Multilinear And Multiparameter Pseudo- Differential Operators And Trudinger-Moser Inequalities

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1 Wayne State University Wayne State University Dissertations --06 Multilinear And Multiparameter Pseudo- Differential Operators And Trudinger-Moser Inequalities Lu Zhang Wayne State University, Follow this and additional works at: Part of the Mathematics Commons Recommended Citation Zhang, Lu, "Multilinear And Multiparameter Pseudo-Differential Operators And Trudinger-Moser Inequalities" 06. Wayne State University Dissertations. Paper 6. This Open Access Embargo is brought to you for free and open access by It has been accepted for inclusion in Wayne State University Dissertations by an authorized administrator of

2 MULTIPARAMETER AD MULTILIEAR PSEUDO-DIFFERETIAL OPERATORS AD SHARP TRUDIGER-MOSER IEQUALITIES by LU ZHAG DISSERTATIO Submitted to the Graduate School of Wayne State University, Detroit, Michigan in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY 06 MAJOR: MATHEMATICS Approved By: Advisor Date

3 DEDICATIO To my grandmother and my adorable Tuanzi. ii

4 ACKOWLEDGEMETS First and foremost, I would like to thank my PhD advisor, Professor Guozhen Lu. Prof. Lu brought me into the areas of harmonic analysis and PDEs, and introduced me a variety of interesting subjects including Multilinear and Multiparameter Fourier analysis and Trudinger-Moser inequalities. During these five years, Prof. Lu keeps giving me fantastic guidance and advice, answering my endless questions, helping me overcome difficulties and offering me all kinds of generous help on my research development with ceaseless devotion. And I have benefitted quite a lot from his enthusiasm and extensive knowledge for mathematics. Without Prof. Lu this work would not have been possible. I am taking this opportunity to thank Prof. Pei-yong Wang, Prof. Pao-Liu Chow, Prof. William Cohn and Prof. Jing Hua for serving in my committee. I must thank guyen Lam, my dear friend and collaborator. He is very generous sharing his beautiful ideas on mathematics with me and gives me countless suggestions when working with him. For sure I cannot accomplish my research work on Trudinger-Moser inequalities without the helpful and enlightending discussions with him. I would like also to thank my friends, Xiaoyue Cui, Mengxia Dong, Xiaolong Han, Jungang Li, Hongwei Mei, Jiawei Shen, Yayuan Xiao and Jiuyi Zhu, who have given me generous help on both academic research and life in WSU. Moreover, I owe my thanks to the entire Department of Mathematics. During my graduate study, the professors here have contributed to my solid knowledge, especially Prof. Guozhen Lu, Prof. Pei-yong Wang, Prof. Pao-Liu Chow and Prof. Tao Mei, while the staffs have trained me and supported me on teaching and many other aspects. I would like to express iii

5 my gratitude to all of them. Also, I want to thank the SF grant DMS which has supported a lot on my research. Finally, I wish to thank my family. I am indebted to my parents and grandparents for their endless care and love. In particular, I m grateful to my girl friend Huan for coming into my life, as well as the endless love, happiness and constant support by her. They are the reasons I finished my graduate studies. iv

6 TABLE OF COTETS DEDICATIO ii ACKOWLEDGEMETS iii Chapter : ITRODUCTIO Brief Background and Introduction Trilinear Pseudo-differential Operators with Flag Symbols Bi-parameter and Bilinear Calderón-Vaillancourt Theorem Sharp Trudinger-Moser Inequalities Chapter : L p ESTIMATE FOR A TRILIEAR PSEUDO-DIFFERETIAL OPERATOR Introduction otations and Preliminaries Reduction to A Localized Version Reduction of the Localized Operator Estimates for T H,0,0 ab Estimates for T F,0,0 ab + T G,0,0 ab Estimates for T E,0,0 ab Proof of Theorem Estimates for k 0 =00 k 0 M T M,k0 f, g, hx Estimates for T f, g, hx Estimates for k 0 =00 k 0 l T l,k0 f, g, hx Chapter 3: BI-PARAMETER AD BILIEAR CALDERÓ-VAILLACOURT THEOREM WITH SUBCRITICAL ORDER Introduction v

7 3. The Boundedness on L L L The Boundedness of the Operator T σ f, g : L L L Proof of the Main Theorem Chapter 4: EQUIVALECE OF TRUDIGER-MOSER IEQUALITIES Introduction Trudinger-Moser Inequalities Adams Inequalities Our Main Results Some Lemata Trudinger-Moser Inequalities of Adachi-Tanaka Type Adams Inequalities Sharp Adams Inequalities on W, R Adams Inequalities on W γ, γ R -Proof of Theorem Chapter 5: TRUDIGER-MOSER IEQUALITIES WITH EXACT GROWTH AD THEIR EXTREMALS Introduction Some Useful Results Trudinger-Moser Inequalities with Exact Growth-Proof of Theorem Sharpness Proof of Theorem REFERECES ABSTRACT AUTOBIOGRAPHICAL STATEMET vi

8 CHAPTER ITRODUCTIO. Brief Background and Introduction Pseudo-differential operators play important roles in harmonic analysis, several complex variables, partial differential equations and other branches of modern mathematics. We s- tudied some types of multilinear and multiparameter Pseudo-differential operators. They include a class of trilinear Pseudo-differential operators, where the symbols are in the form of products of Hörmader symbols defined on lower dimensions, and we established the Hölder type L p estimates for such operators. Such operators derive from the trilinear Coifman-Meyer type operators with flag singularities. And we also studied a class of bilinear bi-parameter Pseudo-differential operators, where the symbols are taken from the general Hörmander class, and we studied the restriction for the order of the symbols which could imply the Hölder type L p estimates. Such types of operators are motivated by the Calderón-Vaillancourt theorem in single parameter setting. Trudinger-Moser inequalities can be treated as the limiting case of the Sobolev embeddings. Sharp Trudinger-Moser inequalities on the first order Sobolev spaces and their analogous Adams inequalities on high order Sobolev spaces play an important role in geometric analysis, partial differential equations and other branches of modern mathematics. Such geometric inequalities have been studied extensively by many authors in recent years and there is a vast literature. There are two types of such optimal inequalities: critical and subcritical sharp inequalities, both are with best constants. Critical sharp inequalities are under the restriction of the full Sobolev norms for the functions under consideration, while the subcritical inequalities are under the restriction of the partial Sobolev norms for the

9 functions under consideration. There are subtle differences between these two type of inequalities. Surprisingly, we proved that these critical and subcritical Trudinger-Moser and Adams inequalities are actually equivalent.. Trilinear Pseudo-differential Operators with Flag Symbols Definition.. For n we denote by MR n the set of all bounded symbols m L R n, smooth away from the origin and satisfying the classical Marcinkiewcz-Mikhlin-Hörmander condition α mξ ξ α for every ξ R n \{0} and sufficiently many multi-indices α. Definition.. We define the Fourier transform of a Schwartz function f SR n to be fξ := fxe πx ξ dx. R n Definition.3. For m > 0, 0 ρ, δ, we say that a smooth function σx, ξ on R n R n belongs to the Hörmander class S m ρ,δ if α x β ξ σx, ξ C α,β + ξ m+δ α ρ β for all multi-indices α, β and some positive constants C α,β depending on α, β. Definition.4. The classical linear Pseudo-differential operators are defined to consist of operators in the form T σ fx = σx, ξ fξ e πixξ dξ R n

