CORRELATION STRUCTURES, MANY-BODY SCATTERING PROCESSES AND THE DERIVATION OF THE GROSS-PITAEVSKII HIERARCHY

Transcription

1 CORRELATIO STRUCTURES, MAY-BODY SCATTERIG PROCESSES AD THE DERIVATIO OF THE GROSS-PITAEVSKII HIERARCHY XUWE CHE AD JUSTI HOLMER A. We consider the dynamics of bosons in three dimensions. We assume the pair interaction is given by β V β ). By studying an associated many-body wave operator, we introduce a BBGKY hierarchy which takes into account all of the interparticle singular correlation structures developed by the many-body evolution from the beginning. Assuming energy conditions on the -body wave function, for β, ], we derive the Gross-Pitaevskii hierarchy with -body interaction. In particular, we establish that, in the limit, all k-body scattering processes vanishes if k and thus provide a direct answer to a question raised by Erdös, Schlein, and Yau in []. Moreover, this new BBGKY hierarchy shares the limit points with the ordinary BBGKY hierarchy strongly for β, ) and weakly for β =. Since this new BBGKY hierarchy converts the problem from a two-body estimate to a weaker three-body estimate for which we have the estimates to achieve β <, it then allows us to prove that all limit points of the ordinary BBGKY hierarchy satisfy the space-time bound conjectured by Klainerman and Machedon in [47] for β, ). C. Introduction.. Organization of the Paper 9.. Acknowledgements. The BBGKY Hierarchy with Singular Correlation Structure. Proof of Theorem. 5.. Estimate for the Potential Term 6.. Estimate for the k-body Interaction Part 9 4. Proof of Theorem. 4.. Estimate for the k-body Interaction Part Estimate for the Potential Part 5. Collapsing and Strichartz Estimates Appendix A. The Topology on the Density Matrices 45 References 46 Mathematics Subject Classification. Primary 5Q55, 5A, 8V7; Secondary 5A, 5B45. Key words and phrases. BBGKY Hierarchy, -particle Schrödinger Equation, Klainerman-Machedon Space-time Bound, Quantum Kac s Program.

2 XUWE CHE AD JUSTI HOLMER. I A Bose-Einstein condensate BEC), is a peculiar gaseous state in which particles of integer spin bosons) occupy a macroscopic quantum state. Though the existence of a BEC was first predicted theoretically by Einstein for non-interacting particles in 95, it was not verified experimentally until the obel prize winning first observation of Bose-Einstein condensate BEC) for interacting atoms in low temperature in 995 [4, 6] using laser cooling techniques. Since then, this new state of matter has attracted a lot of attention in physics and mathematics as it can be used to explore fundamental questions in quantum mechanics, such as the emergence of interference, decoherence, superfluidity and quantized vortices. Investigating various condensates has become one of the most active areas of contemporary research. As in the study of any time-dependent interacting -body system, the main diffi culty in the theory of BEC is that the governing PDE is impossible to solve or simulate when is large. For BEC, the time-evolution of a boson system without trapping in R is governed by the many-body Schrödinger equation.) i t ψ = H ψ where the -body Hamiltonian is given by.) H = xj + j= i<j β V β x i x j )) with β, ]. Here, x,..., x ) R is the position vector of particles in R, we choose ψ ) L R ) =, and we assume the interparticle interaction is given by β V β ). On the one hand,.) V ) = β V β ) is an approximation of the Dirac δ-function as and hence matches the Gross-Pitaevskii description that the many-body effect should be modeled by an on-site strong self interaction. On the other hand, if we denote by scatw ) the D scattering length of the potential W, then we have scat V )) which is the Gross-Pitaevskii scaling condition introduced by Lieb, Seiringer and Yngvason in [5]. In the current experiments, we have 4 which already makes equation.) unrealistic to solve. In fact, according to the references in [5], the largest system one could simulate at the moment has. Hence, it is necessary to find reductions or approximations. It is widely believed that the mean-field approximation / limit of equation.) is given by the cubic nonlinear Schrödinger equation LS).4) i t φ = φ + c φ φ, From here on out, we consider the β > case solely. For β = Hartree dynamics), see [4, 9, 48, 5, 5, 9, 4, 7,,, 8].

