# Weighted Norm Inequalities for Multilinear Fourier Multipliers with Mixed Norm

Authors: Mai Fujita
Journal: Abstract and Applied Analysis
 (2021)

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## Abstract

In this paper, weighted norm inequalities for multilinear Fourier multipliers satisfying Sobolev regularity with mixed norm are discussed. Our result can be understood as a generalization of the result by Fujita and Tomita by using theLr-based Sobolev space, 1<r≤2 with mixed norm.


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## Body

## 1. Introduction

Form∈L∞ℝNn, the N-linear Fourier multiplier operator Tm is defined by
(1)Tmf1,⋯,fNx=12πNn∫ℝnNeix·ξ1+⋯+ξNmξf1^ξ1⋯fN^ξNdξ,for f1,⋯,fN∈Sℝn, where x∈ℝn, ξ=ξ1,⋯,ξN∈ℝnN and dξ=dξ1⋯dξN. Let Ψ∈Sℝd be such that
(2)suppΨ⊂ξ∈ℝd:12≤ξ≤2,∑k∈ℤΨξ/2k=1,ξ∈ℝd\0,and set
(3)mjξ1,⋯,ξN=m2jξ1,⋯,2jξNΨξ1,⋯,ξN,j∈ℤ,where Ψ is as in (2) with d=Nn. We denote by TmLp1w1×⋯×LpNwN⟶Lpw the smallest constant C satisfying
(4)Tmf1,⋯,fNLpw≤C∏i=1NfiLpiwi,f1,⋯,fN∈Sℝn.See Section2 for the definition of function spaces.In the unweighted case, Tomita [1] proved a Hörmander-type multiplier theorem for multilinear operators, namely, if s>Nn/2 then
(5)TmLp1ℝn×⋯×LpNℝn⟶Lpℝn≲supj∈ℤmjHs2ℝNn,for 1<p1,⋯,pN,p<∞ satisfying 1/p1+⋯+1/pN=1/p, where Hs2ℝNn is the L2-based Sobolev space of usual type. Grafakos and Si [2] extended this result to the case p≤1 by using the Lr-based Sobolev space, 1<r≤2. For further results in this direction, see [3–7]. Let 1<p1,⋯,pN<∞ and 1/p1+⋯+1/pN=1/p. In the weighted case, Fujita and Tomita [8] proved that if n/2<si≤n, pi>n/si, and wi∈Apisi/n for all i=1,⋯,N, then
(6)TmLp1w1×⋯×LpNwN⟶Lpw≲supj∈ℤmjHs⟶2ℝNn,where w=w1p/p1⋯wNp/pN and Hs⟶2ℝNn is the L2-based Sobolev space of product type. This result can also be obtained from another approach of [9]. See [10, 11] for the endpoint cases.The following is our main result which can be understood as a generalization of the result by Fujita and Tomita [8]. Taking ri=2 for all i=1,⋯,N in (8), we have (6). Si [12] obtained some weighted estimates for multilinear Fourier multipliers with the Lr-based Sobolev regularity, 1<r≤2.Theorem 1.
Let1<p1,⋯,pN<∞, 1/p1+⋯+1/pN=1/p, r⟶=r1,⋯,rN∈1,2N, rN≤rN−1≤⋯≤r2≤r1, s⟶=s1,⋯,sN∈ℝN, and n/ri<si≤n for all i=1,⋯,N. Assume
(7)pi>nsi,wi∈Apisi/nforalli=1,⋯,N.
Then,(8)TmLp1w1×⋯×LpNwN⟶Lpw≲supj∈ℤmjHs⟶r⟶ℝnN,where w=w1p/p1⋯wNp/pN and Hs⟶r⟶ℝnN is the Sobolev space of product type with mixed norm.


## 2. Preliminaries

### 2.1. Notations

Letn∈ℕ be the fixed dimension of the Euclidean space, and ℤ+n is defined by 0,1,2,⋯n. The Lebesgue measure on ℝn is denoted by dx (see, for example, Chapters 1 and 2 of [13]). Let N be a natural number, N≥2. An operator T acting on N-tuples of functions defined on ℝn is called the N-linear operator. For two nonnegative quantities A and B, the notation A≲B means that A≤CB for some unspecified constant C>0 independent of A and B, and the notation A≈B means that A≲B and B≲A. If x∈ℝd, we denote 1+x21/2 by x. Let Sℝn and S′ℝn be the Schwartz class of all rapidly decreasing smooth functions and tempered distributions, respectively. We define the Fourier transform Ff and the inverse Fourier transform F−1f of f∈Sℝn by
(9)Ffξ=f^ξ=∫ℝne−ix·ξfxdx,F−1fx=12πn∫ℝneix·ξfξdξ.See, for example, Chapter 1 of [14]. To distinguish linear and multilinear operators, for m∈L∞ℝn, we denote the linear Fourier multiplier operator by mD defined by
(10)mDfx=F−1mξf^ξx=12πn∫ℝneix·ξmξf^ξdξ,for f∈Sℝn, where x,ξ∈ℝn. Let 0<p<∞ and w≥0. The weighted Lebesgue space Lpw consists of all measurable functions f on ℝn such that
(11)fLpw=fLpℝn,wxdx=∫ℝnfxpwxdx1/p<∞.Let1<p<∞. We say that a weight w belongs to the Muckenhoupt class Ap if
(12)supB1B∫Bwxdx1B∫Bwx1−p′dxp−1<∞,where the supremum is taken over all balls B in ℝn, B is the Lebesgue measure of B, and p′ is the conjugate exponent of p, that is, 1/p+1/p′=1. It is well known that the Hardy-Littlewood maximal operator M is bounded on Lpw if and only if w∈Ap (see Theorem 7.3 of [14]).


