Function spaces and its applications to problems on Harmonic Analysis

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Function spaces and its applications to problems on Harmonic Analysis

Shohei Nakamura

January 23, 2019

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Preface

1.1 Short introduction to Harmonic Analysis

In this thesis, we collect and summarize the author’s works [7, 6, 55, 56, 80] with more historical backgrounds. As a preface, in this first part, we overview recent progress of Harmonic Analysis and explain more details of the title of this thesis where the standing position of the author’s works in Harmonic Analysis will be clarified.

The main object of the field of Harmonic Analysis is the Fourier transform ^∧ which is formally defined for a functionf :R^d→Cby

fˆ(ξ) = Z

R^d

e^−ix·ξf(x) dx, ξ∈R^d,

whered∈Ndenotes the dimension in this whole thesis. The history of the Fourier transform can be tracked back to the 18th century due to Joseph Fourier. We do not intend to pursue the detailed histories of the Fourier transform here but instead, we will exhibit recent related topics where one can realize the utility of the Fourier transform. The definition of the Fourier transform given in the above is quite vague, for instance if one choosesf as a constant function on the whole space, then ˆf does not make any sense and moreover it will be out of the framework of “functions”. So, it is important to clarify for whichf we can be sure that ˆf makes sense as a function. To describe it, the notion of function spaces occur naturally and one typical example is theL^p space, 1≤p≤ ∞which is the set of all measurable functions f such that

kfk_Lp(R^d)=kfkL^p=kfkp= Z

R^d

|f(x)|^pdx1/p

<∞.

Then one can see that ˆf makes sense as long asf ∈L¹ since iff ∈L¹, then the integral in the definition of the Fourier transform at least converges:

|fˆ(x)| ≤ Z

R^d

|f(x)|dx=kfk_L1<∞.

Moreover, one can define ˆf for any f ∈ L² thanks to Plancherel’s theorem which is one of the fundamental theorems in Harmonic Analysis. In this sense, one can say thatL¹ orL²are suitable function spaces to capture the Fourier transform. Let us see further varieties of the operators induced by the Fourier transform. One of such operators is so called the “Fourier

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multiplier operator”. To give its definition, we need to take a multiplierσ:R^d→Cfirst and then the Fourier multiplier operatorTσ is formally defined for a functionf by

T_σf(x) = Z

R^d

e^ix·ξσ(ξ) ˆf(ξ) dξ, x∈R^d.

Again, we need to clarify the class off for whichTσf makes sense for eachσ. The well-known example ofσis

σj(ξ) =−iξj/|ξ|

for j = 1, . . . , d, where the corresponding Fourier multiplier operators are called the Riesz transform denoted byRj:

Rjf(x) =Tσ_jf(x) = Z

R^d

e^ix·ξ(−iξj/|ξ|) ˆf(ξ) dξ

and whend= 1 it is called the Hilbert transform denoted byH. By using Plancherel’s theorem, one can again ensure thatRjf makes sense as a function iff ∈L². However, one can extend the function space further. Indeed,Rjf exists as a function for anyf ∈L^pand any 1< p <∞ and this is a consequence of the so-called Calder´on-Zygmund theory. So, in that sense, L^p, 1< p < ∞is a suitable function space to handle the Riesz transform, but unlike the case of the Fourier transform,L¹is not suitable. To overcome this problem, the Hardy spaceH¹ was introduced as a substitute of L¹ and it was proved by Fefferman-Stein [39] that Rjf makes sense as long asf ∈H¹, where we do not give the precise definition ofH¹; see Part I. The main purpose of Part I is to develop the Calder´on-Zygmund theory further so that the multilinear version of the Fourier multiplier operator can be handled for the Hardy space inputs. The Hilbert transform and the Riesz transform naturally appear in many field of mathematics, complex analysis, Navier-Stokes equation as well as signal processing. Here, let us see how the Riesz (Hilbert) transform emerges from the question about Fourier transform itself. Perhaps, the most imaginable example of σ would be a constant function, σ(ξ) = 1 for allξ ∈R^d. In such a case, a formal computation shows

Tσf(x) = Z

R^d

e^ix·ξ1 ˆf(ξ) dξ=f(x),

which follows from the Fourier inversion formula. However, again we meet the problem of what is the function space for which the above Fourier inversion formula makes sense? In particular, when does the integral

Z

R^d

e^ix·ξ1 ˆf(x) dξ

converge? For instance, if ˆf ∈L¹ (namely,f in “Fourier-L¹”), then clearly the above Fourier inversion formula does make sense. But, is the FourierL¹ the most suitable function space for that purpose? To tackle this problem, we may employ the summability method. By using a notation QR to denote a cube in R^d center at the origin and whose length 2R > 0, namely QR= [−R, R]^d, we may consider the truncated Fourier integral: for eachR >0,

TσRf(x) = Z

R^d

e^ix·ξ1QR(ξ) ˆf(ξ) dξ, σR(ξ) = 1QR(ξ).

