˆ* _{k,j}(l,s)(ξ, η)−σ*ˆ

*1*

_{k,j}(l,s−*)(ξ, η)| ≤B*2

_{|}_{a}kB*l* *Ll(ξ)|−αl/B|bsjBQs(η)|βs/B;*
(v) *|σ*ˆ* _{k,j}(l,s)(ξ, η)−σ*ˆ

*1*

_{k,j}(l−*,s)(ξ, η)−σ*ˆ

*1*

_{k,j}(l,s−*)(ξ, η)+σ*ˆ

*1*

_{k,j}(l−*,s−*1

*)(ξ, η)| ≤B*2

_{|}_{a}kB*l* *Ll(ξ)|αl/B×*

*|bjBs* *Qs(η)|βs/B;*

(vi) *|σ*ˆ* _{k,j}(l,s−1)(ξ, η)−σ*ˆ

*2*

_{k,j}(l−1,s−1)(ξ, η)| ≤B

_{|}_{a}kB*l* *Ll(ξ)|αl/B;*
(vii) *|σ*ˆ* _{k,j}(l−1,s)(ξ, η)−σ*ˆ

*2*

_{k,j}(l−1,s−1)(ξ, η)| ≤B

_{|}_{b}jB*s* *Qs(η)|βs/B;*
(viii) *for arbitrary functionsgk,j* *on*R*n×*R*m,*

*k,j∈Z*

*σ _{k,j}(l,s)∗gk,j*

2

1*/*2

*p*0

*≤B*2

*k,j∈Z*
*gk,j*2

1*/*2

*p*0

*.* (2.4)

*Then forp*0*< p < p0, wherep*0*is the conjugate exponent ofp0, there exists a positive*

*constantCpsuch that*

*k,j∈Z*

*σ _{k,j}(N,M)∗f*

_{L}_{p}_{(}R*n _{×R}m)*

*≤CpB*2*fLp _{(}*

_{R}

*n*(2.5)

_{×R}m),

*k,j∈Z*

*σ _{k,j}(N,M)∗f*2

*(*1*/*2*)*

_{L}p_{(}_{R}*n _{×R}m)*

*≤CpB*2*fLp _{(}*

_{R}

*n*(2.6)

_{×R}m_{)}*hold for allf* *inLp _{(}*

_{R}

*n*

_{×}_{R}

*m*

_{)}_{. The constant}_{C}*pis independent of the linear *

It is clear that inequality (2.4) is one of the key elements inTheorem 2.4. In particular,
the range of the parameter*p*where (2.5) and (2.6) hold is completely determined by
the largest*p0*where (2.4) holds. Clearly, if (2.4) holds for large*p0→ ∞*, then (2.5) and
(2.6) hold for all 1*< p <∞*. It turns out that to prove our results, we will indeed run
into the case where we need to obtain (2.5) and (2.6) for all 1*< p <∞*. However, in our
case this obstacle can be resolved. In fact, we will show that inequality (2.4) holds for
all*p0=*4*,*8*,*16*, . . . .*Our main tools to achieve this areLemma 2.5andTheorem 2.6.

By a quick investigation of the proof of [7, Lemma 1], we have the following.

**Lemma2.5.** *Let{ν _{k,j}*:

*k, j∈*Z

*}be a sequence of Borel measures in*R

*n×*R

*mand let*

*ν∗(f )=*sup

*k,j∈Z||νk,j|∗f|. Suppose that for someq >*1

*andA >*0

*,*

_{ν}∗_{(f )}

*q≤Afq* (2.7)

*for everyfinLq _{(}*

_{R}

*n*

_{×}_{R}

*m*

_{)}_{. Then the vector-valued inequality}

*k,j∈Z*

*νk,j∗gk,j*

2

1*/*2
_{p}

0

*≤A*sup
*k,j∈Z*

*νk,j*

*k,j∈Z*
*gk,j*

2

1*/*2
_{p}

0

(2.8)

*holds for|*1*/p0−*1*/*2*| =*1*/*2*qand for arbitrary functions{gk,j}on*R*n×*R*m.*

Clearly, if inequality (2.7) holds for all 1*< q <∞*, then inequality (2.8) holds for
all *p0* *=*4*,*8*,*16*, . . .* which is the case that we will need to prove our results. But in
many applications including the ones in this paper inequality (2.7) is not always freely
available for all 1*< q <∞*. However, this problem can be resolved by repeated use of

Theorem 2.4andLemma 2.5along with a certain bootstrapping argument (see (2.15)– (2.22)). To be more speciﬁc, we prove the following theorem.