3 THE KLAIERMA-MACHEDO COJECTURE FOR β, ) where the coupling constant c is exactly given by 8π scat V )). That is, if we define the k-particle marginal densities associated with ψ by.5) γ k) t, x k; x k) = ψ t, x k, x k )ψ t, x k, x k )dx k, x k, x k R k, and assume γ k), x k, x k) k φ x j ) φ x j) as where x k = x,..., x j ) R k, then we have the propagation of chaos, namely,.6) γ k) t, x k, x k) j= k φt, x j ) φt, x j) as j= and φt, x j ) is given by.4) subject to the initial φ, x j ) = φ x j ). aturally, to prove.6), one studies the limit of the Bogoliubov Born Green Kirkwood Yvon { } BBGKY) hierarchy of the many-body system.) satisfied by : [ ].7) i t γ k) + xk, γ k) = i<jk + k [ γ k) V x i x j ), γ k) k [ ] Tr k+ V x j x k+ ), γ k+) if we do not distinguish γ k) as a kernel and the operator it defines. Here the operator V x) represents multiplication by the function V x) and Tr k+ means taking the k + trace, for example, Tr k+ V x j x k+ ) γ k+) = j= V x j x k+ ) γ k+) t, x k, x k+ ; x k, x k+ )dx k+. Such an approach for deriving mean-field type equations by studying the limit of the BBGKY hierarchy was proposed by Kac in the classical setting and demonstrated by Landford s work on the Boltzmann equation. In the current quantum setting, it was suggested by Spohn [54] and has been proven to be successful by Erdös, Schlein, and Yau in their fundamental papers [,,, ] which have inspired many works by many authors [47, 45,, 8,, 9, 7,,, 8,, 56, ]. This paper, like the aforementioned work, is inspired by the work of Erdös, Schlein, and Yau. The first main part of this paper deals with a problem raised on [, p.56]. To motivate and state the problem, we first notice the formal limit of hierarchy.7):.8) i t γ k) + [ xk, γ k)] k [ = b Tr k+ δxj x k+ ), γ k+)] where j= b = V x)dx. R ]

4 4 XUWE CHE AD JUSTI HOLMER We make such an observation because V ) R V x)dx ) δ ). If we plug.9) γ k) t, x k, x k) = k φt, x j ) φt, x j) j= into.8) and assume φ solves.4), then.9) is a solution to.8) if and only if the coupling constant c in.4) equals to b. Since 8π lim scat V )) = b for β, ), the formal limit.8) checks the prediction. It also has been proven in [] for β, /). However, this formal limit does not meet the prediction when β = because 8π scat V )) = 8π scatv ) 8πa for β = which is usually a number smaller than b. In [,, ], Erdös, Schlein and Yau have established rigorously that the real limit of the BBGKY hierarchy.7) associated with.) matches the prediction and is given by.) i t γ k) + [ xk, γ k)] k [ = 8πa Tr k+ δxj x k+ ), γ k+)]. j= The reasoning given is that one has to take into account the correlation between the particles. To be specific, as in [5,,, ], let w be the solution to We scale w by so that w is the solution to.) + β V )w x) = V, lim w x) =. x w x) = β w β x ) + ) w x)) =, lim x) x =. The papers [,, ] then suggest that, instead of considering the limit of hierarchy.7) directly, one should investigate the limit of the following hierarchy.) i t γ k) + x k γ k) x γk) k = ) Ṽ x i x j )γ k),i,j Ṽx i x j)γ k),i,j i<jk + k k j= Tr k+ Ṽ x j x k+ )γ k+),j,k+ Ṽx j x k+)γ k+),j,k+) ), which has the singular correlations between particles built in. Here Ṽ ) = V ) w )),