Thanks to H¨older’s inequality, the integral in the definition ofTσ_Rf converges as long asf ∈ L¹∪L². Since lim_R→∞1Q_R= 1 pointwisely, our problem can be formulated as follows: for any f ∈L¹∪L², does

R→∞lim Tσ_Rf(x) =f(x) (1.1.1)

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hold in a certain sense? For the sake of simplicity, let us consider the case d= 1. Then the Hilbert transformHreveals naturally:

T_σ_Rf(x) =1

2 ie^−iRxH[eîR·f]−ieîRxH[e^−iR·f] +e^−iRxfˆ(−R) +eîRxfˆ(R) ,

although the extra computations are necessary to see this. The important point of this relation is that the problem (1.1.1) can be reduced (by a further argument) to the mapping property of H : L^p → L^p and this is exactly a consequence of the Calder´on-Zygmund theory as we mentioned before. Hence, the answer to (1.1.1) is positive for allf ∈L^p with 1< p≤2. The situations for higher dimension are similar, namely the problem (1.1.1) can be reduced to the mapping properties of Riesz transformsRj :L^p →L^p for allj = 1, . . . , dbecause Tσ_R can be represented by combinations fo certain modulations andRj. Once one notices these relations, it is quite straightforward to see one possibility of generalization by replacing QR by more general polygonsPN,R whose “radius” isRand which is constructed byN edges. As before,

Tσ_R,Nf(x) = Z

R^d

e^ix·ξ1P_R,N(ξ) ˆf(ξ) dξ, σR,N(ξ) = 1P_R,N(ξ)

can be represented by combinations of modulation, rotation and Rj. But now, to represent Tσ_R,N byRj, we need to useRj roughlyNtimes. So, as long asN is fixed, we can carry on the argument for the caseQR but we need to make use of the mapping properties of Rj roughly N times. Then it is natural to ask ourselves about the case N → ∞ which means the “ball multiplier”: lim_N→∞PR,N =BR, where BR stands the ball in R^d centered at the origin and radiusR. In other words, for allf ∈L^p, 1< p≤2, does

R→∞lim Z

R^d

e^ix·ξ1B_R(ξ) ˆf(ξ) dξ=f(x) (1.1.2) hold in certain sense? Iff ∈L², then the answer is positive because of the Plancherel theorem.

Surprisingly, it turns out thatp= 2 is the only case for which (1.1.2) holds true due to Charles Fefferman [38]. He employed the Kakeya-Besicovitch set which is a subset ofR^d containing all line segments orienting in every direction and Lebesgue measure zero to give a counterexample for the casep6= 2. This is a point of contact between Harmonic Analysis and Geometry. Also, the next example of the Fourier multiplier has the same flavor with this Kakeya-Besicovitch set.

Let us go to the second example of the multiplier generated by σt(ξ) =e^−it|ξ|²,t ∈R. In this case,Tσ_t is known as the Schr¨odinger propagator and denoted bye^it∆:

Tσ_tf(x) =e^it∆f(x) = Z

R^d

e^{i(x·ξ−t|ξ|}²⁾fˆ(ξ) dξ sincee^it∆f is the solution to the free Schr¨odinger equation

i∂tu(t, x) =−∆xu(t, x), (t, x)∈R^d+1, u(0, x) =f(x),

which describes the behavior of one particle following Quantum Mechanics. Remark thate^it∆f makes sense for eacht∈Rwheneverf ∈L²thanks to the Plancherel theorem. The Schr¨odinger propagator has an important property called “dispersivity”:

sup

x∈R^d

|e^it∆f(x)| ≤C|t|^−d/2kfk_L1(R^d), t∈R

which insists that the free solution of the Schr¨odinger equation decays in time with order−d/2.