**Theorem** **2.6.** *Letm, n, M, N* *∈*N*,* *B >*1*,* *a, b≥*2*,* *α, β >*0*, and letL*:R*n→*R*N*

*andQ*:R*m _{→}*

_{R}

*M*

_{be linear transformations. Let}_{{}_{λ}(l,s)*k,j* :*k, j∈*Z*, l=*1*,*2*, s=*1*,*2*}be*

*a sequence of nonnegative Borel measures on*R*n _{×}*

_{R}

*m*

_{with}_{}_{λ}(l,s)*k,j* * ≤B*2*,* 1*≤l, s≤*2*.*

*Suppose that*

(i) *|*ˆ*λ(k,j*2*,*2*)(ξ, η)| ≤B*2*|akBL(ξ)|−α/B|bjBQ(η)|−β/B;*
(ii) *|*ˆ* _{λ}(*2

*,*2

*)*

*k,j* *(ξ, η)−λ*ˆ
*(*1*,*2*)*

*k,j* *(ξ, η)| ≤B*2*|akBL(ξ)|α/B|bjBQ(η)|−β/B;*
(iii) *|*ˆ* _{λ}(*2

*,*2

*)*

*k,j* *(ξ, η)−λ*ˆ
*(*2*,*1*)*

*k,j* *(ξ, η)| ≤B*2*|akBL(ξ)|−α/B|bjBQ(η)|β/B;*
(iv) *|*ˆ* _{λ}(*2

*,*2

*)*

*k,j* *(ξ, η)−λ*ˆ
*(*1*,*2*)*

*k,j* *(ξ, η)−*ˆ*λ*
*(*2*,*1*)*

*k,j* *(ξ, η)+*ˆ*λ*
*(*1*,*1*)*

*k,j* *(ξ, η)| ≤B*2*|akBL(ξ)|α/B|bjBQ(η)|β/B;*
(v) *|*ˆ* _{λ}(*2

*,*1

*)*

*k,j* *(ξ, η)−λ*ˆ
*(*1*,*1*)*

*k,j* *(ξ, η)| ≤B*2*|akBL(ξ)|α/B;*
(vi) *|*ˆ* _{λ}(*1

*,*2

*)*

*k,j* *(ξ, η)−λ*ˆ
*(*1*,*1*)*

*k,j* *(ξ, η)| ≤B*2*|bjBQ(η)|β/B.*

*Suppose also that the maximal functionsM(l,s) _{(f )}_{=}*

_{sup}

*k,j∈Z||λ(l,s)k,j* *|∗f|,*1*≤l,s≤*2*,*

*satisfy*

*M(l,s) _{(f )}*

*p≤B*2*fp* (2.9)

*Then the inequality*

* _{M}(*2

*,*2

*)*

_{(f )}*p≤CB*2*fp* (2.10)

*holds for all*1*< p ∞, andf* *inLp _{(}*

_{R}

*n*

_{×}_{R}

*m*

_{)}_{. The constant}_{C}

_{is independent of}_{B}_{and}*the linear transformationsLandQ.*

**Proof.** _{Let}_{d}_{be a ﬁxed positive integer. For 1}*≤ _{u}≤_{d}*

_{, we let}

_{π}_{u}d_{:}R

*d*

*→*R

*u*

_{be}the projection operator. By a similar argument to that in [10], we may assume that

*N≤n*,

*M≤m*,

*L=πNn*, and

*Q=πMm*. Choose and ﬁx a Schwartz function

*φ∈*(R

*)*such that

*(φ)(t)*ˆ

*≡*1 if

*|t| ≤*1

*/*2 and

*(φ)(t)*ˆ

*≡*0 if

*|t| ≥*1. Deﬁne

*ϕk*onR

*N*and

*ψj*on R

*M*

_{by}

_{(}_{ϕ}_{ˆ}

*k)(w)=(φ)(*ˆ *|akBw|*2*)*and *(ψ*ˆ*j)(z)=(φ)(*ˆ *|bjBz|*2*)*. Deﬁne the sequence of
measures*{Γk,j}*by

ˆ

Γ*k,j(ξ, η)=λ*ˆ*(k,j*2*,*2*)(ξ, η)−*ˆ*λ*
*(*1*,*2*)*
*k,j* *(ξ, η)*

ˆ
*ϕkπNnξ*

*−*ˆ* _{λ}(*2

*,*1

*)*

*k,j* *(ξ, η)*

*×ψ*ˆ*j*

*πMmη*

*+*ˆ* _{λ}(*1

*,*1

*)*

*k,j* *(ξ, η)*

ˆ
*ϕk*

*πNnξ*

ˆ
*ψj*

*πMmη*

*.* (2.11)

Then one can easily verify that

ˆ_{Γ}* _{k,j}_{(ξ, η)}_{≤}_{CB}*2

*2*

_{a}kB_{L(ξ)}±α/*B*

*bjBQ(η)±β/*2*B* (2.12)

for*(ξ, η)∈*R*n _{×}*

_{R}

*m*

_{. Let}

_{g(f )}_{=}_{(}*k,j∈Z|Γk,j∗f|*2*)*
1/2

andΓ*∗ _{(f )}_{=}*

_{sup}

*k,j∈Z||Γk,j| ∗f|*.
By (2.11) we have

*M(*2*,*2*) _{f (x, y)}_{≤}_{g(f )(x, y)}_{+}_{C}*

_{ᏹ}

R*N⊗id*_{R}*n−N⊗id*_{R}*MM(*1*,*2*)f (x, y)*

*+Cid*_{R}*N⊗*ᏹ_{R}*M⊗id*_{R}*m−MM(*2*,*1*)f (x, y)*

*+C*ᏹ_{R}*N⊗id*_{R}*n−N⊗*ᏹ_{R}*M⊗id*_{R}*m−MM(*1*,*1*)f (x, y),*

(2.13)