10 3 initially defined for Schwartz class SR n, where σx, ξ S m ρ,δ. Definition.5. For d, m > 0, 0 ρ, δ, we say that a smooth function σx, ξ on R n R dn belongs to the multilinear Hörmander class BS m ρ,δ if α x β ξ σx, ξ C α,β + ξ m+δ α ρ β for all multi-indices α, β and some positive constants C α,β depending on α, β. Definition.6. The classical trilinear Pseudo-differential operators are initially defined for Schwartz functions f, g, h SR n as T σ f, g, hx = σx, ξ, η, ζ fξĝηĥζ eπixξ+η+ζ dξdηdζ R 3n for σx, ξ, η BS,0, 0 where x, ξ, η, ζ R n. We study the following type of trilinear Pseudo-differential operators with flag type symbols. Let ax, ξ, η, bx, η, ζ BS,0 0 be symbols satisfying the conditions l x α ξ β η ax, ξ, η + ξ + η α+β l x β η γ ζ bx, η, ζ + η + ζ β+γ for every x, ξ, η, ζ R and sufficiently many indices α, β and γ, define the operator T ab f, g, hx := ax, ξ, ηbx, η, ζ fξĝηĥζeπixξ+η+ζ dξdηdζ. R 3 We established its Hölder s type L p estimate for such operators T ab f, g, h.

11 4 Theorem.7. The operator T ab defined as. is bounded from L p L p L p 3 to L r for < p, p, p 3 with /p + /p + /p 3 = /r and 0 < r <, provided that p, p, and p, p 3,. The idea of the proof is to reduce the trilinear Pseudo-differential operator with the symbol of flag type to a localized version and takes advantage of the flag paraproducts from Muscalu s work [7] on the L p estimates for the Fourier multipliers with symbols of flag singularities. The work of such types of operators are motivated by the following trilinear Coifman- Meyer type operator with flag singularities studied by C. Muscalu [7], where the multiplier involved is a product of two symbols and has flag singularities..3 Bi-parameter and Bilinear Calderón-Vaillancourt Theorem Then we introduce the bi-parameter Pseudo-differential operators with the symbols taken from the Hörmander class BS0,0. m In the single parameter case, the following operator has been studied by Miyachi and Tomita in [70] Definition.8. Let f, g SR n and for σx, ξ, η BS0,0, m define T σ f, gx = σx, ξ, η fξ ĝη e πixξ+η dξdη R n R n where x, ξ, η R n. In bi-parameter setting, let m R and 0 ρ, δ. We first define the bi-parameter Hörmander class as Definition.9. For m > 0, 0 ρ, δ, the bi-parameter bilinear Hörmander symbols

12 5 BBS m ρ,δ consist of smooth functions on Rn R n R n that satisfy α x α x β ξ γ η β ξ γ η σx, ξ, η C α,β,γ + ξ + η m +δ α ρ β + γ + ξ + η m +δ α ρ β + γ. for all multi-indices α = α, α, β = β, β, γ = γ, γ,. We study the following type of bi-parameter bilinear Pseudo-differential operators defined for f, g SR n with σx, ξ, η BBS m ρ,δ. T σ f, g = σx, ξ, η fξ ĝη e πixξ+η dξdη R n R n where x = x, x, ξ = ξ, ξ, η = η, η R n R n and we denote the class of such operators by OpBBSρ,δ m. It is clear that the estimates for the bi-parameter and bilinear symbols σx, ξ, η are weaker than the classical single parameter bilinear symbol. It is these estimates which make the substantial difference between the bilinear Pseudo-differential operators and the bi-parameter and bilinear Pseudo-differential operators. The result is the following: Theorem.0. Let m R, p, q, r, and + =. p q r a All the operators of class OpBBS m 0,0 are bounded in L p L q L r if m < mp, q = n max{, p, q, r }

13 6 b If the operators of class OpBBS m 0,0 are bounded in L p L q L r, then we must have m mp, q = n max{, p, q, r } The index mp, q in the above theorem can be interpreted as being subcritical in the sense that if m < mp, q then any operators with symbols in the class BBS0,0 m must be bounded from L p R n L q R n to L r R n for any p, q, r satisfying p, q, r and + =, while p q r if m > mp, q then there exist operators with symbols in BBS0,0 m such that they fail to be bounded from L p R n L q R n to L r R n. The proof of the theorem mainly consists of two parts: the boundedness of L L L when m < n, and the boundedness of L L L when m < n, and then our theorem follows from the duality interpolation argument..4 Sharp Trudinger-Moser Inequalities The Trudinger-Moser and Adams inequalities are the replacements for the Sobolev embeddings in the limiting case. When Ω R is a bounded domain and kp <, it is well-known that W k,p 0 Ω L q Ω for all q p. However, by counterexamples, kp W k, k 0 Ω L Ω. In this situation, Trudinger [90] proved that W, 0 Ω L ϕ Ω where L ϕ Ω is the Orlicz space associated with the Young function ϕ t = exp α t / for some α > 0. Theorem Trudinger-967. Let Ω be a domain with finite measure in Euclidean

14 7 space R,. Then there exists a constant α > 0, such that exp α u dx c 0 Ω Ω for any u W, 0 Ω with Ω u dx. We note when the volume of Ω is infinite, there are mainly two types of inequalities: subcritical and critical inequalities. Theorem Adachi-Tanaka, 999 []. For any α < α, there exists a positive constant C,α such that u W, R, u : R φ α u dx C,α u,. where φ t = e t t j j!. j=0 The constant α is sharp in the sense that the supremum is infinity when α α. The above inequality fails at the critical case α = α. So it is natural to ask when the above can be true when α = α. This is done in [8], [6] Theorem Ruf, 005 [8]; Li-Ruf, 008 [6]. For all 0 α α : sup φ α u dx <.3 u R where u = R u + u dx /.

15 8 Moreover, this constant α is sharp in the sense that if α > α, then the supremum is infinity. For our work related to the equivalence of the above two types of inequalities, we begin with an improved sharp subcritical Trudinger-Moser inequality: Theorem.. Let, α =, 0 β < and 0 α < α. Denote π Γ + AT α, β = sup u u β R φ α β u dx x. β Then there exist positive constants c = c, β and C = C, β such that when α is close enough to α : c, β β/ α α AT α, β C, β..4 β/ α α Moreover, the constant α is sharp in the sense that AT α, β =. Then we can provide another proof to the sharp critical Trudinger-Moser inequality using Theorem 4. only. Theorem.. Let, 0 β <, 0 < a, b. Denote MT a,b β = sup φ α β u dx u a + u b R x ; β MT β = MT, β. Then MT a,b β < if and only if b. The constant α is sharp. Moreover, we have the

16 9 following identity: MT a,b β = sup α 0,α α a α α b α β b AT α, β..5 In particular, MT β < and MT β = sup α 0,α α α α α β AT α, β. ow consider the sharp subcritical and critical Adams inequalities on W, R, 3. Our first result is the following sharp subcritical Adams inequality: Theorem.3. Let 3, 0 β < and 0 α < β,. Denote AT A α, β = φ, t = sup u j :j u β t j j!. R φ, α β u dx; x β Then there exist positive constants c = c, β and C = C, β such that when α is close enough to β, : [ c, β α β, ] β AT A α, β [ C, β α β, ] β..6 Moreover, the constant β, is sharp in the sence that AT α, β =.