5 and As, one formally has and THE KLAIERMA-MACHEDO COJECTURE FOR β, ) 5 γ k),i,j = γ k) w x i x j )). γ k) γk),i,j, Ṽ ) 8πa δ ), hence one obtains.) as the limit of the many-body dynamic.). One immediate question to this delicate limiting process is: aside from physical motivation, is there a more mathematical explanation for why.8) is not the limit of.) when β =? An answer is that the "usual" energy condition:.) sup t where Tr S k+) γ k+) + Tr S S S k) γ k) S j = xj ) and S k) = ) C k for k, k S j S j, first proved in [8, ] for β, 5) and later in [45,, 8, 9,, ], is not true when β =. This can be proved by contradiction: assume that.) does hold when β =, then with a simple argument in [45] which is first hinted in [] and used in [45,, 8, 9,, ], one easily proves that hierarchy.7) converges to the wrong limit.8) and reaches a contradiction. Another immediate but much deeper question is that, if the singular correlation structure between particles is so crucial, then why would one only take a pair into account at a time? For example, when considering the term V x x )γ k) why would one only put in the singular correlation structure between particles x and x and why not put in the singular correlation structure between particles x and x or x and x? That is, why not consider a term like j= [Ṽ x x ) w x x ))] [ γ k),, w x x )) The above expression corresponds to a three-body interaction. Basically, the question is: why can this case be dropped? This is actually a problem raised on [, p.56]. Problem [, p.56]). One should rigorously establish the fact that all three-body scattering processes are negligible in the limit. In the first main part of this paper, we provide a direct answer to Problem. We take into account all of the interparticle singular correlation structures developed by the many-body ]?

6 6 XUWE CHE AD JUSTI HOLMER evolution from the beginning. We rigorously establish that, in the limit, all k-body scattering processes vanishes if k. To be specific, we have the following theorem. Theorem. Main Theorem I). Define.4) α k) t, x k, x k) = def where G k) x k) ) ) x k) k) γ t, x k, x k), G k).5) G k) x k) def = w x i x j )). i<j k Suppose β, ]. Assume the energy bound :.6) sup t Tr S ) α ) + Tr S S S ) α ) ) C. Moreover, denote L k the space of Hilbert-Schmidt operators on L R k ). Then every limit point Γ = { γ k)} { } } {Γ of k= t) = α k) in k C [, T ], L k ) with respect to the product k= topology τ prod defined in Appendix A), if there is any, satisfies the cubic Gross-Pitaevskii hierarchy: k [.7) i t γ k) = xj, γ k)] k [ + c Tr k+ δxj x k+ ), γ k+)], j= where the coupling constant c is given by { V x)dx if β, ), R.8) c = 8πa if β =. j= An important feature of α k) is that, considered as bounded operators, αk) and γk) share the same limit for β, ), if there is any. 4 We will prove this simple fact in Lemma.,. Hence, Theorem. and its proof give us a better understanding of the limiting process and allow us to solve an open problem, raised by Klainerman and Machedon in 8, for β, ) in the second main part of this paper. After reading Theorem., an alert reader can easily tell that one needs to prove a uniqueness theorem of solutions to hierarchy.7) before concluding that equation.4) is the mean-field limit to the -body dynamic.). In the second main part of this paper, we solve an open problem about an a-priori bound on the limit points which leads to uniqueness of.7), conjectured by Klainerman and Machedon [47] in 8 for β, ). Though this conjecture was not stated explicitly in [47], as we will explain after stating Theorem., this Klainerman-Machedon a-priori In the Fock space version of the problem, there is another way to insert all of the correlation structures using the metaplectic representation / Bogoliubov transform. See [7]. We remind the readers that the "usual" energy condition.) is not true when β =. The energy conditions.6) and.9) we impose on Theorems. and. have been proven for k =, or with spatial cut-offs for general k in [, ]. 4 The same thing is weakly true for β = but we omit the proof at the moment since Theorem. applies only to β <.