Γ*∗ _{f (x, y)}_{≤}_{g(f )(x, y)}_{+}*

_{2}

_{C}_{ᏹ}

R*N⊗id*_{R}*n−N⊗id*_{R}*MM(*1*,*2*)f (x, y)*

*+*2*Cid*_{R}*N⊗*ᏹ_{R}*M⊗id*_{R}*m−MM(*2*,*1*)f (x, y)*

*+*2*C*ᏹ_{R}*N⊗id*_{R}*n−N⊗*ᏹ_{R}*M⊗id*_{R}*m−MM(*1*,*1*)f (x, y),*

(2.14)

whereᏹ_{R}*d* is the classical Hardy-Littlewood maximal function onR*d*.
By Plancherel’s theorem and (2.12), we get

_{g(f )}_{L}_{2}* _{≤}_{CB}*2

_{f}*L*2 (2.15)

which implies by (2.9) and (2.14) that
_{Γ}*∗ _{(f )}*

*L*2*≤CB*2*fL*2*.* (2.16)

By applyingLemma 2.5(for*q=*2) along with the trivial estimate*Γk,j ≤CB*2, we get

*k,j∈Z*

_{Γ}_{k,j}_{∗}gk,j

2

1*/*2
_{p}

0

*≤Cp*0*B*

2

*k,j∈Z*
*gk,j*

2

1*/*2
_{p}

0

for all*p0*satisfying 1*/*4*= |*1*/p0−*1*/*2*|*. ByTheorem 2.4, (2.12), and (2.17), we obtain

* _{g(f )}_{L}_{p}_{≤}_{C}_{p}_{B}*2

_{}_{f}_{}*Lp* for4

3*< p <*4 (2.18)

which implies by (2.9) and (2.14) that

_{Γ}*∗ _{(f )}*

*Lp≤CB*2*fLp* for4

3*< p <*4*.* (2.19)

Reasoning as above, we get

*g(f )Lp≤CpB*2*fLp* for8

7*< p <*8*.* (2.20)

By repeating the above argument we eventually get

*g(f ) _{L}p≤CpB*2

*fLp*for 1

*< p <∞*(2.21)

which when combined with (2.9) and (2.13) implies that

*M(*2*,*2*) _{(f )}*

*Lp≤CpB*2*fLp* for 1*< p <∞.* (2.22)

For*p= ∞*, the inequality holds trivially. The proof of the theorem is complete.

For*ρ≥*2,*k, j∈*Z*−*, let*D(k, j, ρ)= {(u, v)∈*R*n×*R*m*:*ρk−*1*≤ |u|< ρk, ρj−*1*≤ |v|<*
*ρj _{}}*

_{. For suitable mappings}

_{Γ}

_{:}

_{R}

*n*

_{→}_{R}

*N*

_{,}

_{Λ}

_{:}

_{R}

*m*

_{→}_{R}

*M*

_{, and ˜}

_{b}_{:}

_{S}*n−*1

_{×}

_{S}*m−*1

_{→}_{R}

_{, we}deﬁne the measures

*{∆˜b,*Γ

*,*Λ

*,k,j,ρ*:

*k, j∈*Z

*−}*and the related maximal operator∆

*∗b,*˜Γ

*,*Λ

*,ρ*onR

*N*

_{×}_{R}

*M*

_{by}

R*N _{×R}Mf d*∆˜

*b,*Γ

*,*Λ

*,k,j,ρ=*

*D(k,j,ρ)f*

Γ*(x),*Λ*(y)b*˜*x, y* *|x|−n _{|}_{y}_{|}−m_{dx dy,}*

∆*∗*
˜

*b,*Γ*,*Λ*,ρf (x, y)=* sup
*k,j∈Z−*

_{∆}_{b,}_{˜}

Γ*,*Λ*,k,j,ρ∗f (x, y).*

(2.23)

For*l∈*N, letᏭ*l*denote the class of polynomials of*l*variables with real coeﬃcients.
For*d∈*Nand*=(1, . . . ,**d)∈(Ꮽ1)d*, deﬁne the maximal functionᏹ*f*onR*d*by

ᏹ*f (x)=*sup
*r >*0
1
*r*

*r*

*−r*

*fx−*(t)*dt.* (2.24)

The following result can be found in [15].