17 0 Theorem.4. Let 3, 0 β <, 0 < a, b. We denote: A a,b β = φ u a + u b R sup, β, β x β u dx; A, β = A β ; Then A a,b β < if and only if b. The constant β, is sharp. Moreover, we have the following identity: A a,b β = sup α 0,β, α β, α β, b a β b AT A α, β..7 In particular, A β < and A β = sup α 0,β, α β, α β, β AT A α, β. Finally, we study the following improved sharp critical Adams inequality under the assumption that a version of the sharp subcritical Adams inequality holds: Theorem.5. Let 0 < γ < be an arbitrary real positive number, p = γ, 0 α < β 0, γ = [ ω π γ Γ γ Γ γ ] p p, 0 β <, 0 < a, b. We note GAT A α, β = sup u W γ,p R : γ u p u p β p R φ,γ α β p u p dx; x β

18 GA a,b β = sup u W γ,p R : γ u a p + u b p R φ,γ β 0, γ β p u p dx x β where φ,γ t = j :j p t j j!. Assume that GAT A α, β < and there exists a constant C, γ, β > 0 such that GAT A α, β C, γ, β α β 0,γ p p.8 Then when b p, we have GA a,b β <. In particular GA p,p β <. Though we have to assume a sharp subcritical Adams inequality 4.0, the main idea of Theorem 4.5 is that since GAT A α, β is actually subcritical, i.e. α is strictly less than the critical level β 0, γ, it is easier to study than GA a,b β. Hence, it suggests a new approach in the study of GA a,b β. To achieve the best constant under the restriction of the semi-norm, we can also study the following Trudinger-Moser inequality with exact growth. Theorem.6. Let λ > 0, 0 β <, q >, 0 < α α and p > q. Denote T ME p,q,,α,β = sup u D, R L q R : u u q β q R Φ,q,β α β u p + u β dx. x β Then T ME p,q,,α,β can be attained in any of the following cases a β > 0 and all 0 < α α, b β = 0, q / and all 0 < α α,

19 c β = 0, q, p > and all 0 < α α, d β = 0, q, p, p < q and α = α.

20 3 CHAPTER L p ESTIMATE FOR A TRILIEAR PSEUDO-DIFFERETIAL OPERATOR. Introduction For n we denote by MR n the set of all bounded symbols m L R n, smooth away from the origin and satisfying the classical Marcinkiewcz-Mikhlin-Hörmander condition α mξ ξ α for every ξ R n \{0} and sufficiently many multi-indices α. Denote by T m by the n-linear operator T m f,..., f n x := mξ f ξ f n ξ n e πiξ + +ξ n x dξ, R n where ξ = ξ,..., ξ n R n and f,..., f n are Schwartz functions on R, denoted by SR. From the classical Coifman-Meyer theorem we know T m extends to a bounded n-linear operator from L p R L pn R to L r R for < p,..., p n and /p + + /p n = /r > 0. In fact this property holds for the high dimensions when f i L p i R d, i =,..., n and m MR nd, see [5, 34, 43]. The case p was proved by Coifman and Meyer [5] and was extended to p < by Grafakos and Torres [34] and Kenig and Stein [43]. Moreover, in the multiparameter setting, the same boundedness property is true, see [73 75], and also see [6] for a weaker restriction for the multiplier. For the corresponding pseudo-differential variant of the classical Coifman-Meyer theorem, let the symbol σx, ξ belong to the bilinear Hörmander symbol class BS,0, 0 that is, σ satisfies

21 4 the condition l x α ξ σx, ξ + ξ α. for any x R, ξ = ξ,..., ξ n R n and sufficiently many indices l, α. We have the following Theorem.. The operator T σ f,..., f n x := σx, ξ f ξ f n ξ n e πiξ + +ξ n x dξ R n. is bounded from L p R L pn R to L r R for < p,..., p n and /p + +/p n = /r > 0, where f,..., f n SR and σ satisfies.. For the proof of the above theorem, see [6] for bilinear, high dimensional case and [73] for one dimensional, n-linear case. Also, this boundedness property holds in the multi-parameter setting, see [6, 73]. For the trilinear Coifman-Meyer type theorem, Muscalu [7] proved the following theorem where the multiplier involved is a product of two symbols and has f lag singularities, that is, for m, m MR satisfying α ξ β η m ξ, η β η γ ζ m η, ζ ξ + η α+β η + ζ β+γ.3 for every ξ, η, ζ R and sufficiently many indices α, β and γ, we define T m,m f, f, f 3 x := m ξ, ηm η, ζ f ξ f η f 3 ζe πiξ+η+ζ x dξdηdζ, R 3.4

22 5 where f, f, f 3 SR. Then we have Theorem.. [7] The operator defined in.4 maps L p L p L p 3 L r for < p, p, p 3 < with /p + /p + /p 3 = /r and 0 < r <. In addition, T m,m also maps L L p L q L s, L p L L q L s, L L t L L t for every < p, q, t < and /p + /q = /s. Moreover, for the above theorem, the estimates like L L L t L t or L L L L are false, and these can be checked if we set f to be identically. Our main purpose is to consider a pseudo-differential operator corresponding to the above theorem, that is, let ax, ξ, η, bx, η, ζ BS,0 0 be symbols satisfying the conditions l x α ξ β η ax, ξ, η + ξ + η α+β l x β η γ ζ bx, η, ζ + η + ζ β+γ.5 for every x, ξ, η, ζ R and sufficiently many indices α, β and γ, define the operator T ab f, g, hx := ax, ξ, ηbx, η, ζ fξĝηĥζeπixξ+η+ζ dξdηdζ. R 3 It s easy to see that the symbol ax, ξ, η bx, η, ζ satisfies a less restrictive condition than the condition. for the symbol σ in Theorem.. Our main result on this is the following Theorem.3. The operator T ab defined as. is bounded from L p L p L p 3 to L r for < p, p, p 3 < with /p + /p + /p 3 = /r and 0 < r <. In addition, T ab also maps L L p L q L s, L p L L q L s, L L t L L t for every < p, q, t < and

23 6 /p + /q = /s. The proof of Theorem.3 is to reduce the trilinear pseudo-differential operator with the symbol of flag singularity to a localized version and takes advantage of the flag paraproducts from Muscalu s work [7] on the L p estimates for the Fourier multipliers with symbols of flag singularity. amely, we need to prove an equivalent localized version Theorem.9 of Theorem.3 see [73], and also [6] for the multi-parameter setting. Moreover, the key to prove the localized result is that, conditions.5 allow us to only consider the dyadic intervals with lengths at most in the flag paraproducts. More precisely, in section.3 we show that our main theorem can be reduced to an estimate for a localized operator T 0,0 ab f, g, hx = R 3 a 0 ξ, ηb 0 η, ζ fξĝηĥζeπixξ+η+ζ dξdηdζϕ 0 x, where ϕ 0 x is a Schwartz function supported near the origin and a 0, b 0 satisfy a stronger decay condition than the classical Hörmander-Mikhlin condition. In section.4, we will decompose the operator T 0,0 ab to some operators of different forms. Among these operators, some of them could be reduced to the classical pseudo-differential operator in Theorem., and the others could be written as flag paraproducts, which are used in the proof of Theorem., in the forms of T f, g, h ϕ 0 x = f, φ I B I I I I g, h, φ I φ 3 Iϕ 0 where BI g, h = g, φ J h, φ J φ 3 J, J J J, ωj 3 w I