This property yields the Strichartz estimate which guarantees that for all 2≤q, r≤ ∞ such

that 2

q +d r = d

2, (q, r, d)6= (2,∞,2),

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the inequality ke^it∆fk_L^q

tL^r_x(R^d+1)= Z

R

Z

R^d

|e^it∆f(x)|^rdx^q/r dt

^1/q

≤Ckfk_L2(R^d) (1.1.3) holds true for all f ∈ L²(R^d). This was established by Strichartz [90] for the diagonal case q = r = 2(d+ 2)/d, Ginibre-Velo [46] for the non-diagonal and non-endpoint case (q, r) 6=

(2,2d/(d−2)) and then the endpoint estimate was proved by the monumental work due to Keel-Tao [64], see also [78, 96, 99] and [93] for the application to nonlinear PDEs. In part II, we will generalize and improve these Strichartz estimates to handle infinitely many quantum particles.

This Schr¨odinger propagator has another perspective to look at. Suppose supp ˆf ⊂B1 for the sake of simplicity. Then, if we denote the truncated paraboloid P^d ={(ξ, ξd+1)∈R^d+1 :

|ξ| ≤1, ξd+1=−|ξ|²} and the surface measure onP^dby dσ, then e^it∆f(x) =

Z

(ξ,ξd+1)∈P^d

ei(x,t)·(ξ,ξ_d+1)fˆ(ξ) dσ(ξ, ξd+1) =gdσ(x, t),d

where g : P^d → C is defined by g(ξ, ξd+1) = f(ξ). The operator gdσd is called the Fourier extension operator which is the dual operator of the Fourier restriction operator Rwhich is formally defined by forF :R^d+1→C,

RF(ξ, ξd+1) = ˆF(ξ, ξd+1), (ξ, ξd+1)∈P^d.

Of course,RFmay not be well defined sinceP^dis a measure zero inR^d+1but it does for instance ifF ∈L¹(R^d+1) since ˆF is a continuous function by the Riemann–Lebesgue theorem. It was posed by Stein that what is the class ofF for whichRF makes sense. From Stein’s maximum principle [87] (if the surface has nice symmetry like a sphere), this question is equivalent to the L^p-L^q bound ofRfor some 1≤p, q,≤ ∞: for allF ∈L^p(R^d+1),

kRFkL^q(P^d)≤CkFkL^p(R^d+1), or equivalently (from the duality) for allg∈L^q⁰(P^d),

kgdσkd _Lp0(R^d+1)≤Ckgk_Lq0(P^d). (1.1.4) Applying (1.1.4) with the Knapp example g = 1_B_δ_∩Pd and taking δ → 0, we learn that q ≤ dp⁰/(d+ 2) is necessary. Also, the asymptotic behavior of Bessel function reveals1dσ(x)d ∼ hxi^−d/2and hence,p⁰>2(d+ 1)/dis also necessary. Based on this observation, Stein gave the following Fourier restriction conjecture.

Conjecture 1.1.1. Letn=d+ 1≥2. Then (1.1.4) holds true as long asp⁰>2n/(n−1) and q≤(n−1)p⁰/(n+ 1).

This conjecture was proved by Fefferman-C´ordoba whenn = 2 since in that case the end point becomes (p⁰, q⁰) = (4,4) where one can exploit the orthogonality by regardingkgdσkd L⁴ as k|dgdσ|²k_L2. For the higher dimensionn≥3, this conjecture is still an open problem regardless of the recent progress via the bilinear/multilinear approach and the polynomial partitioning method [4, 17, 49, 50, 60, 94, 98]. It is notable that in the caseq⁰= 2 and hencep⁰ = 2(d+ 2)/d the extension estimate (1.1.4) due to Stein-Tomas corresponds to the Strichartz estimate (1.1.3) withq=r= 2(d+ 2)/d. This exhibits the close connection between the Strichartz estimates, applications to non-liner PDEs and the Fourier restriction conjecture. As a further example, we

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give the connection between the bilinear restriction estimate and the Besov type improvement of the Strichartz estimates. Let us consider the case n = 2 (and so d = 1). The bilinear restriction estimate takes a form that

[g1dσg[2dσ _L_q₍

R²)≤Ckg1k_L2(P¹)kg2k_L2(P¹) (1.1.5) for all g1, g2 whose supports are separated and so the constant C depends on the distance between supp(g1) and supp(g2). The sharp exponent is q = 2 in that case and it needs a few lines of computations to see (1.1.5) with q = 2 by using Plancherel’s theorem, complex interpolation and fact that (dσ₁)∗(dσ₂)∈L^∞(R²) if and only if the supports of dσ₁ and dσ₂ are separated. For the higher dimension, the sharp bilinear restriction estimate is no longer easy to prove but Tao [92] could manage to show that. Let us formally putg₁=g₂=g in (1.1.5) withq= 2 although this is prohibited by the assumption of the separated support condition.