**Lemma2.7.** *For*1*< p≤ ∞, there exists a positive constantC _{p}such that*

_{ᏹ}_{}*f _{p}≤Cpfp* (2.25)

*forf∈Lp _{(}*

_{R}

*d*

_{)}_{. The constant}_{C}*pmay depend on the degrees of the polynomials*1, . . . ,

**Lemma2.8.** *Let*Φ:*B _{n}(*0

*,*1

*)→*R

*N*

*and*Ψ:

*B*0

_{m}(*,*1

*)→*R

*M*

*beC∞*

*mappings and let,*ᏼ

*=(P1, . . . , PN)*:R

*n→*R

*Nand*ᏽ

*=(Q1, . . . , QM)*:R

*m→*R

*Mbe polynomial mappings. Let*˜

*b(·,·)be a function on***S***n−1 _{×}*

_{S}*m−1*(i) ˜

_{satisfying the following conditions:}

_{b}_{}_{L}_{q}*(Sn−*1_{×}_{S}*m−*1*)≤ |I|−1/q* *for someq >*1*and for some capIon***S***n−1×***S***m−1;*
(ii) ˜* _{b}_{}_{L}*1

*1*

_{(S}n−

_{×}

_{S}*m−*1

*1*

_{)}≤*. Letρ=*2

*[*log

*(*1

*/|I|)]andB(I, ρ)=*log

*(*1

*/|I|)if|I|< e−*1

*and*

*letρ=*2*andB(I, ρ)=*1*if|I| ≥e−1, where[·]is the greatest integer function.*
*If*Φ*and*Ψ*are of ﬁnite type at*0*, then for*1*< p≤ ∞andf∈Lp _{(}*

_{R}

*N*

_{×}_{R}

*M*

_{)}_{there exists}*a positive constantCpwhich is independent of*˜*bsuch that*

∆*∗*

˜

*b,*ᏼ*,*Ψ*,ρ(f )*

_{L}_{p}_{(}

R*N _{×R}M)≤Cp*

*B(I, ρ)*2*fLp _{(}*

_{R}

*N*(2.26)

_{×R}M_{)},
∆*∗*

˜

*b,*Φ*,*ᏽ*,ρ(f )*

_{L}p_{(}_{R}*N _{×R}M_{)}≤Cp*

*B(I, ρ)*2*fLp _{(}*

_{R}

*N*(2.27)

_{×R}M_{)}.**Proof.** We will only present the proof of (2.26). By the deﬁnition of∆*∗*_{˜}

*b,*ᏼ*,*Ψ*,ρ*we notice
that∆*∗*

˜

*b,*ᏼ*,*Ψ*,ρf (x, y)*is dominated by

sup
*j∈Z−*

*ρj−*1* _{≤|}_{v}_{|}_{<ρ}j*
1

*|v|m*

**S***n−*1

˜_{b(u, v)}_{ᏹ}_{ᏼ}_{,ρ,u}_{f}_{·}_{, y}_{−}_{Ψ}_{(v)}_{(x)}_{dσ (u)dv,}_{(2.28)}

where

ᏹᏼ*,ρ,uh(x)=*sup
*k∈Z−*

*ρk*

*ρk−*1

*hx−*ᏼ(tu)*dt*

*t* *.* (2.29)

ByLemma 2.7we immediately get

∆*∗*

˜

*b,*ᏼ*,*Ψ*,ρ(f )*

_{L}p_{(}_{R}*N _{×R}M)≤Cp*

*B(I, ρ)*

R*M*

_{Ᏼ}_{Ψ}* _{,}*˜

_{b}0*f (·, y)*
*p*
*Lp _{(}*

_{R}

*N*

_{)}dy1*/p*

*,* (2.30)

where

Ᏼ_{Ψ}*,*˜*b*0*g(y)=*sup
*j∈Z−*

*ρj−*1_{≤|}_{v}_{|}_{<ρ}j

*gy−*Ψ*(v)b0(v)*˜

*|v|m* *dv* (2.31)

and ˜*b0*is a function on**S***m−1*_{deﬁned by ˜}_{b0(v)}_{=}

**S***n−*1*|b(u, v)*˜ *|dσ (u)*. By the arguments
in the proof of the*Lp*_{boundedness of the corresponding maximal function in the }
one-parameter setting in [1, Theorem 3.8], we obtain (2.26). This ends the proof of our
theorem.

ByLemma 2.7we immediately get the following.

**Lemma2.9.** *Let*ᏼ*=(P1, . . . , P _{N})*:R

*n→*R

*N*

*and*ᏽ

*=(Q1, . . . , Q*:R

_{M})*m→*R

*M*

*be*

*poly-nomial mappings. Letb(*˜ *·,·)be as inLemma 2.8. Then for* 1*< p≤ ∞, there exists a*
*constantCpsuch that*

∆*∗*

˜

*b,*ᏼ*,*ᏽ*,ρ(f )*

_{p}≤Cp

*B(I, ρ)*2*fLp _{(}*

_{R}

*N*(2.32)

_{×R}M_{)}**3. Certain Fourier transform estimates.** We will need the following two lemmas
from [8].