24 7 but with dyadic intervals have lengths at most. Then by taking advantage of the flag paraproducts mentioned above, we will be able to prove the desired estimate for the localized version of our theorem in section 5.. otations and Preliminaries Let SR denote the Schwartz space of rapidly decreasing, C functions in R. Define the Fourier transform of a function f in SR as F fξ = fξ = fxe πix ξ dx R extended in the usual way to the space of tempered distribution S R, which is the dual space of SR. We use A B to represent that there exists a universal constant C > so that A CB, and use the notation A B to denote that A B and B A. We call the intervals in the form of [ k n, k n + ] in R to be dyadic intervals, where k, n Z. We denote by D the set of all such dyadic intervals. Definition.4. For I D, we define the approximate cutoff function as χ I x := + distx, I 00.6 I Definition.5. Let I R be an arbitrary interval. A smooth function ϕ is said to be a bump adapted to I if and only if one has ϕ l C l C M I l + x x I / I M

25 8 for every integer M and sufficiently many derivatives l, where x I denotes the center of I and I is the length of I. If ϕ I is a bump adapted to I, we say that I /p ϕ I is an L p -normalized bump adapted to I, for p. Definition.6. A sequence of L -normalized bumps Φ I I D adapted to dyadic intervals I D is called a non-lacunary sequence if and only if for each I D there exists an interval ω I = ω I symmetric with respect to the origin so that supp Φ I ω I and ω I I. Definition.7. A sequence of L -normalized bumps Φ I I D adapted to dyadic intervals I D is called a lacunary sequence if and only if for each I D there exists an interval ω I = ω I so that supp Φ I ω I, ω I I dist0, ω I and 0 / 5ω I. Definition.8. Let I, J D be two families of dyadic intervals with lengths at most. Suppose that φ j I I I for j =,, 3 are three families of L -normalized bump functions such that the family φ I I I is non-lacunary while the families φ j I I I for j are both lacunary, and φ j J J J for j =,, 3 are three families of L -normalized bump functions, where at least two of the three are lacunary. We define as in [7] the discrete model operators T and T,k0 for a positive integer k 0 by T f, g, h = f, φ I B I I I I g, h, φ I φ 3 I.7 where BI g, h = g, φ J h, φ J φ 3 J J.8 J J, ωj 3 w I T,k0 f, g, h = f, φ I B I I I I,k 0 g, h, φ I φ 3 I.9 where BI,k 0 g, h = g, φ J h, φ J φ 3 J J.0 J J, k 0 ωj 3 w I

26 9.3 Reduction to A Localized Version To prove the theorem, we proceed as follows. First pick a sequence of smooth functions ϕ n n Z such that supp ϕ n [n, n + ] and ϕ n =. n Z Then we can decompose the operator T ab in. as T ab = n Z T n ab where T n abf, g, hx := T ab f, g, hxϕ n x. Suppose we can prove the estimate T n abf, g, h r f χ In p g χ In p h χ In p3,. where I n is the interval [n, n + ], and χ In is defined as in.6. Then our main Theorem.3 can be proved by the following estimate T ab f, g, h r n Z T n abf, g, h r r /r n Z f χ In r p g χ In r p h χ In r p 3 /r f χ In p p /p g χ In p p /p h χ In p 3 p 3 /p 3 n Z n Z n Z f p g p h p3.

27 0 Thus, we only need to prove.. Consider that for a fixed n 0 Z, we have T n 0 ab f, g, hx = R 3 ax, ξ, η ϕ n0 xbx, η, ζ ϕ n0 xϕ n0 x fξĝηĥζeπixξ+η+ζ dξdηdζ, where ϕ n0 is a smooth function supported on the interval [n 0, n 0 + ] and equals on the support of ϕ n0. Then we rewrite the symbols ax, ξ, η ϕ n0 x and bx, η, ζ ϕ n0 x by using Fourier series with respect to the x variable ax, ξ, η ϕ n0 x = l Z a l ξ, ηe πixl bx, η, ζ ϕ n0 x = l Z b l ξ, ηe πixl, where by taking advantage of conditions.5 we can have α,β ξ,η a l ξ, η β,γ η,ζ b l η, ζ + l M + ξ + η α+β + l M + η + γ β+γ for a large number M and sufficiently many indices α, β, γ. ote the decay in l, l means we only need to consider the case for l, l = 0, which is given by T n 0,0,0 ab f, g, hx = R 3 a 0 ξ, ηb 0 η, ζ fξĝηĥζeπixξ+η+ζ dξdηdζϕ n0 x,

28 where symbols a 0, b 0 satisfy the following conditions α,β ξ,η a 0ξ, η β,γ η,ζ b 0η, ζ + ξ + η α+β.. + η + γ β+γ Using the translation invariance, we only need to prove the following localized result for n 0 = 0 Theorem.9. The operator T 0,0 ab f, g, hx = R 3 a 0 ξ, ηb 0 η, ζ fξĝηĥζeπixξ+η+ζ dξdηdζϕ 0 x.3 has the following boundedness property T 0,0 ab f, g, h r f χ I0 p g χ I0 p h χ I0 p3.4 for < p, p, p 3 < and /p +/p +/p 3 = /r, where ϕ 0 is a smooth function supported within [, ] and a 0, b 0 satisfy the conditions.. In addition, this estimate also holds for the cases where at most one p i = for i =,, 3 or p, p 3 =, < p <. ow we are ready to do some decompositions to the operator in.3..4 Reduction of the Localized Operator In this section, we will mainly show the problem can be reduced to some operators or paraproducts that we are familiar with.

29 Let ϕ SR be a Schwartz function such that supp ϕ [, ] and ϕξ = on [ /, /]. Define ψ SR be the Schwartz function satisfying ψξ := ϕξ/ ϕξ, and let ψ k = ψ / k and ψ = ϕ. ote that = k ψ k, where supp ψ [ k+, k ] [ k, k+ ] for k 0. Then for any m, n Z, we use m n to denote m n > 00 and m n to denote m n 00. Consider the decomposition = k ξ, η, ζ ψk ξ ψ k η k k ψk η ψ k ζ..5 k Without loss of generality, we consider + k k k ψk ξ ψ k k := A + B + C + D, η = ψk k k ψ k ξ ψ k η + ξ ψ k η ψk k k k >00,or k >00 ξ ψ k η + ψk k k,k,k 00 ξ ψ k η.6