### 2.2. Function Spaces

To distinguish spaces of usual type and mixed type concerning integrable indices, we useℝNn and ℝnN, respectively.We recall the definition ofLp-spaces with mixed norm [15]. Let p⟶=p1,⋯,pN∈0,∞N. The Lebesgue spaces with mixed norm Lp⟶ℝnN consists of all measurable functions F on ℝNn such that
(13)FLp⟶ℝnN=Fx1,⋯,xNLp1ℝn,dx1⋯LpNℝn,dxN<∞,where x1,⋯,xN∈ℝnN and dxi is the Lebesgue measure with respect to the variable xi for all i=1,⋯,N. In particular, if each pi is equal to p∈0,∞, then we have FLp⟶ℝnN=FLpℝNn. For r⟶=r1,⋯,rN∈1,∞N and s⟶=s1,⋯,sN∈ℝN, the norm of the Sobolev space of product type with mixed norm Hs⟶r⟶ℝnN for F∈S′ℝNn is defined by
(14)FHs⟶r⟶ℝnN=F−1ξ1s1⋯ξNsNF^ξ1,⋯,ξNLr⟶ℝnN,where ξi=1+ξi21/2 for i=1,⋯,N and F−1 is the inverse Fourier transform of ℝNn. Taking ri=2 for all i=1,⋯,N, we obtain the L2-based Sobolev space of product type Hs⟶2ℝNn, namely, FHs⟶2ℝNn=ξ1s1⋯ξNsNF^ξ1,⋯,ξNL2ℝNn. It should be remarked that if si=s/N,s≥0 for all i=1,⋯,N,
(15)Hs2ℝNn↪Hs⟶2ℝNn,where Hs2ℝNn is the L2-based Sobolev space of usual type, that is to say, FHs2ℝNn=ξsF^L2ℝNn, where ξ∈ℝNn.Forp⟶=p1,⋯,pN∈1,∞N and s⟶=s1,⋯,sN∈ℝN, the norm of the weighted Lebesgue space with mixed norm Ls⟶p⟶ℝnN for F∈S′ℝNn is also defined by
(16)FLs⟶p⟶ℝnN=Fx1,⋯,xNLp1ℝn,x1s1dx1⋯LpNℝn,xNsNdxN,where x1,⋯,xN∈ℝnN and xisi=1+xi2si/2 for all i=1,⋯,N. For accuracy, we will frequently write Ls1,⋯,sNp1,⋯,pNℝnN instead of Ls⟶p⟶ℝnN in the proof.Forp⟶=p1,⋯,pN, q⟶=q1,⋯,qN∈0,∞N, we shall agree that if a~b is a relation between numbers a and b, then p⟶~q⟶ means that pi~qi holds for each i.


### 2.3. Cut-Off Functions

We collect cut-off functions which will be used later on [8]. Let ϕ0 be a C∞-function on 0,∞ satisfying
(17)ϕ0t=1on0,14N,suppϕ0⊂0,12N.We setϕ1t=1−ϕ0t. For i1,⋯,iN∈0,1N, we define the function Φi1,⋯,iN on ℝNn\0 by
(18)Φi1,⋯,iNξ=ϕi1ξ1ξ⋯ϕiNξNξ,where ξ=ξ1,⋯,ξN∈ℝnN and ξ=ξ12+⋯+ξN2. Note that
(19)Φ0,⋯,0ξ=0.According to the notation of [5] or [6], we also set A0, A1: A0 denotes the set of φ∈Sℝn for which suppφ is compact and φ=1 on some neighborhood of the origin; A1 denotes the set of ψ~∈Sℝn for which suppψ~ is a compact subset of ℝn\0.


### 2.4. Lemmas

The following lemmas will be used in the proof of Theorem1.Lemma 2 (see Lemma 3.1 of [8]).
LetΦii,⋯,iN be the same as in (18). Then, the following are true:
(1)
Forξ1,⋯,ξN∈ℝn×⋯×ℝn\0,⋯,0,
(20)∑i1,⋯,iN∈0,1N,i1,⋯,iN≠0,⋯,0Φi1,⋯,iNξ1,⋯,ξN=1.(2)
Fori1,⋯,iN∈0,1N and α1,⋯,αN∈ℤ+n×⋯×ℤ+n, then there exists a constant Ci1,⋯,iNα1,⋯,αN>0 such that
(21)∂ξ1α1⋯∂ξNαNΦi1,⋯,iNξ≤Ci1,⋯,iNα1,⋯,αNξ1+⋯+ξN−α1+⋯+αN,for all ξ=ξ1,⋯,ξN∈ℝn×⋯×ℝn\0,⋯,0.(3)
Ifij=1 for some j=1,⋯,N and ik=0 for all k=1,⋯,N with j≠k, then suppΦi1,⋯,iN⊂ξ1,⋯,ξN:ξk≤ξj/Nfork≠j. If ij=ij′=1 for some j,j′=1,⋯,N with j≠j′, then suppΦi1,⋯,iN⊂ξ1,⋯,ξN:ξj/4N≤ξj′≤4Nξj,ξk≤4Nξjfork≠j,j′.Lemma 3 (see Chapter 7 of [14]).
Let1<p<∞ and w∈Ap. Then, there exists ε>0 such that w∈Ap−ε.Lemma 4 (see [16]).
Letψ∈Sℝn be such that suppψ⊂η∈ℝn:1/r≤η≤r for some r>1. If 1<p<∞ and w∈Ap, then
(22)∑j∈ℤψD2jf21/2LpwfLpw,where ψD/2jf=F−1ψ·/2jf^.Lemma 5 (see [17]).
Let1<p,q<∞ and w∈Ap. Then,
(23)∑k∈ZMfkq1/qLpw≲∑k∈ℤfkq1/qLpw.Lemma 6 (see Proposition 2.7 of [14]).
Letϕ be a function which is positive, radial, decreasing (as a function on 0,∞), and integrable. Set ϕtx=1/tnϕx/t for t>0. Then,
(24)supt>0ϕt∗fx≲Mfx,for x∈ℝn.Lemma 7 (see Theorem1 of Section 10 of [15]).
Letp⟶=p1,⋯,pN∈1,∞N, then
(25)F∗GLp⟶ℝnN≤FLp⟶ℝnNGL1ℝNn.Lemma 8 (see Theorem1 of Section 12 of [15]).
Letp⟶=p1,⋯,pN∈1,2N and pN≤pN−1≤⋯≤p1, then
(26)F^L⟶p′ℝnN≤FLp⟶ℝnN,where ⟶p′=p1′,⋯,pN′ and p′i is the conjugate exponent of pi for i=1,⋯,N.