**Lemma3.1.** *Let*Φ:*B(*0*,*1*)→*R*dbe a smooth mapping and let*Ω*be a homogeneous*

*function on*R*n* _{of degree}_{0}_{. Suppose that}_{Φ} _{is of ﬁnite type at}_{0}_{and}_{Ω}_{∈}_{L}q_{(}_{S}*n−1 _{)}_{for}*

*someq >*1*. Then there areN0∈*N*,δ∈(*0*,*1*],C >*0*, andj0∈*Z*−such that*

2*j−*1* _{≤|}y|<*2

*je*

*−iξ·*Φ*(y)*Ω*(y)*

*|y|ndy*

*≤CΩLq _{(S}n−*1

*2*

_{)}*N*0

*j|ξ|−δ*(3.1)

*for allj≤j0andξ∈*R*d _{.}*

**Lemma3.2.** *Letm∈*N*and letR(·)be a real-valued polynomial on*R*nwith*deg*(R) *
*m−*1*. Suppose thatP (y)=|α|=maαy*

*α*

*+R(y),*Ω*is a homogeneous function of degree*
*zero, and*Ω*∈Lq _{(}*

_{S}*n−1*

_{)}_{for some}_{q >}_{1}

_{. Then there exists a constant}_{C >}_{0}

_{such that}

2*j−*1_{≤|}_{y}_{|}_{<}_{2}*je*

*−iP (y)*Ω*(y)*

*|y|ndy*

*≤CΩLq _{(S}n−*1

*)*

2*mj*

*|α|=m*

*aα*

−1*/*2*qm*

(3.2)

*holds for allj∈*Z*−andaα∈*R*.*

**Lemma _{3.3}.**

*Φ*

_{Let}_{:}

_{B}_{n}_{(}_{0}

_{,}_{1}

*R*

_{)}→*N*

*Ψ*

_{and}_{:}

_{B}_{m}_{(}_{0}

_{,}_{1}

*R*

_{)}→*M*

_{be}

_{C}∞*˜*

_{mappings and let}*b(·,·),I,ρ, andB(I, ρ)be as inLemma 2.8. Suppose that*Φ*and*Ψ *are of ﬁnite type at*

0*. Then there areN0, M0∈*N*,δ∈(*0*,*1*],C >*0*, andj0, k0∈*Z*−such that*

_{∆˜}ˆ_{b,}_{Φ}_{,}_{Ψ}* _{,k,j,ρ}(ξ, η)≤CB(I, ρ)*2

*ρN*0

*k|*−

_{ξ}|*δ/B(I,ρ)*0

_{ρ}M*j|*−

_{η}|*δ/B(I,ρ)*

_{(3.3)}

*for allk≤k0,j≤j0, and(ξ, η)∈*R*N _{×}*

_{R}

*M*

_{.}**Proof.** _{We start by the proof of (}_{3.3}_{) for the case} *| _{I}|_{< e}−1*

_{. By the deﬁnition of}∆˜

*b,*Φ

*,*Ψ

*,k,j,ρ*, we get

_{∆}ˆ_{˜}

*b,*Φ*,*Ψ*,k,j,ρ(ξ, η)≤C*

log

_{1}
*|I|*

**S***m−*1*Sk(y, ξ)dσ (y),* (3.4)

where

*Sk(y, ξ)=*

*ρk−*1_{≤|}x|<ρke

*−iξ·*Φ*(x)b(x, y)*˜

*|x|n* *dx*

*.* (3.5)

Now, byLemma 3.1,

*Sk(y, ξ)≤*

*[*log 1_{}*/|I|]−1*

*s=0*

**S***m−*1

*ρ(k−*1)_{2}*s _{≤|}_{x}_{|}_{<ρ}(k−*1)

_{2}

*(s+*1)

*e*

*−iξ·*Φ*(x)*˜*b(x, y)*

*|x|n* *dx*

*≤C*

*[*log 1_{}*/|I|]−*1

*s=*0

**S***m−*1

* _{b(·, y)}*˜

*Lq*1

_{(S}n−*)*

*ρN*0*(k−*1*)*_{2}*N*0*(s+*1*)|ξ|*−*δ*

*≤CρδN*0

**S***m−*1

* _{b(}*˜

_{·}_{, y)}*Lq _{(S}n−*1

*0*

_{)}ρN*k|ξ|*−

*δ.*

Therefore, byLemma 2.8(i) and Hölder’s inequality, we have

_{∆˜}ˆ_{b,}

Φ*,*Ψ*,k,j,ρ(ξ, η)≤CρδN|I|−*1*/q(ρN*0*k|ξ|)−δ* (3.7)

which, when combined with the trivial bound *|˜*ˆ_{b,}_{Φ}_{,}_{Ψ}* _{,k,j,ρ}(ξ, η)| ≤C[*log

*(*1

*/|I|)]*2

_{, }im-plies

_{∆}ˆ_{˜}

*b,*Φ*,*Ψ*,k,j,ρ(ξ, η)≤C*

log

_{1}
*|I|*

2_{}

*ρN0k _{|}_{ξ}_{|}−δ/[*log

*(*1

*/|I|)]*

_{.}_{(3.8)}

Similarly, we have

_{∆}ˆ_{˜}

*b,*Φ*,*Ψ*,k,j,ρ(ξ, η)≤C*

log

_{1}
*|I|*

2_{}

*ρM*0*j| _{η}|−δ/[*log

*(*1

*/|I|)]*

_{.}_{(3.9)}

Combining estimates (3.8) and (3.9) yields the estimate in (3.3) when*|I|< e−1*.