30 3 where term D can be written out specifically, which contains finite number of terms: D = ϕξ ϕη + Others To estimate C, note in this case actually both k and k are at least. Suppose k > 00, we have: and then ψk ξ ψ k η = k k,k >00 C = k>00 k>00 ψ k ξ ψk η + k>00 ψ k ξ ψk η ψ k ξ ψ k η where supp ψk [ k+0, k 0 ] [ k 0, k+0 ]. Estimates for A and B are quite similar: A = k B = k k <k 00 ψk ψk k <k 00 η ψ k ξ = k 00 k 00 ψ k ξ ϕ k η.7 ξ ψ k η = ϕ k ξ ψ k η,.8 where ϕ k is a Schwartz function with supp ϕ k [ k 00, k+00 ]. For k 0 we call the families like ψ k k to be Ψ type functions, whose Fourier transform have almost disjoint supports for different scales and call the families like ϕ k k to be Φ type functions, whose Fourier transforms have overlapping supports for different scales. In the rest of work, for convenience purpose we don t distinguish between ψ k and ψ k, since they are of the same type and have comparative scales for the supports of their Fourier transforms, and we always use ψ k to represent such Ψ type functions. Similarly we always use ϕ k to represent a Φ type

31 4 function. With such notations we can write.6 as = k 00 k ψk ξ ψ k ξ ϕ k η + k 00 k ψk η ϕ k ξ ψ k η + k>00 ψ k ξ ψ k η + D..9 Later from the proof, we will see in.9 the three summations work similarly, since what we really need is at least one lacunary family in each summation. And all the functions in D play a same role as ϕξ ϕξ, which means we actually can replace.9 by an equivalently version, which is k 0 φ k ξ φ k η + ϕξ ϕξ,.0 where at least one of the families φ k ξ k and φ k ξ k is Ψ type. ow to deal with.5, it s equivalent to consider ξ, η, ζ = k ψk ξ ψ k η k k ψk η ψ k ζ k k φ k ξ φ k η + ϕξ ϕη k φ k η φ k ζ + ϕη ϕζ = k φ k ξ φ k η k φ k η φ k ζ + k φ k ξ φ k η ϕη ϕζ + k φ k η φ k ζ ϕξ ϕη + ϕξ ϕη ϕη ϕζ := E + F + G + H,. where for convenience purpose the symbol is used to show the equivalence, and we will simply treat ξ, η, ζ = E + F + G + H in the rest of the work.

32 5 Then by using the above and.3, we can decompose the localized operator as T 0,0 ab f, g, hx = R a 0 ξ, ηb 0 η, ζ fξĝηĥζeπixξ+η+ζ dξdηdζϕ 0 x 3 = a 0 ξ, ηb 0 η, ζe + F + G + H fξĝηĥζeπixξ+η+ζ dξdηdζϕ 0 x R 3 := T E,0,0 ab + T F,0,0 ab + T G,0,0 ab + T H,0,0 ab...4. Estimates for T H,0,0 ab Recall T H,0,0 ab f, g, hx = a 0 ξ, ηb 0 η, ζ ϕξ ϕη ϕη ϕζ R 3 fξĝηĥζeπixξ+η+ζ dξdηdζϕ 0 x, where note that m H ξ, η, ζ := a 0 ξ, ηb 0 η, ζ ϕξ ϕη ϕη φζ satisfies the condition α ξ β η γ ζ m Hξ, η, ζ + ξ + η + ζ α+β+γ for sufficiently many indices α, β, γ. Then our desired localized estimate follows from Theorem., since we find the operator T H,0,0 ab is just the localized operator used in the proof of Theorem., see [6, 73]..4. Estimates for T F,0,0 ab Recall + T G,0,0 ab F = k φ k ξ φ k η ϕη ϕζ, where at least one of the families φ k k and φ k k is Ψ type.

33 6 When φ k k is Ψ type, ote that to make k φ k η ϕη 0, k will have a upper bound for the summation, say k 00. Then desired estimate under this situation can be done by using the same way as in T H,0,0 ab, since only finite number of terms are involved. When φ k k is Φ type, we must have φ k k is Ψ type. Recall T F,0,0 ab f, g, hx = k R 3 a 0 ξ, η φ k ξ φ k ηb 0 η, ζ ϕη ϕζ fξĝηĥζeπixξ+η+ζ dξdηdζϕ 0 x,.3 then we can use Fourier series to write a 0 ξ, η φ k ξ φ k η = n,n Z C k n,n e πin ξ/ k e πin η/ k,.4 where the Fourier coefficients C k n,n are given by C k n,n = a k 0 ξ, η φ k ξ φ k ηe πin ξ/k e πin η/k. R By the decay condition. and the advantage that φ k k is Ψ type, we can get the following by integration by parts sufficiently many times C k n,n + n + n M. ote by the decay in n, n we only need to consider the case when n, n = 0, see [73] and the proof in section.5 for more details, and similar things can be done for b 0 η, ζ ϕη ϕζ. Then, we can use Hölder s inequality and take advantage the fact that ϕ is a bump function

34 7 adapted to [, ] to prove the localized result for.3, that is, k R 3 φ k ξ φ k η ϕη ϕζ fξĝηĥζeπixξ+η+ζ dξdηdζϕ 0 x r k R 3 φ k ξ ϕη ϕζ fξĝηĥζeπixξ+η+ζ dξdηdζϕ 0 x r = k φ k fxϕ 0 xϕ gx ϕ 0 xϕ hx ϕ 0 x r k φ k fxϕ 0 x p ϕ gx ϕ 0 x p ϕ hx ϕ 0 x p3 f χ I0 p g χ I0 p h χ I0 p3, where we take φ 0 to be on supp φ 0 and supported in a slightly larger interval containing supp φ 0. The last inequality is true since ϕ k k is Ψ type. Also, in the above we can simply write k φ k η ϕη = ϕη in the above since k is positive..4.3 Estimates for T E,0,0 ab Recall E = k 0 φ k ξ φ k η k 0 φ k η φ k ζ, where at least one of the families φ k k and φ k k is Ψ type and at least one of the families φ k k and φ k k is Ψ type. Also we consider the corresponding localized operator T E,0,0 ab f, g, hx = φ k ξ φ k ηa 0 ξ, η φ k η φ k ζb 0 η, ζ R 3 k k fξĝηĥζeπixξ+η+ζ dξdηdζϕ 0 x.

35 8 By using Fourier series as before, we only need to consider the following operator φ k ξ φ k η φ k η φ k ζ fξĝηĥζeπixξ+η+ζ dξdηdζϕ 0 x. R 3 k k As usual we consider three cases of E E = + + k k k k := I + J + K, k k φ k ξ φ k η φ k η φ k ζ and decompose T E,0,0 ab := T I,0,0 ab + T J,0,0 ab + T K,0,0 ab. ote K is actually a symbol in BS 0,0, since k is positive. That is, T K,0,0 ab f, g, hx = m K ξ, η, ζ fξĝηĥζeπixξ+η+ζ dξdηdζϕ 0 x, R 3 where m K ξ, η, ζ satisfies the condition as.. Thus, the desired localized estimate follows from the proof of Theorem., just as T H,0,0 ab. T I,0,0 ab and T J,0,0 ab are similar, we define T I ab by the following equality Tabf, I g, hx ϕ 0 x =: T I,0,0 ab f, g, hx = φ k ξ φ k η φ k η φ k ζ fξĝηĥζeπixξ+η+ζ dξdηdζϕ 0 x..5 R 3 k k From [7, 73], we know Tab I can be written by using paraproducts, which is the following