## 2.1. Notations

Letn∈ℕ be the fixed dimension of the Euclidean space, and ℤ+n is defined by 0,1,2,⋯n. The Lebesgue measure on ℝn is denoted by dx (see, for example, Chapters 1 and 2 of [13]). Let N be a natural number, N≥2. An operator T acting on N-tuples of functions defined on ℝn is called the N-linear operator. For two nonnegative quantities A and B, the notation A≲B means that A≤CB for some unspecified constant C>0 independent of A and B, and the notation A≈B means that A≲B and B≲A. If x∈ℝd, we denote 1+x21/2 by x. Let Sℝn and S′ℝn be the Schwartz class of all rapidly decreasing smooth functions and tempered distributions, respectively. We define the Fourier transform Ff and the inverse Fourier transform F−1f of f∈Sℝn by
(9)Ffξ=f^ξ=∫ℝne−ix·ξfxdx,F−1fx=12πn∫ℝneix·ξfξdξ.See, for example, Chapter 1 of [14]. To distinguish linear and multilinear operators, for m∈L∞ℝn, we denote the linear Fourier multiplier operator by mD defined by
(10)mDfx=F−1mξf^ξx=12πn∫ℝneix·ξmξf^ξdξ,for f∈Sℝn, where x,ξ∈ℝn. Let 0<p<∞ and w≥0. The weighted Lebesgue space Lpw consists of all measurable functions f on ℝn such that
(11)fLpw=fLpℝn,wxdx=∫ℝnfxpwxdx1/p<∞.Let1<p<∞. We say that a weight w belongs to the Muckenhoupt class Ap if
(12)supB1B∫Bwxdx1B∫Bwx1−p′dxp−1<∞,where the supremum is taken over all balls B in ℝn, B is the Lebesgue measure of B, and p′ is the conjugate exponent of p, that is, 1/p+1/p′=1. It is well known that the Hardy-Littlewood maximal operator M is bounded on Lpw if and only if w∈Ap (see Theorem 7.3 of [14]).


## 2.2. Function Spaces

To distinguish spaces of usual type and mixed type concerning integrable indices, we useℝNn and ℝnN, respectively.We recall the definition ofLp-spaces with mixed norm [15]. Let p⟶=p1,⋯,pN∈0,∞N. The Lebesgue spaces with mixed norm Lp⟶ℝnN consists of all measurable functions F on ℝNn such that
(13)FLp⟶ℝnN=Fx1,⋯,xNLp1ℝn,dx1⋯LpNℝn,dxN<∞,where x1,⋯,xN∈ℝnN and dxi is the Lebesgue measure with respect to the variable xi for all i=1,⋯,N. In particular, if each pi is equal to p∈0,∞, then we have FLp⟶ℝnN=FLpℝNn. For r⟶=r1,⋯,rN∈1,∞N and s⟶=s1,⋯,sN∈ℝN, the norm of the Sobolev space of product type with mixed norm Hs⟶r⟶ℝnN for F∈S′ℝNn is defined by
(14)FHs⟶r⟶ℝnN=F−1ξ1s1⋯ξNsNF^ξ1,⋯,ξNLr⟶ℝnN,where ξi=1+ξi21/2 for i=1,⋯,N and F−1 is the inverse Fourier transform of ℝNn. Taking ri=2 for all i=1,⋯,N, we obtain the L2-based Sobolev space of product type Hs⟶2ℝNn, namely, FHs⟶2ℝNn=ξ1s1⋯ξNsNF^ξ1,⋯,ξNL2ℝNn. It should be remarked that if si=s/N,s≥0 for all i=1,⋯,N,
(15)Hs2ℝNn↪Hs⟶2ℝNn,where Hs2ℝNn is the L2-based Sobolev space of usual type, that is to say, FHs2ℝNn=ξsF^L2ℝNn, where ξ∈ℝNn.Forp⟶=p1,⋯,pN∈1,∞N and s⟶=s1,⋯,sN∈ℝN, the norm of the weighted Lebesgue space with mixed norm Ls⟶p⟶ℝnN for F∈S′ℝNn is also defined by
(16)FLs⟶p⟶ℝnN=Fx1,⋯,xNLp1ℝn,x1s1dx1⋯LpNℝn,xNsNdxN,where x1,⋯,xN∈ℝnN and xisi=1+xi2si/2 for all i=1,⋯,N. For accuracy, we will frequently write Ls1,⋯,sNp1,⋯,pNℝnN instead of Ls⟶p⟶ℝnN in the proof.Forp⟶=p1,⋯,pN, q⟶=q1,⋯,qN∈0,∞N, we shall agree that if a~b is a relation between numbers a and b, then p⟶~q⟶ means that pi~qi holds for each i.