The proof of (3.3) for the case*|I| ≥e−1* follows by exactly the same argument as
that for the case*|I|< e−*1_{but this time we replace}_{ρ}_{and log}_{(}_{1}_{/}_{|}_{I}_{|}_{)}_{by 2 and log}_{(}_{2}_{)}_{,}
respectively, and use the observation that*|I|−1/q* _{≤}_{e}_{. This concludes the proof of our}
lemma.

ByLemma 3.2and the same argument employed in the proof ofLemma 3.3, we get the following.

**Lemma _{3.4}.**

*N*

_{Let}_{N0, M0}∈*˜*

_{, and let}

_{b(}·_{,}·_{)}_{,}_{I}_{,}_{ρ}_{, and}_{B(I, ρ)}_{be as in}_{Lemma 2.8}_{. Let}*R1(·)andR2(·)be real-valued polynomials on*R

*n*

_{and}_{R}

*m*

_{, respectively, with}_{deg}

_{(R1) }*N0−*1*and*deg*(R2) M0−*1*. LetP (x)=|α|=N*0*aαx*
*α*

*+R1(x)andQ(y)=|β|=M*0*bβy*
*β*

*+*
*R2(y). Then there exists a constantC >*0*such that for allk, j∈*Z*andaα, bβ∈*R*,*

*D(k,j,ρ)e*

*i(P (x)+Q(y))* *b(x, y)*˜

*|x|n _{|}_{y}_{|}mdx dy*

*≤CB(I, ρ)*2

*ρN*0*k*

*|α|=N0*

*a _{α}*

−1*/*2´*qN*0*B(I,ρ)*

*ρM*0*j*

*|β|=M*0

*bβ*

−1*/*2´*qM*0*B(I,ρ)*

*.*
(3.10)

**Theorem3.5.** *Let*Φ:*B _{n}(*0

*,*1

*)→*R

*N*

*and*Ψ:

*B*0

_{m}(*,*1

*)→*R

*M*

*beC∞mappings and let*˜

*b(·,·),I,ρ, andB(I, ρ)be as inLemma 2.8. Suppose that*Φ*and*Ψ *are of ﬁnite type at*

0*. Then for*1*< p≤ ∞andf∈Lp _{(}*

_{R}

*N*

_{×}_{R}

*M*

_{)}_{, there exists a positive constant}_{C}*pwhich is*

*independent ofb*˜*such that*

_{∆}*∗*
˜

*b,*Φ*,*Ψ*,ρ(f )p≤Cp*

**Proof.** Without loss of generality, we may assume that ˜*b* *≥*0. Let *N0, M0∈* N,
*δ∈(*0*,*1*]*, *C >*0, and *k0, j0∈*Z*−* be as inLemma 3.3. ForΦ*=(*Φ1, . . . ,Φ*N)*and Ψ*=*

*(*Ψ1, . . . ,Ψ*M)*, we letᏼ*=(P1, . . . , PN)*andᏽ*=(Q1, . . . , QM)*, where

*Pl(x)=*

*|α|=N*0*−1*
1
*α*!

*∂α*Φ*l*

*∂xα* *(*0*)x*
*α*

*,*

*Qs(y)=*

*|β|M*0*−1*
1
*β*!

*∂β*Ψ*s*

*∂yβ(*0*)y*
*β*

*,*

(3.12)

for 1* s M*and 1* l N*. Then,

_{∆}ˆ_{˜}

*b,*Φ*,*Ψ*,k,j,ρ(ξ, η)−*∆ˆ˜*b,*ᏼ*,*Ψ*,k,j,ρ(ξ, η)≤C*

*ρN*0*k| _{ξ}|*

**S***n−*1*Hj(x, η)dσ (x),* (3.13)

where

*Hj(x, η)=*

*ρj−*1_{≤|}_{y}_{|}_{<ρ}je

*−iη·*Ψ*(y)b(x, y)*˜

*|y|m* *dy*

*.* (3.14)

ByLemma 3.1and the argument in the proof of (3.3), we get

_{∆}ˆ_{˜}_{b,}

Φ*,*Ψ*,k,j,ρ(ξ, η)−*∆ˆ˜*b,*ᏼ*,*Ψ*,k,j,ρ(ξ, η)*

*≤CB(I, ρ)*2*ρN*0*k| _{ξ}|δ/[B(I,ρ)]_{ρ}M*0

*j|*

_{η}|−δ/[B(I,ρ)]_{.}(3.15)