Then eventually one has

kgdσkd _L4(R²)≤Ckgk_L2(P¹)

which is “better” than the linear Fourier restriction conjecture. Indeed, the best thing for the linear one is

k1_B_Rgdσkd _L4(R²)≤Clog(R)kgk_L4(P¹)

which is clearly weaker. This means the constantC in (1.1.5) should blow up as the distance between the supports of g1, g2 tends to 0. But, one can track the dependence on separation of the constant carefully and then it turns out that it is possible to deduce the linear Fourier restriction estimate (1.1.4) from the bilinear one (1.1.5) due to Tao-Vargas-Vega [94].

Let us go back to the Schr¨odinger propagator. For the sake of simplicity, let us consider the case d = 2 (not d = 1 now). The reason of this is the Stein-Tomas exponent becomes 2(d+ 2)/2 = 4 in this case. Then, the Strichartz estimate (1.1.3) can be stated by

ke^it∆fk_L⁴

t,x(R²⁺¹)≤CkfkL²(R²). (1.1.6) The bilinear analogue of this estimate is also possible as follows, see [11]. Forf1, f2 such that supp( ˆf1)⊂ {ξ∈ R² : |ξ| ≤2M} and supp( ˆf1)⊂ {ξ∈ R² : |ξ| ∼N}, where 0 < M ≤N are fixed number,

e^it∆f1e^it∆f2

_L2

t,x(R²⁺¹)≤C(M/N)^1/2kf1k_L2(R²)kf2k_L2(R²). (1.1.7) Now, we are allowed to overlap the supports of two functions sinceq= 2 is no longer endpoint of the casen=d+ 1 = 3. Indeed, if one takesf₁=f₂=f, then this bilinear estimate corresponds to the linear one (1.1.6). The key point of these estimates is the explicit dependence of the separation M, N. By making use of this information skillfully, it is possible to improve the linear Strichartz (1.1.6) in terms of another function space, the Besov space ˙B^s_p,q (we do not give the precise definition here):

ke^it∆fk_L4

t,x(R²⁺¹)≤CkfkB˙_2,4⁰ (R²), (1.1.8) which improves (1.1.6) because of the well-known embedding L²(R²) ⊂ B˙⁰_2,4(R²). The idea of implication from the bilinear estimate to the refined Strichartz (1.1.8) is to exploit the orthogonality of the disjoint support as follows. Let us breakf =P

j∈ZPjf, wherePj is usual Littlewood-Payley projection. Then first use the triangle inequality

keît∆fk²₄=keît∆f eît∆fk2=

X

j∈Z

e^it∆Pjf X

k∈Z

e^it∆Pkf ₂

≤X

l≥0

X

k∈Z

e^it∆Pk+lf·e^it∆Pkf ₂

.

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Notice that the Fourier supports ofeît∆Pk₁+lf·eît∆Pk₁f andeît∆Pk₂+lf·eît∆Pk₂f are disjoint ifk16=k2andl≥1. Since the norm is nowL², we can exploit the orthogonality and then use (1.1.7) and Cauchy-Schwarz to have

ke^it∆fk²₄≤CX

l≥1

X

k∈Z

e^it∆Pk+lf·e^it∆Pkf

2 2

1/2

+{The term l= 0}

≤CX

l≥1

2^−l/2

X

k∈Z

kPk+lfk²₂kPkfk²₂ ^1/2

+{The term l= 0}

≤CX

l≥1

2^−l/2

X

k∈Z

kPk+lfk⁴₂^1/2 X

k∈Z

kPkfk⁴₂^1/2 ^1/2

+{The term l= 0}

∼ kfk²_B_˙0 2,4

+{The term l= 0}.

Such an improvement by using Besov spaces provides powerful tools. One such example is the so-called “profile decomposition” which has a lot of applications to nonlinear PDEs as well as the problem to identify the maximizer of the (linear) Strichartz estimate, see for instance [84] and references therein. Our object in Chapter 3 will have a similar flavor of such a Besov improvement.