## 2.3. Cut-Off Functions

We collect cut-off functions which will be used later on [8]. Let ϕ0 be a C∞-function on 0,∞ satisfying
(17)ϕ0t=1on0,14N,suppϕ0⊂0,12N.We setϕ1t=1−ϕ0t. For i1,⋯,iN∈0,1N, we define the function Φi1,⋯,iN on ℝNn\0 by
(18)Φi1,⋯,iNξ=ϕi1ξ1ξ⋯ϕiNξNξ,where ξ=ξ1,⋯,ξN∈ℝnN and ξ=ξ12+⋯+ξN2. Note that
(19)Φ0,⋯,0ξ=0.According to the notation of [5] or [6], we also set A0, A1: A0 denotes the set of φ∈Sℝn for which suppφ is compact and φ=1 on some neighborhood of the origin; A1 denotes the set of ψ~∈Sℝn for which suppψ~ is a compact subset of ℝn\0.


## 2.4. Lemmas

The following lemmas will be used in the proof of Theorem1.Lemma 2 (see Lemma 3.1 of [8]).
LetΦii,⋯,iN be the same as in (18). Then, the following are true:
(1)
Forξ1,⋯,ξN∈ℝn×⋯×ℝn\0,⋯,0,
(20)∑i1,⋯,iN∈0,1N,i1,⋯,iN≠0,⋯,0Φi1,⋯,iNξ1,⋯,ξN=1.(2)
Fori1,⋯,iN∈0,1N and α1,⋯,αN∈ℤ+n×⋯×ℤ+n, then there exists a constant Ci1,⋯,iNα1,⋯,αN>0 such that
(21)∂ξ1α1⋯∂ξNαNΦi1,⋯,iNξ≤Ci1,⋯,iNα1,⋯,αNξ1+⋯+ξN−α1+⋯+αN,for all ξ=ξ1,⋯,ξN∈ℝn×⋯×ℝn\0,⋯,0.(3)
Ifij=1 for some j=1,⋯,N and ik=0 for all k=1,⋯,N with j≠k, then suppΦi1,⋯,iN⊂ξ1,⋯,ξN:ξk≤ξj/Nfork≠j. If ij=ij′=1 for some j,j′=1,⋯,N with j≠j′, then suppΦi1,⋯,iN⊂ξ1,⋯,ξN:ξj/4N≤ξj′≤4Nξj,ξk≤4Nξjfork≠j,j′.Lemma 3 (see Chapter 7 of [14]).
Let1<p<∞ and w∈Ap. Then, there exists ε>0 such that w∈Ap−ε.Lemma 4 (see [16]).
Letψ∈Sℝn be such that suppψ⊂η∈ℝn:1/r≤η≤r for some r>1. If 1<p<∞ and w∈Ap, then
(22)∑j∈ℤψD2jf21/2LpwfLpw,where ψD/2jf=F−1ψ·/2jf^.Lemma 5 (see [17]).
Let1<p,q<∞ and w∈Ap. Then,
(23)∑k∈ZMfkq1/qLpw≲∑k∈ℤfkq1/qLpw.Lemma 6 (see Proposition 2.7 of [14]).
Letϕ be a function which is positive, radial, decreasing (as a function on 0,∞), and integrable. Set ϕtx=1/tnϕx/t for t>0. Then,
(24)supt>0ϕt∗fx≲Mfx,for x∈ℝn.Lemma 7 (see Theorem1 of Section 10 of [15]).
Letp⟶=p1,⋯,pN∈1,∞N, then
(25)F∗GLp⟶ℝnN≤FLp⟶ℝnNGL1ℝNn.Lemma 8 (see Theorem1 of Section 12 of [15]).
Letp⟶=p1,⋯,pN∈1,2N and pN≤pN−1≤⋯≤p1, then
(26)F^L⟶p′ℝnN≤FLp⟶ℝnN,where ⟶p′=p1′,⋯,pN′ and p′i is the conjugate exponent of pi for i=1,⋯,N.