7 THE KLAIERMA-MACHEDO COJECTURE FOR β, ) 7 bound is necessary to implement Klainerman-Machedon s powerful and flexible approach in the most involved part of proving the cubic nonlinear Schrödinger equation LS) as the limit of quantum -body dynamics. Kirkpatrick-Schlein-Staffi lani [45] completely solved the T version of the conjecture with a trace theorem and were the first to successfully implement such an approach. However, the R version of the conjecture as stated inside Theorem., was fully open until recently. T. Chen and Pavlović [] have been able to prove the conjecture for β, /4). In [9], X.C simplified and extended the result to the range of β, /7]. X.C. and J.H. [] then extended the β, /7] result by X.C. to β, /). In the second main part of this paper, we prove it for β, ). In particular, away from the β = case, the conjecture is now resolved. To be specific, we prove the following theorem. Theorem. Main Theorem II). Define and R k) = k xj x j, j= B j,k+ γ k+) = Tr k+ [ δxj x k+ ), γ k+)], Suppose β, ). Assume the energy bound:.9) sup t Tr S k+) α k+) + Tr S S S k) α k) then every limit point Γ = { γ k)} { {Γ of k= t) = { { } } hence of γ k) k= = conjectured by Klainerman-Machedon [47] in 8:.) α k) ) } k= C k+ for k, } obtained in Theorem. and because they have the same limit), satisfies the space-time bound T R k) B j,k+ γ k+) t,, ) L x,x dt C k. In particular, there is only one limit point due to the Klainerman-Machedon uniqueness theorem [47, Theorem.]. In 7, Erdös, Schlein, and Yau obtained the first uniqueness theorem of solutions [, Theorem 9.] to hierarchy.7). The proof is surprisingly delicate it spans 6 pages and uses complicated Feynman diagram techniques. The main diffi culty is that hierarchy.7) is a system of infinitely coupled equations. Briefly, [, Theorem 9.] is the following: Theorem. Erdös-Schlein-Yau uniqueness [, Theorem 9.]). There is at most one nonnegative symmetric operator sequence { γ k)} that solves hierarchy.7) subject to the k= energy condition.) sup Tr S k) γ k) C k. t [,T ]

8 8 XUWE CHE AD JUSTI HOLMER In [47], based on their null form paper [46], Klainerman and Machedon gave a different uniqueness theorem of hierarchy.7) in a space different from that used in [, Theorem 9.]. The proof is shorter pages) than the proof of [, Theorem 9.]. Briefly, [47, Theorem.] is the following: Theorem.4 Klainerman-Machedon uniqueness [47, Theorem.]). There is at most one symmetric operator sequence { γ k)} that solves hierarchy.7) subject to the space-time k= bound.). When propagation of chaos.6) happens, condition.) is actually.) sup x φ L C, t [,T ] while condition.) means.) T x φ φ ) L dt C. When φ satisfies LS.4), both are known. Due to the Strichartz estimate [4],.) implies.), that is, condition.) seems to be a bit weaker than condition.). The proof of [47, Theorem.] pages) is also considerably shorter than the proof of [, Theorem 9.] 6 pages). It is then natural to wonder whether [47, Theorem.] provides a simple proof of uniqueness. To answer such a question it is necessary to know whether the limit points in Theorem. satisfy condition.). Away from curiosity, there are realistic reasons to study the Klainerman-Machedon bound.). In the LS literature, uniqueness subject to condition.) is called unconditional uniqueness while uniqueness subject to condition.) is called conditional uniqueness. While the conditional uniqueness theorems usually come for free with the uniqueness conditions verified naturally in LS theory because they are parts of the existence argument, the unconditional uniqueness theorems usually do not yield any information of existence. Recently, using a version of the quantum de Finetti theorem from [49], T. Chen, Hainzl, Pavlović, and Seiringer [5] provided an alternative pages proof to [, Theorem 9.] and confirmed that it is an unconditional uniqueness result in the sense of LS theory. 5 Therefore, the general existence theory of the Gross-Pitaevskii hierarchy.7) subject to general initial datum has to require that the limits of the BBGKY hierarchy.7) lie in the space in which the space-time bound.) holds. See [,,, 4]. Moreover, while [, Theorem 9.] is a powerful theorem, it is very diffi cult to adapt such an argument to various other interesting and colorful settings: a different spatial dimension, a three-body interaction instead of a pair interaction, or the Hermite operator instead of the Laplacian. The last situation mentioned is physically important. On the one hand, all the known experiments of BEC use harmonic trapping to stabilize the condensate [4, 6, 9, 44, 55]. On the other hand, different trapping strength produces quantum behaviors which do not exist in the Boltzmann limit of classical particles nor in the quantum case when the trapping is missing and have been experimentally observed [5, 57, 5, 4, 7]. The Klainerman-Machedon 5 See also [56, 4, 4].