As we saw in the above, it is important to choose suitable function spaces depending on each problem and hence one needs to be familiar with variable function spaces, not only L^p spaces but also weightedL^p spaces, Hardy spaces, Lorentz spaces, Besov spaces etc. Perhaps, reflecting such a circumstance, there seem to exist at least two approaches in the research area of Harmonic Analysis. The first one is to investigate inequalities itself related to the Fourier multiplier or the Fourier restriction conjecture and the invented argument in there has sometimes fruitful applications to the nonlinear PDEs. The second one is to focus mainly on the functions spaces. The idea for the second direction is that one can capture the sensitive behavior of functions by using several function spaces and hence it is also helpful to understand the feature of each function space more deeply. Of course, it is impossible to separate these two directions explicitly but it is also true that there is a little distance between these two directions.

The main feature of the author’s work exhibited in the rest of this thesis is to bridge these two directions. For example, in Parts I and II below, our main subject would be more likely the first direction (Fourier multiplier and Strichartz estimates). However, in the proof, implicitly, we will heavily depend on the skillful use of several functions spaces, which is possible thanks to the deep understanding of the function spaces.

The organization of this thesis is as follows. In Chapter 2, we will exhibit results about the multilinear Fourier multipliers, which are based on [55, 56]. There, we will state our results involving motivations, examples and applications, however, we do not give the proof of these theorems and refer to [55, 56] for it. In Chapter 3, we will discuss the orthonormal Strichartz estimates including the whole proofs and the results in this chapter are based on [6, 7, 80].

I should menthon that [55, 56] are joint works with Professors Loukas Grafakos, Hahn Van Nguyen and Yoshihiro Sawano and [6, 7] are joint works with Professors Neal Bez, Younghun Hong, Sanghyuk Lee and Yoshihiro Sawano, while [80] is completely an own work of the author.

1.2 Acknowledgements

First of all, the author deeply appreciates to his family Masahiko, Misako, Gento, Etsuko. They have been supporting him patiently and constantly from his childhoods to now.

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He shows his gratitude to his supervisor, Yoshihiro Sawano. Thanks to his great lecture, the author began to be interested in the field of analysis and his critical and helpful comments pushed his research a lot.

He is grateful to Neal Bez who introduced to him the fruitful world of Harmonic analysis, the author learns the behavior as a mathematician from him.

He is also deeply thankful to Sanghyuk Lee who gave him necessary knowledges to establish the results of this thesis, the author learned the attitude to the mathematics from him.

He has a great pleasure to work with a lot of collaborators, Jonathan Bennett, Loukas Grafakos, Denny Ivanal Hakim, Younghun Hong, Takeshi Iida, Hanh Van Nguyen, Takahiro Noi and Hitoshi Tanaka.

Deep appreciate to his friends, Nozomi Iyonaga, he had a lot of chance to refresh from mathematical stuff thanks to her “unpredictable suggestions of events”, Mizuki Tateishi, he shared impressive mathematical experiences with him in high school, and Yujiro Nogi who is a mathematical brother of him somehow. All of their kindly friendships more or less contributed to push his career and he would not be able to achieve and complete the PhD course unless such a great chance encounters. Thanks to a lot of places, coast in Hirado, youme-town in Munakata, Kyushu university in Hakozaki, Tokyo metropolitan university in Mianami-osawa, lofty peak of Mt.Yatsugatake and marvelous onsen in Nagano, Rodina-sabou in Kunitachi and Sainsbury in Birmingham.

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Chapter 2

Multilinear Calde´ on-Zygmund theory on Hardy spaces

2.1 Conditions for the boundedness into Hardy spaces

In this section, we obtain the boundedness for multilinear singular operators of various types from products of Lebesgue or Hardy spaces into Hardy spaces, under suitable cancellation conditions. This particular line of investigation was initiated in the work of Coifman, Lions, Meyer and Semmes [28] who showed that certain bilinear operators with vanishing integral map L^q×L^q⁰ into the Hardy spaceH¹ for 1< q <∞withq⁰=q/(q−1). This result was extended by Dobyinksi [35] for Coifman-Meyer multiplier operators and by Coifman and Grafakos [30] for finite sums of products of Calder´on-Zygmund operators. In [30] the boundedness was extended toH^p¹×H^p² →H^p for the entire range 0< p₁, p₂, p <∞and 1/p= 1/p₁+ 1/p₂, under the necessary cancellation conditions.