## 3. Lemmas

In this section, we prove lemmas which play important roles in the proof of Theorem1. The proof of the following lemma is based on the argument of Proposition 1.3.2 of [18] or Lemma 3.3 of [1].Lemma 9.
Letr>0, p⟶=p1,⋯,pN, q⟶=q1,⋯,qN∈1,∞N, s⟶=s1,⋯,sN∈ℝ≥0N, and p⟶≤q⟶. Then, the estimate
(27)ξ1s1⋯ξNsNF^ξ1,⋯,ξNLq⟶ℝnN≲ξ1s1⋯ξNsNF^ξ1,⋯,ξNLp⟶ℝnN,holds, where suppF⊂ξ=ξ1,⋯,ξN∈ℝnN:ξ≤r.Proof.
We consider only the caseN=2. Let ϕ∈Sℝn be such that
(28)ϕy=1ony∈ℝn:y≤r,suppϕ⊂y∈ℝn:y≤2r.
SincesuppF⊂x=x1,x2∈ℝn2:xi≤r,i=1,2, we see that Fx1,x2=ϕx1ϕx2Fx1,x2. Then, it is easy to see that
(29)F^ξ1,ξ2=∫ℝn2ϕ^ξ1−η1ϕ^ξ2−η2F^η1,η2dη1dη2.
For fixedξ2∈ℝn, by Minkowski’s inequality for integrals, it follows that
(30)ξ1s1F^ξ1,ξ2Lq1ℝξ1n≤∫ℝη2nϕ^ξ2−η2ξ1s1∫ℝη1nϕ^ξ1−η1F^η1,η2dη1Lq1ℝξ1ndη2.
Sinceξ1s1≲ξ1−η1s1η1s1, for fixed η2∈ℝ, we obtain
(31)ξ1s1∫ℝη1nϕ^ξ1−η1F^η1,η2dη1Lq1ℝξ1nq1=∫ℝξ1n∫ℝη1nξ1s1ϕ^ξ1−η1F^η1,η2dη1q1dξ1≲∫ℝξ1n∫ℝη1nξ1−η1s1η1s1ϕ^ξ1−η1F^η1,η2dη1q1dξ1≤supξ1∈ℝn∫ℝη1nξ1−η1s1η1s1ϕ^ξ1−η1F^η1,η2dη1q1−p1×∫ℝξ1n∫ℝη1nξ1−η1s1η1s1ϕ^ξ1−η1F^η1,η2dη1p1dξ1,where we have used the fact p1≤q1.
For the first term on the right-hand side of (31), by Hölder’s inequality and a change of variables, we have
(32)∫ℝη1nξ1−η1s1η1s1ϕ^ξ1−η1F^η1,η2dη1≤·s1ϕ^Lp1′ℝη1n·s1F^·,η2Lp1ℝη1n,where ξ1∈ℝn. Thus, we see that
(33)supξ1∈ℝn∫ℝη1nξ1−η1s1η1s1ϕ^ξ1−η1F^η1,η2dη1q1−p1≤·s1ϕ^Lp1′ℝη1nq1−p1·s1F^·,η2Lp1ℝη1nq1−p1.
For the second term on the right-hand side of (31), by Young’s inequality, we obtain
(34)∫ℝξ1n∫ℝη1nξ1−η1s1η1s1ϕ^ξ1−η1F^η1,η2dη1p1dξ1≤·s1ϕ^L1ℝξ1np1·s1F^·,η2Lp1ℝξ1np1.
By (33) and (34), it follows that
(35)ξ1s1∫ℝη1nϕ^ξ1−η1F^η1,η2dη1Lq1ℝξ1n≲·s1F^·,η2Lp1ℝη1n,where we have used the fact that ·s1ϕ^Lp1′,·s1ϕ^L1<∞. By the same way for ξ2∈ℝn, we have the desired estimate with N=2.☐The following is a key lemma in the proof of Theorem1. Fujita and Tomita (Proposition A.2 of [6]) proved (6) using the fact that H2s⟶ℝNn is a multiplication algebra when si>n/2 for all i=1,⋯,N. Instead of this, we shall use the following lemma.Lemma 10.
LetN0∈ℕ, r⟶=r1,⋯,rN∈1,2N, rN≤rN−1≤⋯≤r2≤r1, s⟶=s1,⋯,sN∈ℝN, n/ri<si≤n, and n/si<qi<ri for all i=1,⋯,N. Then, the estimate
(36)Fm2j·Ψ·/2kLs1q1′,⋯,sNqN′q1′,⋯,qN′ℝnN≲supj∈ℤmjHs⟶r⟶ℝnN,holds for all j∈ℤ, −N0≤k≤N0, and m∈Hs⟶r⟶ℝnN.Proof.
We consider only the caseN=2. By the change of variables and k satisfies −N0≤k≤N0, we see that
(37)Fm2j·Ψ·2kLs1q1′,s2q2′q1′,q2′ℝn2≲mj+k^Ls1q1′,s2q2′q1′,q2′ℝn2,where mj is defined by (3). By Lemma 9 and Hausdorff-Young’s inequality with mixed type (Lemma 8), it follows that
(38)ξ1s1ξ2s2mj+k^ξ1,ξ2L⟶q′ℝn2≲ξ1s1ξ2s2mj+k^ξ1,ξ2L⟶r′ℝn2≤F−1ξ1s1ξ2s2mj+k^ξ1,ξ2Lr⟶ℝn2,where we have used the fact q⟶≤r⟶. This completes the proof with N=2.☐The following lemma is known, but we shall give a proof for the reader’s convenience.Lemma 11.
Letr>0, 1<qi<∞, and n/qi<si<∞ for all i=1,⋯,N. Then, the estimate
(39)Tm·/2jf1,⋯,fNx≲∥m^∥Ls1q1′,⋯,sNqN′q1′,⋯,qN′ℝnN∏i=1NMfiqix1/qi,holds for all x∈ℝn, j∈ℤ, m∈Hs⟶r⟶ℝnN with suppm⊂ξ=ξ1,⋯,ξN∈ℝnN:∣ξ∣≤r and f1,⋯,fN∈Sℝn.Proof.
We consider only the caseN=2. For all x∈ℝn and j∈ℤ, by Fubini’s theorem and the change of variables, we see that
(40)Tm·/2jf1,f2x=2jn2∫ℝn2F−1m2jx−y1,2jx−y2f1y1f2y2dy1dy2,where F−1 is the inverse Fourier transform of ℝn2. For fixed y2∈ℝn, by Hölder’s inequality and Lemma 6 with ϕx=1+∣x∣−s1q1, it follows that
(41)∫ℝy1n1+2j∣x−y1∣s1F−1m2jx−y1,2jx−y21+2j∣x−y1∣−s1f1y12jndy1≤∫ℝy1n1+2j∣x−y1∣s1q1′F−1m2jx−y1,2jx−y2q1′2jndy11/q1′×∫ℝy1nf1y1q11+2j∣x−y1∣s1q12jndy11/q1≲∫ℝy1n1+2jx−y1s1q1′F−1m2jx−y1,2jx−y2q1′2jndy11/q1′Mf1q1x1/q1,where we have used the fact that s1q1>n. By the same way for y2 and the change of variables, we have the desired estimate with N=2.☐