36 9 lemma. Lemma.0. Define T I ab as in.5, then we can write T f, g, hx + M l= k 0 =00 T I abf, g, hx = k 0 l T l,k0 f, g, hx + k 0 =00 k 0 M T M,k0 f, g, hx where T f, g, h = I I with BI g, h = J J ω J 3 w I T l,k0 f, g, h = I I with B l I,k 0 g, h = f, φ I B I I g, h, φ I φ 3 I g, φ J h, φ J φ 3 J J f, φ I B I I,k l 0 g, h, φ I φ 3 I J J k 0 ω 3 J w I g, φ J h, φ J φ 3 J J In the above, a T f, g, h and BI g, h are defined as.7 and.8 in definition.8. b For each l, T l f, g, h and BI l g, h are of the type.9 and.0 in definition.8. l here is actually involved in the families φ I I and φ J J, but it won t affect our proof since it does not change the types of those functions. c M is a large positive integer, and the multiplier m M,k0 ξ, η, ζ in T M,k0 satisfies the

37 30 condition α ξ β η γ ζ m M,k 0 ξ, η, ζ k 0 α+β+γ + ξ + η + ζ α+β+γ.6 for sufficiently many indices α, β, γ d All the dyadic intervals in T and T l,k0 have lengths at most for all k 0 00, l M. Proof. We follow closely the work [7], where the Fourier expansions of φ k η are used to get the desired forms of paraproducts. The only two statements we need to show are that all the dyadic intervals there have lengths at most one and the decay number in the denominator from.6. Actually both of them follow from the fact k, k 0. So far we have reduced Theorem.9 to the estimate of the operator T I,0,0 ab..5 Proof of Theorem.9 In this section by using the decomposition in Lemma.0, we are able to prove the localized estimate for T I,0,0 ab, which will complete the proof of Theorem Estimates for k 0 =00 k 0 M T M,k0 f, g, hx For this part, note that the condition.6 is almost the classical case. Then by repeating the work in [6, 73] we will see this condition can provide an estimate T M,k0 f, g, hϕ 0 x C 0k 0 f χ I0 p g χ I0 p h χ I0 p3 which is accepted since we can choose M large enough.

38 3.5. Estimates for T f, g, hx Taking advantage of that I, we can split T f, g, hx = f, φ I B I 5I 0 I I g, h, φ I φ 3 I + f, φ I B I 5I 0 I I g, h, φ I φ 3 I c = I + II..7 For Part I, we do the following decompositions first f = n fχ In, n gχ In, n 3 hχ In3, where I ni = [n i, n i + ], i =,, 3, n i Z. Then we can write T f, g, hx = n T fχ In, gχ In, hχ In3 x. n n 3 When n, n, n 3 0, the desired estimate follows from Theorem. T fχ In, gχ In, hχ In3 x ϕ 0 x r n 0 n 0 n 3 0 fχ In p gχ In p hχ In3 p3 n 0 n 0 n 3 0 f χ I0 p g χ I0 p h χ I0 p3, where the last inequality holds from χ [,] χ I0 x.

39 3 When n, n, n 3 > 0, we write = I I J J ω 3 J ω I I T fχ In, gχ In, hχ In3 x ϕ 0 x r fχ J In, φ I gχ In, φ J hχ In3, φ 3 J φ I, φ 3 J φ 3 Ixϕ 0 x r. Then we use Hölder s inequality to get fχ I J In, φ I gχ In, φ J hχ In3, φ 3 J φ I, φ 3 J φ 3 Ixϕ 0 x r I J + disti n, I M fχ In p I p p + disti n, J I J gχ In p J p p + disti n 3, J hχ In3 p3 J p 3 p 3 J I distx, I r + M distx, J + 3 dx R I J J + p p 3 + disti n, I M + disti n, J I J I I + R distx, I M + I + disti n 3, J J distx, J 3 dx fχ In p gχ In p hχ In3 p3,.8 J where M j, j are sufficiently large integers and φ j I, φj J are L -normalized bump functions adapted to I, J for j =,, 3. We first consider the case when disti, J 3. Recall we have the restriction that ωj 3 ωi, which implies that I / J. By using the subadditivity of r r we have T fχ In, gχ In, hχ In3 x ϕ 0 x r r i,j 0 I 5I 0,J 9I 0 I = i, J = j + R I + disti n, I I distx, I M + I M + disti n, J J + disti n 3, J J distx, J 3 dx fχ In p gχ In p hχ In3 p3 r J

40 33 i,j 0 I 5I 0,J 9I 0 I = i, J = j fχ In p gχ In p hχ In3 p3 r i + i n 6 M + j n 9 + j n 3 9 n 6 M n 9 n 3 9 fχ In p gχ In p hχ In3 p3 r. Observe that for large enough integers M,, we have χ In n 6 M χ I0, χ In n 9 χ I0, χ In3 n 3 9 χ I0. Thus, n >0 n >0 n 3 >0 n >0 n >0 n 3 >0 fχ In p gχ In p hχ In3 p3 r n >0 n >0 n 3 >0 f χ I0 p g χ I0 p h χ I0 p3 r T fχ In, gχ In, hχ In3 x ϕ 0 x r r n 6 M n 9 n 3 9 n 6 M n 9 n 3 9 f χ I0 p g χ I0 p h χ I0 p3 r. For the other possibility, that is, when disti, J > 3, we consider whether J is close to I n or I n3. Without loss of generality, we assume distj, I n, distj, I n3 >, and other cases will follow in the similar way. Using the notation J m = [m, m + ], m Z and.8 we can get T fχ In, gχ In, hχ In3 x ϕ 0 x r r

41 34 i,j 0 I 5I 0 I = i m >3 + disti n 3, J J J Jm, J = j distj,in,distj,in 3 > R + fχ In p gχ In p hχ In3 p3 r i,j 0 I 5I 0 I = i m >3 J Jm J = j distj,in,distj,in 3 > fχ In p gχ In p hχ In3 p3 r i,j 0 I 5I 0 I = i m >3 J Jm J = j distj,in,distj,in 3 > fχ In p gχ In p hχ In3 p3 r, I + disti n, I I distx, I M + I M + disti n, J J distx, J 3 dx J i + i n 6 M + j m n 3 m 0 i + i n 6 M + j m n 3 n 0 where 0 = min{m, 3 } is sufficiently large and we use m n. ow we take the sum over n, n, n 3 and get n >0 n >0 n 3 >0 n >0 n >0 n 3 >0 fχ In p gχ In p hχ In3 p3 r n >0 n >0 n 3 >0 f χ I0 p g χ I0 p h χ I0 p3 r T fχ In, gχ In, hχ In3 x ϕ 0 x r r n 6 M n 0 n 3 3 n 6 M 4 n 0 n3 3 4 f χ I0 p g χ I0 p h χ I0 p3 r. For other possible chooses of n, n, n 3, they will be treated in different ways. Among these cases, when n > 0, we can do similar things as the above to get our desired estimate directly, by considering whether J is close to I or not. ote in the case we are free