9 THE KLAIERMA-MACHEDO COJECTURE FOR β, ) 9 approach applies easily in these meaningful situations [45,, 8, 9,, 6,, ] ). Thus proving the Klainerman-Machedon bound.) actually helps to advance the study of quantum many-body dynamic and the mean-field approximation in the sense that it provides a flexible and powerful tool in D... { Organization } of the Paper. We will first compute the { BBGKY } hierarchy satisfied by α k), defined in.4), in. Due to the definition of α k), the BBGKY hierarchy { } of α k), written as.), takes into account all of the singular correlation structures developed by the many-body evolution from the beginning. The differences between hierarchy.) and hierarchy.7) are obvious: hierarchy.) for α k) has k body interactions where k =,..., k, but most importantly, for the purpose of Theorems. and., hierarchy.) does not have -body interactions not under an integral sign. We will call the key new terms the potential terms, which consist of three-body interactions, and the k-body interaction terms, which consist of k-body interaction for { all k}. With the BBGKY hierarchy satisfied by computed in, we prove Theorem. α k) in as a "warm up" first and then establish Theorem. in 4. The gut of the proof of Theorems. and. is the careful application of the D and 6D retarded endpoint Strichartz estimates [4] and the Littlewood-Paley theory. One of the effects of considering the singular interparticle correlation structures developed by the many-body evolution is to replace the potential.4) V x i x j ) = β V β x i x j )) with the new potential.5) xl G,i,l ) xl G,j,l ), i j, i l, j l among other terms)..5) could be considered as a three-body interaction, since it is only nontrivial if all three x i, x j, and x l are within β. One might wonder why a three-body interaction is better then a two-body interaction because a three-body interaction is more complicated. For the purposes of estimates, the original potential.4) has the behavior.6) V x i x j ) β β x i x j ) For the new potential, we have effectively.7) xl G,i,l ) xl G,j,l ) 4β β x i x l ) β x j x l ) ote that if β = and i = j, then.7) and.6) are effectively the same, and there is no gain in going from.4) to.5). However, i = j in.5) and hence.5), a three-body interaction, actually offers more localization than.4), a two-body interaction. It is then natural to use the 6D endpoint Strichartz estimate when one wants to estimate a term like tk [ U k) t k t k+ ) Here U k) t k ) = e it k xj e it k x j. xl G,i,l ) xl G,j,l )α k) t k+) ] dt k+ L T L x,x.

10 XUWE CHE AD JUSTI HOLMER Using the Littlewood-Paley theory or frequency localization effectively gains one derivative in the analysis. That is, we avoid a β in the estimates. Heuristically speaking, it sort of averages the best and the worst estimates. Here, the "best" means no derivatives hits V and the "worst" means that two derivatives hit V. For example, say one would like to look at.8) PMP M x x V x x ) L. Here, PM i is the Littlewood-Paley projection onto frequencies M, acting on functions of x i R. There are two ways to look at.8), namely and M V x x ) L β P MP M V ) x x ) L. Then depending on the sizes of β and M, one is better than the other. As we will see in the proof of Theorem. in 4, such a consideration will effectively avoid a β in the estimates... Acknowledgements. J.H. was supported in part by SF grant DMS T BBGKY H S C S Recall.5) G k) x,, x ) = i<j k where w is defined via.). We decompose G k) G k) = i<j k and define the multiplication operator Y k) w x i x j )), as follows: G,i,j, G,i,j = w x i x j ).) Y k) ) ψ )x,, x ) def = G k) x,, x k )ψ x,, x ). An immediate property of Y k) by is the following. Lemma.. Let α k) be defined as in.4). For β, ), f L R k), lim α k) f) γk) f) L C f lim Y k) op =. Here α k) f) and γk) operator norm. f) means the operators αk) and γk) act on f, and op means the Proof. We have α k) f γk) f L = Y k) γk) Y k) Y k) γk) Y k) Y k) op γ k) f γk) f L f γk) Y k) f L + γ k) Y k) f L + op op Y k) γ k) f γk) f L Y k) op f L. op

11 THE KLAIERMA-MACHEDO COJECTURE FOR β, ) otice that the Hilbert-Schmidt norm of γ k) is uniformly bounded by because we assume ψ ) L R ) =. Moreover, since β <, we have lim Y k) op =. In fact, consider fx, x ) ω x x ) fx, x ) where ω x x ) ω x x ) ω x x ) dx = ω x x ) = β ω β x x )) β ω β x x )) C β as fx, x ) dx because ω L R ) and β <. So we conclude that lim α k) f) γk) f) L C f lim { To compute the BBGKY hierarchy of α k) Y k) op =. }, we first give the following lemma. Lemma.. We have.) H k) where H k), A k) and E k) Y k) is the ordinary Laplacian ) = Y k) H k) ) H k), + Ak) k, = xj, j= is the zeroth order operator of multiplication by is the first order operator i,j,l k i,j,l distinct j,l k j l xl G,l,i xl G,l,j G,l,i G,l,j, xl G,j,l G,j,l xl. + Ek) ), Before proceeding to the proof, let us note that the terms A k) and Ek) should be thought of as error terms. Indeed, A k) involves only three-body interaction it is only nontrivial if x i, x j, and x l are within β of each other.