## 4. Proof of Theorem1

In this section, we prove Theorem1. Let 1<p1,⋯,pN<∞, 1/p1+⋯+1/pN=1/p, r⟶=r1,⋯,rN∈1,2N, rN≤rN−1≤⋯≤r2≤r1, s⟶=s1,⋯,sN∈ℝN, and n/ri<si≤n for all i=1,⋯,N. Assume pi>n/si and wi∈Apisi/n for all i=1,⋯,N and set w=w1p/p1⋯wNp/pN. We also assume that m∈L∞ℝNn satisfies
(42)supj∈ℤ∥mj∥Hs⟶r⟶ℝnN<∞.Sincen/si<minri,pi and wi∈Api/n/si for all i=1,⋯,N, by Lemma 3, we can take n/si<qi<minri,pi satisfying wi∈Api/qi for all i=1,⋯,N. By Lemma 2 (1), we decompose m as follows:
(43)mξ=∑i1,⋯,iN∈0,1N,i1,⋯,iN≠0,⋯,0Φi1,⋯,iNξmξ=∑i1,⋯,iN∈0,1N,i1,⋯,iN≠0,⋯,0mi1,⋯,iNξ.


### 4.1. Estimate form1,0,⋯,0 Type

We first consider the case wherei1,⋯,iN satisfies #j:ij=1=1. Without loss of generality, we may assume that i1=1. We simply write m instead of m1,0,⋯,0. Note that by Lemma 2 (3),
(44)suppm⊂ξ=ξ1,⋯,ξN∈ℝnN:∣ξi∣≤∣ξ1∣/N,i=2,⋯,N.It is easy to see that ifξ=ξ1,⋯,ξN∈suppm, then ∣ξ1+⋯+ξN∣≈∣ξ1∣.Letψ be as in (2) with d=n. Since w∈ANp⊂A∞ (see p. 1232 of [19]), we can use the results of Grafakos and Si (see Lemma 2.4 of [9]) and Fujita and Tomita (see Remark 2.6 of [6]); hence,
(45)Tmf1,⋯,fNLpw≲∑j∈ℤΔjTmf1,⋯,fN21/2Lpw,where Δjg=ψD/2jg.By Fubini’s theorem and the Fourier inversion formula, it is easy to see that(46)ΔjTmf1,⋯,fNx=12πNn∫ℝnNeix·ξ1+⋯+ξNmξψξ1+⋯+ξN/2jf1^ξ1⋯fN^ξNdξ.We shall prove that we can find functionsφ∈A0 and ψ~∈A1 independent of j such that
(47)mξψξ1+⋯+ξN2j=mξψξ1+⋯+ξN2jψ~ξ1/2j2φξ2/2j⋯φξN/2j.Once this is proven, setting(48)mjξ=m2jξψξ1+⋯+ξNψ~ξ1φξ2⋯φξN,we have
(49)ΔjTmf1,⋯,fNx=Tmj·/2jΔ~jf1,f2,⋯,fNx,where Δ~jf1=ψ~D/2jf1. Let ξ=ξ1,⋯,ξN∈suppm satisfying 2j−1≤∣ξ1+⋯+ξN∣≤2j+1. We take functions ψ~∈A1 such that ψ~η=1 on η∈ℝn:N/22N−1≤∣η∣≤2N and φ∈A0 such that φη=1 on η∈ℝn:∣η∣≤2. Hence, we obtain (47).SincesuppΨ·/2ℓ⊂ξ∈ℝnN:2ℓ−1≤∣ξ∣≤2ℓ+1 and
(50)suppψ~ξ1φξ2⋯φξN⊂ξ=ξ1,⋯,ξN∈ℝnN:2−j0≤ξ≤2j0,for some j0∈ℕ, we have
(51)Tmj·/2jΔ~jf1,f2,⋯,fNx=∑k=−N0N0Tmj,k·/2jΔ~jf1,f2,⋯,fNx,for some N0∈ℕ, where mj,kξ=mjξΨξ/2k. By Lemma 11, we see that
(52)Tmj,k·/2jΔ~jf1,f2,⋯,fNx≲∥mj,k^∥Ls1q1′,⋯,sNqN′q1′,⋯,qN′ℝnNMΔ~jf1q1x1/q1∏i=2NMfiqix1/qi.We shall prove that the estimate(53)∥mj,k^∥Ls1q1′,⋯,sNqN′q1′,⋯,qN′ℝnN≲supj∈ℤ∥mj∥Hs⟶r⟶ℝnN,holds. Once this is proved, combining (51), (52), and (53), we have
(54)Tmj·/2jΔ~jf1,f2,⋯,fNx≲supj∈ℤ∥mj∥Hs⟶r⟶ℝnNMΔ~jf1q1x1/q1∏i=2NMfiqix1/qi.Sincemjξ is defined by (48) and we have used m instead of m1,0,⋯,0, by Young’s inequality with mixed type (Lemma 7), we see that
(55)Fmj·Ψ·/2kLs1q1′,⋯,sNqN′q1′,⋯,qN′ℝnN≤Fm2j·Ψ·/2kLs1q1′,⋯,sNqN′q1′,⋯,qN′ℝnN×ξ1s1⋯ξNsNBj^L1ℝNn,where Bjξ1,⋯,ξN=Φ1,0,⋯,02jξψξ1+⋯+ξNψ~ξ1φξ2⋯φξN. By Lemma 10, we obtain (53), where we have used the fact that
(56)ξ1s1⋯ξNsNBj^L1ℝNn<∞.By Hölder’s inequality, we have(57)∑j∈ℤTmj·/2jΔ~jf1,f2,⋯,fN21/2Lpw≲supj∈ℤ∥mj∥Hs⟶r⟶ℝnN×∑j∈ℤMΔ~jf1q12/q11/2Lp1w1∏i=2NMfiqi1/qiLpiwi.For the second term on the right-hand side of (57), since 1<2/q1,p1/q1, and w1∈Ap1/q1, it follows from Lemmas 5 and 4 that
(58)∑j∈ℤMΔ~jf1q12/q11/2Lp1w1=∑j∈ℤMΔ~jf1q12/q1q1/2Lp1/q1w11/q1≲∑j∈ℤΔ~jf12q1/2Lp1/q1w11/q1=∑j∈ℤΔ~jf121/2Lp1w1≲∥f∥Lp1w1.For the third term on the right-hand side of (57), since pi>qi and wi∈Api/qi for all i=2,⋯,N, we see that
(59)∏i=2NMfiqi1/qiLpiwi=∏i=2NMfiqiLpi/qiwi1/qi≲∏i=2NfiqiLpi/qiwi1/qi=∏i=2NfiLpiwi,where we have used the boundedness of M on Lpi/qiwi for all i=2,⋯,N. By (45), (49), (54), (57), (58), and (59), we obtain the desired estimate.