42 35 to take summation over J since we have a decay on i and j i. But when n 0, say n, n 0, n 3 0 things are different. In this situation, the term + distin,i I M in.8 won t give us a decay factor, which means we will have trouble when taking the summation over dyadic intervals I. Actually the decay factors from other terms are with respect to j which can t help since i > j. Recall our desired estimate in this case T fχ In, gχ In, hχ In3 x ϕ 0 x r f χ I0 p g χ I0 p h χ I0 p3..9 n, n 0 n 3 >0 Suppose that from the proof of Theorem. see [7, 73] we can get an additional decay with respect to n 3 such like / n 3 M for sufficiently positive integer M, then we only need to apply Theorem. to get n, n 0 n 3 >0 T fχ In, gχ In, hχ In3 x ϕ 0 x r n 3 M fχ I n p gχ In p hχ In3 p3 f χ I0 p g χ I0 p h χ I0 p3. ow we will see how to get such a decay / n 3 M. As before we consider two possible cases disti, J 3 and disti, J > 3. When disti, J > 3, as before consider the integral R + distx, I M + I distx, J 3 dx. J We can get a decay about m M for J J m, m Z, and see whether J m is close n 3 to

43 36 or not. As before by considering whether J is close to I n3 or not, we will get an additional decay / n 3 M. When disti, J 3, as before we have that J is near the origin J 9I 0. In this case our desired decay comes from the size and energy estimates used in the proof of Theorem., see [7, 73]. Those size and engergy terms corresponding to the function hχ n3 would be defined based on the inner product terms like hχ In3, φ J. ow since J is close to the origin, such inner product will provide a decay about / n 3 M. Or one can see the proof of Lemma.3 or section 8. in [73] to see clearly we can actually get such a decay factor for the size estimate. That means we can get an additional decay from the result of Theorem., since the boundedness there is based on the size and energy estimates. So far we have proved Part I in.7. For Part II, using the intervals I n = [n, n + ], J m = [m, m + ], m, n Z we can write T f, g, hx ϕ 0 x r r = I 5I 0 c J J ω 3 J ω I n 5 m Z I I n J Jm ω J 3 ω I f, φ I g, φ J h, φ 3 J φ I, φ 3 J φ 3 Ixϕ I J 0 x r r f, φ I g, φ J h, φ 3 J φ I, φ 3 J φ 3 Ixϕ I J 0 x r r. We will use Hölder s inequality and take advantage of the decay factors as before to write the above as n 5 m Z i,j 0 I In,J Jm I = i, J = j I f, φ I g, φ J h, φ 3 J φ I, φ 3 J φ 3 Ixϕ J 0 x r r

44 37 n 5 m Z i,j 0 I In,J Jm I = i, J = j I J f χ I n p I p p g χ Jm p J p p h χ Jm p3 J p 3 p 3 I disti, I 0 r + M distx, I 3 + M + I R I i + i n M 3 f χ In p g χ Jm p h χ Jm p3 n 5 m Z i,j 0 I In,J Jm I = i, J = j + R distx, I M + I distx, J 3 dx r J distx, J 3 dx r,.30 J where again M j, j are sufficiently large integers. Then we consider two possible cases, disti n, J m 5 and disti n, J m > 5. When disti n, J m 5, we use the same technique as before n M χin χ I0 and χ In χ Jm, for M sufficiently large. ote that the decay factor for i actually implies a decay for the summation over dyadic intervals J, since i j. Then we can estimate.30 by n 5 n 5 n M 3 f χin p g χ Jm p h χ Jm p3 r n M 3 4 f χ0 p g χ 0 p h χ 0 p3 r f χ I0 p g χ I0 p h χ I0 p3 r, which is the desired estimate.

45 38 When disti n, J m > 5, we need to take advantage of the integral in.30. That is, R + distx, I M + I distx, J 3 dx n m L, J where L = min{m, 3 } is large enough. ow.30 can be written by n 5 m n >5 i,j 0 I In,J Jm I = i, J = j f χ In p g χ Jm p h χ Jm p3 m n L r n 5 i + i n M 3 n M 3 f χ In p g χ Jn p h χ Jn p3 r f χ I0 p g χ I0 p h χ I0 p3 r, where as before the decay factor for i allows us to take the summation over dyadic intervals J, since i j. ow are are done with Part II, which means we have proved the desired estimate for T f, g, hx..5.3 Estimates for k 0 =00 k 0 l T l,k0 f, g, hx There is nothing new in this case, since it will be almost the same as what we did for T f, g, hx. ote for T l,k0 f, g, hx, the only difference is that we have I ω I k 0 J ω 3 J instead of I ω I J ω 3 J in T f, g, hx. That is, let I = i, J = j, we will have i k 0 = j 0, k Recall we only need i j in the proof for T f, g, hx, and the method obviously works for T l,k0 f, g, hx in the setting i k 0 = j 0, k 0 00, which will give us a bound uniformly with respect to k 0. Then we will be able to take the summation over k 0 by using l. In this way we can get the

46 39 estimate for k 0 =00 k 0 l T l,k0 f, g, hx. So far we have proved the desired localized estimate for the operator T E,0,0 ab f, g, hx in., which means Theorem.9 has been proved. Then from this localized result, we can conclude that Theorem.3 is true.

47 40 CHAPTER 3 BI-PARAMETER AD BILIEAR CALDERÓ-VAILLACOURT THEO- REM WITH SUBCRITICAL ORDER 3. Introduction Pseudo-differential operators play important roles in harmonic analysis, several complex variables, partial differential equations and other branches of modern mathematics, see e.g. [3], [79], [44], [85], [83], [87], [89], etc. We first recall that the Hörmander class Sρ,δ m Rn of linear pseudo-differential operators are defined to consist of operators in the form T σ fx = σx, ξ fξ e πixξ dξ 3. R n where x, ξ, η R n and σ satisfies α x β ξ σx, ξ C α,β + ξ m+δ α ρ β for all multi-indices α, β and some positive constants C α,β depending on α, β. The function f is taken initially from the Schwartz class SR n. Hörmander [37, 38] proved the operators with symbols in S 0 ρ,δ are L bounded when 0 δ < ρ. In a celebrated paper, Calderón and Vaillancourt [0] established the L boundedness when 0 δ = ρ <. C. Fefferman [9] further extended to the L p boundedness < p < for operators with symbols in S m ρ,δ with 0 δ < ρ and m n p ρ. The result of C. Fefferman is sharp in the sense that for m < n ρ, then the Lp p boundedness fails. Paivarinta and E. Somersalo later considered the critical case of δ = ρ

48 4 in [78] by establishing h p to h p boundedness for all 0 < p <, where h p is the local Hardy space of Goldberg [35]. The result of [78] strengthens the H to L boundedness of Coifman and Meyer [4] when m = n. We also refer to the more extensive treatment of pseudo- differential operators and their applications in PDEs to [4], [3], [79], [44], [46], [83], [87], [89], etc. The bilinear analogue of such pseudo-differential operators are defined to be the class BS m ρ,δ Rn consisting of operators of the following form: Let f, g SR n and define T σ f, gx = σx, ξ, η fξ ĝη e πixξ+η dξdη R n R n 3. where x, ξ, η R n and σ satisfies α x β ξ γ η σx, ξ, η C α,β,γ + ξ + η m+δ α ρ β + γ 3.3 for all multi-indices α, β, γ and some positive constants C α,β,γ depending on α, β, γ. The first work of bilinear singular integrals and pseudo-differential operators is due to Coifman and Meyer [4, 5] which originated from specific problems about Calderón s commutators. Subsequently, the symbolic calculus for bilinear pseudo-differential operators was studied, e.g., in the works [6, 68] motivated by the bilinear Calderón-Zygmund theory developed [7, 34, 43], etc. and references therein. In particular, critical order for boundedness of bilinear pseudo-differential operators with symbols BS0,0 m has been considered in [6, 70]. The L p estimates of multi-parameter and multi-linear Coifman-Meyer type Fourier multipliers were established in [74]. Recently, Chen and the first author [] gave a different proof