Similarly, it is easy to verify that the following estimates hold:

_{∆}ˆ_{˜}

*b,*Φ*,*Ψ*,k,j,ρ(ξ, η)−*∆ˆ*b,*˜Φ*,*ᏽ*,k,j,ρ(ξ, η)*

*≤CB(I, ρ)*2*ρN*0*k| _{ξ}|*−

*δ/[B(I,ρ)]*0

_{ρ}M*j|*

_{η}|δ/[B(I,ρ)]_{,}_{∆˜}ˆ_{b,}

Φ*,*Ψ*,k,j,ρ(ξ, η)−*∆˜ˆ*b,*ᏼ*,*Ψ*,k,j,ρ(ξ, η)−*∆˜ˆ*b,*Φ*,*ᏽ*,k,j,ρ(ξ, η)+*∆˜ˆ*b,*ᏼ*,*ᏽ*,k,j,ρ(ξ, η)*

*≤CB(I, ρ)*2*ρN*0*k _{|}_{ξ}_{|}δ/[B(I,ρ)]_{(ρ}M*0

*j*

_{|}_{η}_{|}_{)}δ/[B(I,ρ)]_{,}_{∆}ˆ˜_{b,}_{Φ}_{,}_{ᏽ}* _{,k,j,ρ}(ξ, η)−*∆ˆ

*˜*

_{b,}_{ᏼ}

_{,}_{ᏽ}

*2*

_{,k,j,ρ}(ξ, η)≤CB(I, ρ)*ρN*0

*k|ξ|δ/[B(I,ρ)],*

_{∆˜}ˆ

_{b,}ᏼ*,*Ψ*,k,j,ρ(ξ, η)−*∆˜ˆ*b,*ᏼ*,*ᏽ*,k,j,ρ(ξ, η)≤C*

*B(I, ρ)*2*ρM*0*j|η|δ/[B(I,ρ)] _{.}*

(3.16)

By (3.3), (3.15)–(3.16),Theorem 2.6, and Lemmas2.8and2.9, we get (3.11). This con-cludes the proof of the theorem.

**4. Proof ofTheorem 1.3.** By assumption,Ωcan be written asΩ*=∞µ=*1*cµbµ*, where

*cµ∈*C, *bµ* is *q*-block supported on a cap *Iµ* on**S**

*n−*1_{×}_{S}*m−*1_{, and} * _{M}*0

*,*1

(2.3). Every*q*-block function*bµ(·,·)*has a companion function ˜*bµ(·,·)*deﬁned by

˜

*bµ(x, y)=bµ(x, y)−*

**S***n−*1*bµ(u, y)dσ (u)−*

**S***m−*1*bµ(x, v)dσ (v)*

*+*

**S***n−*1_{×}_{S}*m−*1*bµ(u, v)dσ (u)dσ (v).*

(4.1)

It is easy to verify that each ˜*bµ*enjoys the following properties:

**S***n−*1
˜

*bµ(u,·)dσ (u)=*

**S***m−*1
˜

*bµ(·, v)dσ (v)=*0;

* _{b}*˜

*µLq _{(S}n−*1

_{×}

_{S}*m−*1

*)≤*4

*|I|−*1

*/q*; ˜

*bµL*1

*1*

_{(S}n−

_{×}

_{S}*m−*1

*)≤*4

*.*

(4.2)

By the vanishing property onΩwe have

Ω*=*
*∞*

*µ=1*

*cµb*˜*µ* (4.3)

and this yields

_{P}_{Ω}_{,}_{Φ}_{,}_{Ψ}_{f}_{p}_{}_{}*∞*

*µ=*1

_{c}_{µ}_{T˜}_{b}

*µf*

* _{p},* (4.4)

where

*T˜ _{bµ}f (x, y)=*p.v.

*Bn(*0*,*1*)×Bm(*0*,*1*)*

*fx−Φ(u), y−Ψ(v)*˜*bµ(u, v)*

*|u|n _{|}_{v}_{|}mdu dv.* (4.5)

Let*N0*,*M0*,ᏼ, andᏽbe given as in the proof ofTheorem 3.5. For 1* l N*, 1* s M*,
let*al,α=(*1*/α*!*)(∂*

*α*

Φ*l/∂x*
*α*

*)(*0*)*and*bs,β=(*1*/β*!*)(∂*
*β*

Ψ*s/∂y*
*β*

*)(*0*)*. For 0* τ N0*, 0* κ *
*M0*, we deﬁne*Pτ=(P _{l}τ, . . . , PNτ)*and

*Q*

*κ*

*=(Q*1*κ, . . . , Q*

*κ*
*M)*by

*P _{l}τ(x)=*

*|α|τ*

*al,αx*
*α*

*,* for*l=*1*, . . . , N,* 0* τ N0−*1;