### 4.2. Estimate form1,1,i3,⋯,iN Type

Next, we consider the case wherei1,⋯,iN satisfies #j:ij=1≥2. Without loss of generality, we may assume that i1=i2=1. We simply write m instead of m1,1,i3,⋯,iN, where i3,⋯,iN∈0,1. Note that by Lemma 2 (3),
(60)suppm⊂ξ∈ℝnN:ξ1/4N≤ξ2≤4Nξ1,ξi≤4Nξ1,i=3,⋯,N,where ξ=ξ1,⋯,ξN∈ℝnN.Sinceψ is in (2) with d=n, we see that
(61)Tmf1,⋯,fNx=∑j∈ℤ12πNn∫ℝnNeix·ξ1+⋯+ξNmξψξ12jf1^ξ1⋯fN^ξNdξ.We shall prove that we can find functionsφ∈A0 and ψ~∈A1 independent of j such that
(62)mξψξ12j=mξψξ12jψ~ξ12jψ~ξ22j2φξ32j⋯φξN2j.Once this is proved, setting(63)mjξ=m2jξψξ1ψ~ξ2φξ3⋯φξN,we have
(64)Tmf1,⋯,fNx=∑j∈ℤTmj·/2jΔ~jf1,Δ~jf2,f3,⋯,fNx,where Δ~jfi=ψ~D/2jfi for i=1,2. Let ξ=ξ1,⋯,ξN∈suppm and ξ1∈suppψ·/2j. We take functions ψ~∈A1 such that ψ~η=1 on η∈ℝn:1/8N≤∣η∣≤8N and φ∈A0 such that φη=1 on η∈ℝn:∣η∣≤8N. Hence, we obtain (62).As in Section4.1, by Lemmas 11 and 10, and Schwarz’s inequality, we have
(65)∑j∈ℤTmj·/2jΔ~jf1,Δ~jf2,f3,⋯,fNx≲supj∈ℤ∥mj∥Hs⟶r⟶ℝnN∑j∈ℤMΔ~jf1q1x2/q11/2∑j∈ℤMΔ~jf2q2x2/q21/2×∏i=3NMfiqix1/qi.The rest of the proof is similar to that ofm1,0,⋯,0, and we omit it.