49 4 of the L p estimates of [74] and also establish the L p estimates under the limited smoothness of the Fourier symbol; Dai and the first author [6] proved the same L p estimates of [74] for multi-parameter and multi-linear pseudo-differential operators. More recently, Hong and the first author [36] carried out a theory of symbolic calculus for multi-parameter and multi-linear pseudo-differential operators. Let m R and 0 ρ, δ. In this article we will study the following type of biparameter and bilinear pseudo-differential operators defined for f, g SR n : T σ f, g = σx, ξ, η fξ ĝη e πixξ+η dξdη R n R n where x = x, x, ξ = ξ, ξ, η = η, η R n R n and σ satisfies α x α x β ξ γ η β ξ γ η σx, ξ, η C α,β,γ + ξ + η m +δ α ρ β + γ + ξ + η m +δ α ρ β + γ 3.4 for all multi-indices α = α, α, β = β, β, γ = γ, γ, and some positive constants C α,β,γ depending on α, β, γ. We denote the class of such symbols by BBSρ,δ m. We also denote by OpBBSm ρ,δ the class of all operators T σ with σ BBS m ρ,δ. It is clear that the estimates in 3.4 that the bi-parameter and bilinear symbol σx, ξ, η satisfies are weaker than those in 3.3 satisfied by the bilinear symbol. It is these estimates which make the substantial difference between the bilinear pseudo-differential operators and the bi-parameter and bilinear pseudo-differential operators.

50 43 Given the above bi-parameter and bilinear operator T = T σ, we can define its adjoints T and T as follows: T f, g, h = T h, g, f = T f, h, g for all f, g SR n The main result on this is the following: Theorem 3. Main Theorem. Let m R, p, q, r, and p + q = r. a All the operators of class OpBBS m 0,0 are bounded in L p L q L r if m < mp, q = n max{, p, q, r } b If the operators of class OpBBS m 0,0 are bounded in L p L q L r, then we must have m mp, q = n max{, p, q, r } The index mp, q in the above theorem can be interpreted as being subcritical in the sense that if m < mp, q then any operators with symbols in the class BBS0,0 m must be bounded from L p R n L q R n to L r R n for any p, q, r satisfying p, q, r and + =, p q r while if m > mp, q then there exist operators with symbols in BBS0,0 m such that they fail to be bounded from L p R n L q R n to L r R n when p, q, r and p + q = r. We should mention in the bilinear one-parameter case, Bényi, Bernicot, Maldonado, aibo and Torres [6] established the boundedness for m < mp, q and Miyachi and Tomita [70] proved the boundedness at the critical case when m = mp, q.

51 44 The proof of the Main Theorem mainly consists of two parts: the boundedness of L L L when m < n, and the boundedness of L L L when m < n, and then our theorem follows from the interpolation argument. 3. The Boundedness on L L L In this section, we will prove the boundedness of the bi-parameter and bilinear operator T σ on L L L. Actually we can prove the following more general case: Theorem 3.. When m < n ρ, for σ BBS m ρ,δ where 0 δ, ρ, δ <, we then have that T σ : L L L. To prove this theorem, we need the following lemma in the bi-parameter setting see also [36]. A one-parameter version can be found in [6]. Lemma 3.3. Let m R, 0 δ, ρ, σ BBSρ,δ m. a If 0 < R, R and supp σ {x, ξ, η : ξ i + η i R i, i =, } then T σ f, g L R R n f L g L, f, g L. b If R, R and supp σ {x, ξ, η : R i ξ i + η i 4R i, i =, } then T σ f, g L R R ρn+ m f L g L, f, g L. c If 0 < R, R and supp σ {x, ξ, η : ξ + η R, R ξ + η 4R }, then T σ f, g L R n R ρn+ m f L g L, f, g L.

52 45 Proof. Consider T σ f, g = Kx, x y, x zfygzdydz, R n where and F 4n Kx, y, z = σx, ξ, ηe πiξ y e πiη z dξdη = F4n σx,, y, z R n denotes the inverse Fourier transform with respect to ξ, η R R. Then it suffices to show that a sup x R Kx, y, z dydz R R n R n, b sup x R Kx, y, z dydz R R n R ρn+ m, c sup x R Kx, y, z dydz R R n n R ρn+ m. for the corresponding three parts in the lemma. For part a, note σ is a smooth function with compact support. For an 0, we have + y, z Kx, y, z = σx, ξ, η ξ η e πiξ y e πiη z dξdη R n ξ η σx, ξ, ηe iξ y e iη z dξdη, R n which implies Kx, y, z R R n + y, z, and parta is true if we choose > n.

53 46 For part b consider Kx, y, z dydz = Kx, y, z dydz + R n y + z R ρ y + z R ρ + Kx, y, z dydz + y + z R ρ y + z R ρ Kx, y, z dydz y + z R ρ y + z R ρ Kx, y, z dydz. y + z R ρ y + z R ρ By using Cauchy-Schwarz inequality, Plancherel s formula and R, R, we have Kx, y, z dydz y + z R ρ y + z R ρ R R ρn Kx, y, z dydz y + z R ρ y + z R ρ R R ρn σx, ξ, η dξdη ξ + η R ξ + η R R R ρn + ξ + η m + ξ + η m dξdη ξ + η R ξ + η R R R ρn R R m+n = R R ρn+m, Kx, y, z dydz y + z R ρ y + z R ρ y, z y, z Kx, y, z dydz y + z R ρ y + z R ρ dydz y + z R ρ y, z 4 y, z 4 y + z R ρ R R ρ4 n ξ η ξ η σx, ξ, η dξdη ξ + η R ξ + η R R R ρ4 n + ξ + η m ρ4 + ξ + η m ρ4 dξdη ξ + η R ξ + η R R R ρ4 n R R m ρ4+n = R R ρn+m,

54 47 and Kx, y, z dydz y, z Kx, y, z dydz y + z R ρ y + z R ρ y + z R ρ y + z R ρ dydz y + z R ρ y, z 4 y + z R ρ R ρn ρ4 n R ξ η σx, ξ, η dξdη ξ + η R ξ + η R R ρn ρ4 n R + ξ + η m + ξ + η m ρ4 dξdη ξ + η R ξ + η R R ρn R ρ4 n R m+n R m ρ4+n = R R ρn+m. Thus, we are done with part b. For part c we consider Kx, y, z dydz = R n Kx, y, z dydz + y + z R ρ Kx, y, z dydz. y + z R ρ Then: Kx, y, z dydz + y, z Kx, y, z dydz y + z R ρ y + z R ρ ρn R ρn R dydz y + z R + y ρ, z ξ η σx, ξ, η dξdη ξ + η R ξ + η R + ξ + η m + ξ + η m dξdη R ρn R m+n R n = R n R ρn+m.