*Qκs(y)=*

*|β|κ*

*bs,βy*
*β*

*,* for*s=*1*, . . . , M,*0* κ M0−*1; (4.6)

Lemma 3.4and the same argument as in the proofs of (3.3) and (3.15), we get
∆*(τ,κ)*
˜
*bµ,k,j,ρµ*

* CBIµ, ρµ*

2 ;

_{∆}ˆ*(τ,κ)*

˜

*bµ,k,j,ρµ(ξ, η)*

* CBIµ, ρµ*2

*ρτkµ*

*|α|=τ*

*N*

*l=τ*

*al,αξl*

*−ατ/[B(Iµ,ρµ)]*

*×*
ρ*κj*

*µ*

*|β|=κ*

*M*

*s=κ*

*bs,βηs*

*−ακ[B(Iµ,ρµ)]*
;

_{∆}ˆ*(τ,κ)*

˜

*bµ,k,j,ρµ(ξ, η)−*
ˆ
∆*(τ−1,κ)*

˜

*bµ,k,j,ρµ(ξ, η)*

* CBIµ, ρµ*

2_{}
*ρτkµ*

*|α|=τ*

*N*

*l=τ*

*al,αξl*

*ατ[B(Iµ,ρµ)]*
*×*
ρ*κj*
*µ*

*|β|=κ*

*M*

*s=κ*

*bs,βηs*

*−ακ/[B(Iµ,ρµ)]*
;

_{∆}ˆ*(τ,κ)*

˜

*bµ,k,j,ρµ(ξ, η)−*
ˆ
∆*(τ,κ−1)*

˜

*bµ,k,j,ρµ(ξ, η)*

* CBIµ, ρµ*

2_{}
*ρτkµ*

*|α|=τ*

*N*

*l=τ*

*al,αξl*

*−ατ/[B(Iµ,ρµ)]*
*×*
*ρκjµ*

*|β|=κ*

*M*

*s=κ*

*bs,βηs*

*ακ/[B(Iµ,ρµ)]*

;

_{∆}ˆ*(τ,κ)*

˜

*bµ,k,j,ρµ(ξ, η)−*∆ˆ
*(τ−*1*,κ)*

˜

*bµ,k,j,ρµ(ξ, η)−*∆ˆ
*(τ,κ−*1*)*

˜

*bµ,k,j,ρµ(ξ, η)+*∆ˆ
*(τ−*1*,κ−*1*)*

˜

*bµ,k,j,ρµ* *(ξ, η)*

* CBIµ, ρµ*2

ρ*τk*

*µ*

*|α|=τ*

*N*

*l=τ*

*al,αξl*

*−ατ/[B(Iµ,ρµ)]*

*×*
ρ*κj*

*µ*

*|β|=κ*

*M*

*s=κ*

*bs,βηs*

*ακ/[B(Iµ,ρµ)]*

;

_{∆}ˆ*(τ,κ−1)*

˜

*bµ,k,j,ρµ(ξ, η)−*
ˆ

∆*(τ−1,κ−1)*

˜

*bµ,k,j,ρµ* *(ξ, η)*

* CB(Iµ, ρµ)]*2

*ρµτk*

*|α|=τ*

*N*

*l=τ*

*al,αξl*

*ατ/[B(Iµ,ρµ)]*

;

_{∆}ˆ*(τ−*1*,κ)*

˜

*bµ,k,j,ρµ(ξ, η)−*
ˆ

∆*(τ−*1*,κ−*1*)*

˜

*bµ,k,j,ρµ* *(ξ, η)*

* CBIµ, ρµ*2

ρ*κj*

*µ*

*|β|=κ*

*M*

*s=κ*

*bs,βηs*

*ακ/[B(Iµ,ρµ)]*

;

(4.7)

By (3.3), (3.15)-(3.16), (4.7)-(4.8),Lemma 2.5, Theorems2.4and3.5, andLemma 2.9, we get

_{T˜}_{b}

*µfp=*

*k,j∈Z−*

∆*(N*0*,M*0*)*

˜

*bµ,k,j,ρµ∗f*

* _{p}≤CpBIµ, ρµ)*2

*fp,*(4.8)

for every*f∈Lp _{(}*

_{R}

*N*

_{×}_{R}

*M*

_{)}_{,}

_{µ}_{=}_{1}

_{,}_{2}

_{, . . . ,}_{and for all}

_{p}_{, 1}

_{< p <}_{∞}_{. Hence, (}

_{1.9}

_{) follows by}(2.3), (4.4), and (4.8).

Finally, a proof ofTheorem 1.4can be obtained using the above estimates and the techniques in [4]. We omit the details.

**Acknowledgments.** _{The publication of this paper was supported by Yarmouk }
Uni-versity Research Council. The authors of this paper would like to thank the referees for
their valuable remarks.

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Ahmad Al-Salman: Department of Mathematics, Yarmouk University, Irbid, Jordan
*E-mail address*:[email protected]