## 4.1. Estimate form1,0,⋯,0 Type

We first consider the case wherei1,⋯,iN satisfies #j:ij=1=1. Without loss of generality, we may assume that i1=1. We simply write m instead of m1,0,⋯,0. Note that by Lemma 2 (3),
(44)suppm⊂ξ=ξ1,⋯,ξN∈ℝnN:∣ξi∣≤∣ξ1∣/N,i=2,⋯,N.It is easy to see that ifξ=ξ1,⋯,ξN∈suppm, then ∣ξ1+⋯+ξN∣≈∣ξ1∣.Letψ be as in (2) with d=n. Since w∈ANp⊂A∞ (see p. 1232 of [19]), we can use the results of Grafakos and Si (see Lemma 2.4 of [9]) and Fujita and Tomita (see Remark 2.6 of [6]); hence,
(45)Tmf1,⋯,fNLpw≲∑j∈ℤΔjTmf1,⋯,fN21/2Lpw,where Δjg=ψD/2jg.By Fubini’s theorem and the Fourier inversion formula, it is easy to see that(46)ΔjTmf1,⋯,fNx=12πNn∫ℝnNeix·ξ1+⋯+ξNmξψξ1+⋯+ξN/2jf1^ξ1⋯fN^ξNdξ.We shall prove that we can find functionsφ∈A0 and ψ~∈A1 independent of j such that
(47)mξψξ1+⋯+ξN2j=mξψξ1+⋯+ξN2jψ~ξ1/2j2φξ2/2j⋯φξN/2j.Once this is proven, setting(48)mjξ=m2jξψξ1+⋯+ξNψ~ξ1φξ2⋯φξN,we have
(49)ΔjTmf1,⋯,fNx=Tmj·/2jΔ~jf1,f2,⋯,fNx,where Δ~jf1=ψ~D/2jf1. Let ξ=ξ1,⋯,ξN∈suppm satisfying 2j−1≤∣ξ1+⋯+ξN∣≤2j+1. We take functions ψ~∈A1 such that ψ~η=1 on η∈ℝn:N/22N−1≤∣η∣≤2N and φ∈A0 such that φη=1 on η∈ℝn:∣η∣≤2. Hence, we obtain (47).SincesuppΨ·/2ℓ⊂ξ∈ℝnN:2ℓ−1≤∣ξ∣≤2ℓ+1 and
(50)suppψ~ξ1φξ2⋯φξN⊂ξ=ξ1,⋯,ξN∈ℝnN:2−j0≤ξ≤2j0,for some j0∈ℕ, we have
(51)Tmj·/2jΔ~jf1,f2,⋯,fNx=∑k=−N0N0Tmj,k·/2jΔ~jf1,f2,⋯,fNx,for some N0∈ℕ, where mj,kξ=mjξΨξ/2k. By Lemma 11, we see that
(52)Tmj,k·/2jΔ~jf1,f2,⋯,fNx≲∥mj,k^∥Ls1q1′,⋯,sNqN′q1′,⋯,qN′ℝnNMΔ~jf1q1x1/q1∏i=2NMfiqix1/qi.We shall prove that the estimate(53)∥mj,k^∥Ls1q1′,⋯,sNqN′q1′,⋯,qN′ℝnN≲supj∈ℤ∥mj∥Hs⟶r⟶ℝnN,holds. Once this is proved, combining (51), (52), and (53), we have
(54)Tmj·/2jΔ~jf1,f2,⋯,fNx≲supj∈ℤ∥mj∥Hs⟶r⟶ℝnNMΔ~jf1q1x1/q1∏i=2NMfiqix1/qi.Sincemjξ is defined by (48) and we have used m instead of m1,0,⋯,0, by Young’s inequality with mixed type (Lemma 7), we see that
(55)Fmj·Ψ·/2kLs1q1′,⋯,sNqN′q1′,⋯,qN′ℝnN≤Fm2j·Ψ·/2kLs1q1′,⋯,sNqN′q1′,⋯,qN′ℝnN×ξ1s1⋯ξNsNBj^L1ℝNn,where Bjξ1,⋯,ξN=Φ1,0,⋯,02jξψξ1+⋯+ξNψ~ξ1φξ2⋯φξN. By Lemma 10, we obtain (53), where we have used the fact that
(56)ξ1s1⋯ξNsNBj^L1ℝNn<∞.By Hölder’s inequality, we have(57)∑j∈ℤTmj·/2jΔ~jf1,f2,⋯,fN21/2Lpw≲supj∈ℤ∥mj∥Hs⟶r⟶ℝnN×∑j∈ℤMΔ~jf1q12/q11/2Lp1w1∏i=2NMfiqi1/qiLpiwi.For the second term on the right-hand side of (57), since 1<2/q1,p1/q1, and w1∈Ap1/q1, it follows from Lemmas 5 and 4 that
(58)∑j∈ℤMΔ~jf1q12/q11/2Lp1w1=∑j∈ℤMΔ~jf1q12/q1q1/2Lp1/q1w11/q1≲∑j∈ℤΔ~jf12q1/2Lp1/q1w11/q1=∑j∈ℤΔ~jf121/2Lp1w1≲∥f∥Lp1w1.For the third term on the right-hand side of (57), since pi>qi and wi∈Api/qi for all i=2,⋯,N, we see that
(59)∏i=2NMfiqi1/qiLpiwi=∏i=2NMfiqiLpi/qiwi1/qi≲∏i=2NfiqiLpi/qiwi1/qi=∏i=2NfiLpiwi,where we have used the boundedness of M on Lpi/qiwi for all i=2,⋯,N. By (45), (49), (54), (57), (58), and (59), we obtain the desired estimate.


## 4.2. Estimate form1,1,i3,⋯,iN Type

Next, we consider the case wherei1,⋯,iN satisfies #j:ij=1≥2. Without loss of generality, we may assume that i1=i2=1. We simply write m instead of m1,1,i3,⋯,iN, where i3,⋯,iN∈0,1. Note that by Lemma 2 (3),
(60)suppm⊂ξ∈ℝnN:ξ1/4N≤ξ2≤4Nξ1,ξi≤4Nξ1,i=3,⋯,N,where ξ=ξ1,⋯,ξN∈ℝnN.Sinceψ is in (2) with d=n, we see that
(61)Tmf1,⋯,fNx=∑j∈ℤ12πNn∫ℝnNeix·ξ1+⋯+ξNmξψξ12jf1^ξ1⋯fN^ξNdξ.We shall prove that we can find functionsφ∈A0 and ψ~∈A1 independent of j such that
(62)mξψξ12j=mξψξ12jψ~ξ12jψ~ξ22j2φξ32j⋯φξN2j.Once this is proved, setting(63)mjξ=m2jξψξ1ψ~ξ2φξ3⋯φξN,we have
(64)Tmf1,⋯,fNx=∑j∈ℤTmj·/2jΔ~jf1,Δ~jf2,f3,⋯,fNx,where Δ~jfi=ψ~D/2jfi for i=1,2. Let ξ=ξ1,⋯,ξN∈suppm and ξ1∈suppψ·/2j. We take functions ψ~∈A1 such that ψ~η=1 on η∈ℝn:1/8N≤∣η∣≤8N and φ∈A0 such that φη=1 on η∈ℝn:∣η∣≤8N. Hence, we obtain (62).As in Section4.1, by Lemmas 11 and 10, and Schwarz’s inequality, we have
(65)∑j∈ℤTmj·/2jΔ~jf1,Δ~jf2,f3,⋯,fNx≲supj∈ℤ∥mj∥Hs⟶r⟶ℝnN∑j∈ℤMΔ~jf1q1x2/q11/2∑j∈ℤMΔ~jf2q2x2/q21/2×∏i=3NMfiqix1/qi.The rest of the proof is similar to that ofm1,0,⋯,0, and we omit it.



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