Additional approaches to these results were provided by Grafakos and Li [54], Hu and Meng [61], and Huang and Liu [62]. All the aforementioned accounts on this topic are based on different approaches and address two classes of operators but [30], [61], and [62] seem to contain flaws in their proofs; in fact, as of this writing, only the approach in [54] stands, which deals with the case of finite sums of products of Calder´on-Zygmund operators. In this section we investigate this topic via a new method based on (p,∞)-atomic decompositions. Our approach is powerful enough to encompass many types of multilinear operators that include all the previously studied (Coifman-Meyer type and finite sums of products of Calder´on-Zygmund operators) as well as mixed types. An alternative approach to Hardy space estimates for bilinear operators has appeared in the recent work of Hart and Lu [59].

Recall that the Hardy space H^p with 0 < p < ∞ is given as the space of all tempered distributionsf for which

kfkH^p= sup

t>0

|e^t∆f| _L_p

is finite, where e^t∆ denotes the heat semigroup for 0 < p ≤ ∞. Note that H^p and L^p are isomorphic with norm equivalence when 1< p≤ ∞.

In this work we study the boundedness into H^p of the following three types of operators:

• multilinear singular integral operators of Coifman-Meyer type;

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• sums ofm-fold products of linear Calder´on-Zygmund singular integrals;

• multilinear singular integrals of mixed type (i.e., combinations of the previous two types).

Letm, nbe positive integers. For a bounded functionσon (Rⁿ)^mwe consider the multilinear operator

T_σ(f₁, . . . , f_m)(x) = Z

(Rⁿ)^m

σ(ξ₁, . . . , ξ_m)fb₁(ξ₁)· · ·fc_m(ξ_m)e^ix·(ξ¹^+···+ξ^m⁾dξ₁· · ·dξ_m (x∈Rⁿ) forf1, . . . , fm∈ S. HereSis the space of Schwartz functions andfb(ξ) =R

Rⁿf(x)e^−ix·ξdxis the Fourier transform of a given Schwartz functionf onRⁿ. The space of tempered distributions is denoted byS⁰.

Certain conditions on σ imply that Tσ extends to a bounded linear operator from L^p¹ ×

· · · ×L^p^m to L^p as long as 1< p₁, . . . , p_m≤ ∞and 0< p <∞satisfies 1

p= 1 p1

+· · ·+ 1 pm

. (2.1.1)

Such a condition is the following by Coifman-Meyer (modeled after the classical Mihlin linear multiplier condition)

|∂^ασ(ξ1, . . . , ξm)|.(|ξ1|+· · ·+|ξm|)^−|α|, (ξ1, . . . , ξm)∈(Rⁿ)^m\ {0} (2.1.2) for α ∈ (N0n)^m satisfying |α| ≤ M for some large M. Such operators are called m-linear Calder´on-Zygmund operators and there is a rich theory for them analogous to the linear one.

Anm-linear Calder´on-Zygmund operator associated with a Calder´on-Zygmund kernelKon R^mnis defined by

Tσ(f1, . . . , fm)(x) = Z

(Rⁿ)^m

K(x−y1, . . . , x−ym)f1(y1)· · ·fm(ym)dy1· · ·dym, (2.1.3) where σ is the distributional Fourier transform of K on (Rⁿ)^m that satisfies (2.1.2). When m= 1, these operators reduce to classical Calder´on-Zygmund singular integral operators.

We also study another type of m-linear operators so called product type. Before giving general definition, we introduce two examples of such operators. Let σ₁, σ₂ be the classical Mihlin multipliers defined byσ(ξ₁, ξ₂) =σ₁(ξ₁)σ₂(ξ₂). Then the first prominent example is

T_σ(f₁, f₂)(x) = Z

Rⁿ×Rⁿ

σ₁(ξ₁)σ₂(ξ₂) ˆf₁(ξ₁) ˆf₂(ξ₂)e^ix·(ξ¹^+ξ²⁾dξ₁dξ₂=T_σ₁(f₁)(x)T_σ₂(f₂)(x).

Secondly, let n = 2, m = 2 and for ξ = (η, ρ) ∈ R², set σ1(ξ) = _|(η,ρ)|^η , σ2(ξ) = _|(η,ρ)|^ρ . Namely, these are multipliers associated with Riesz transform on R². Then we define for (ξ₁, ξ₂)∈R²×R²,σ(ξ₁, ξ₂) =σ₁(ξ₁)σ₂(ξ₂)−σ₂(ξ₁)σ₁(ξ₂) which implies

Tσ(f1, f2)(x) =Tσ1(f1)(x)Tσ2(f2)(x)− Tσ2(f1)(x)Tσ1(f2)(x). (2.1.4) Clearly, the example (2.1.4) does not satisfy the Coifman-Meyer type condition (2.1.2). The second example was motivated by the determinant of the Jacobian of a map (f, g): J(f, g) =

∂x₁f ∂x₂g−∂x₂f ∂x₁gand the mapping property of the operator into Hardy spaces was studied in [30]. It would be notable to see that for the second example,

Z

R²

Tσ(f₁, f₂)(x)dx= Z

R²

(Tσ1(f₁)(x)Tσ2(f₂)(x)− Tσ2(f₁)(x)Tσ1(f₂)(x))dx= 0.

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Now, we introduce the operator more general form. Anm-linear operator ofproduct type on R^mnis defined by

T

X

ρ=1

T_σ^ρ

1(f1)(x)· · ·T_σ_m^ρ(fm)(x) (x∈Rⁿ), (2.1.5) where theT_σ^ρ

j’s are linear Calder´on-Zygmund operators associated with the multipliersσ_j^ρ. In terms of kernels these operators can be expressed as

T_σ(f₁, . . . , f_m)(x) =

T

X

ρ=1 m

Y

j=1

Z

Rⁿ

K_σ^ρ

j(x−y_j)f_j(y_j)dy_j, whereK₁^ρ, . . . , K_m^ρ are the Calder´on-Zygmund kernels of the operatorT_σ^ρ

1, . . . , T_σ^ρ

m, respectively forρ= 1, . . . , T.

With these two types of multiplier, Coifman-Meyer type andproduct type, in mind, it seems natural to consider these mixture version. For example, as a 6-linear operator, we may consider the following example. LetI1={1,2,3}, I2={4,5}, I3={6}. Also letTσ_I₁, Tσ_I₂ andTσ_I₃ be respectively 3-linear, 2-linear and 1-linear Coifman-Meyer type operators-namely, the multipliers σI₁, σI₂, σI₃ satisfy (2.1.2). Then we define σ(ξ1, . . . , ξ6) = σI₁(ξ1, ξ2, ξ3)σI₂(ξ4, ξ5)σI₃(ξ6) and hence,

Tσ(f1, . . . , f6)(x) =Tσ_I₁(f1, f2, f3)(x)Tσ_I₂(f4, f5)(x)Tσ_I₃(f6)(x).

More generally, we consider operators ofmixed type, i.e., of the form T_σ(f₁, . . . , f_m)(x) =

T

X

ρ=1

X

I₁^ρ,...,I_G(ρ)^ρ G(ρ)

Y

g=1

T_σ

Iρ

g({f_l}_l∈I^ρ_g)(x), (2.1.6) where for eachρ= 1, . . . , T, I₁^ρ, . . . , I_G(ρ)^ρ is a partition of{1, . . . , m} and eachTσ_Iρ

g is an|I_g^ρ|- linear Coifman-Meyer multiplier operator. We writeI₁^ρ+· · ·+I_G(ρ)^ρ ={1, . . . , m} to denote such partitions.

In this section, we study operators of the form (2.1.3), (2.1.5), and (2.2.11). We will be working with indices in the following range

0< p₁, . . . , p_m≤ ∞, 0< p <∞

that satisfy (2.1.1). Throughout this paper we reserve the lettersto denote the following index:

s= [n(1/p−1)]₊ (2.1.7)

and we fixN sa sufficiently large integer, sayN =m(n+ 1 + 2s).

We recall that a (p,∞)-atom is anL^∞-functionathat satisfies|a| ≤χQ, whereQis a cube onRⁿ with sides parallel to the axes and

Z

Rⁿ

x^αa(x)dx= 0

for allαwith|α| ≤N. By convention, whenp=∞,ais called a (∞,∞)-atom ifQ=Rⁿ and kakL^∞ ≤1.No cancellations are required for (∞,∞)-atoms.

Before stating our main theorems, let us compare these three types of operators. One can notice that the singularity of the Coifman-Meyer type multiplier is just one point-namely, the

Function spaces and its applications to problems on Harmonic Analysis