12 XUWE CHE AD JUSTI HOLMER Proof. We start with.) xl ) log G k) Using that we can rewrite.) as xl G k) G k).4) x l G k) G k) = i<j k = x l G k) G k) xl G,i,j G,i,j = xl ) log G k) + x l G k) G k) ) = j k j l j k j l xl G,l,j G,l,j, xl G,l,j G,l,j. On the other hand, we have log G k) = i<j k log G,i,j, and hence.) also reads xl ) log G k) = i<j k = j k j l x l G,i,j G,i,j + xl G,l,j G,l,j + j k j l Plugging this into.4) and expanding the square in.4), xl G,i,j G,i,j xl G,l,j G,l,j ). x l G k) G k) = j k j l xl G,l,j G,l,j i<j k i l, j l xl G,l,i xl G,l,j G,l,i G,l,j We infer from.) that G = V G, so x l G k) G k) = j k j l V,l,j i<j k i l, j l xl G,l,i xl G,l,j G,l,i G,l,j ow summing in l, l k, we obtain H k) Gk) = Gk) i<j k l k i l, j l xl G,l,i xl G,l,j G,l,i G,l,j,

13 THE KLAIERMA-MACHEDO COJECTURE FOR β, ) Here H k) Gk) is considered as Hk) applied to the function Gk). ote that the sum on the right side is perhaps more intuitively written as H k) Gk) = xl G,l,i xl G,l,j Gk) G,l,i G,l,j i,j,l k i,j,l distinct which implies.). { With the above Lemma, we compute the BBGKY hierarchy of the left of the operator equation.), we obtain.5) Y k) Thus Y k) Hk) k) Y ) = H k), + Ak) + Ek) ) α k) }. Applying Y k) could be regarded as an approximation to the wave operator relating Hk) is an exact wave operator relating Hk) although a more precise statement is that Y k) approximation of H k),, namely the operator Hk) not self-adjoint, the wave operator Y k) We now work out the BBGKY hierarchy of, + Ak) is not { unitary. } α k) + Ek). Since Hk), + Ak). We will need to compute Y k) [Hk) to to Hk),, to an is + Ek), γk) k) ]Y. To this end, we use the operator property: given two operators Y, Y, let α = Y γy, then In the above, taking Y = Y k).6) Y k) [Hk), γk) k) ]Y Y [H, γ]y = Y HY )α αy HY ). and Y = Y k) ), and applying.5) give = Hk), + Ak) Moreover, let us introduce the operator W k) + Ek) )α αhk), + Ak) + Ek) ) ) which acts on any kernel Kx k, x k ) by W k) Kx k, x k) = [Y k) Y k ) K]x k, x k) = G x k )G x k )Kx k, x k). With the above notation, the BBGKY hierarchy of equations for the operators { or the corresponding kernels α k) x k, x k ) = transpose) is given by.7) where.8) i t α k) = = W k) H k), ) Hk, + k k l= ) W k) } γk) x k,x k ) G x k )G using that Y k) x k ) { α k) ) ) α k) + A k) ) Ak α k) + E k) ) Ek α k) B,l,k+W k+) B,l,k+W k+) ) α k+) )x k ; x k) k V x l x k+ ) V x l x k+ )) R j= ) α k+) } = Y k) γk) Y k) is equal to its G,j,k+ G,j,k+α k+)...)dx k+

14 4 XUWE CHE AD JUSTI HOLMER where ) is x,..., x k, x k+ ; x,, x k, x k+). We will decompose the terms in.8) to properly set up the Duhamel-Born series. Let L,l,k+ + def = G,l,k+ L,l,k+ + def = G,l,k+ k G,j,k+ G,j,k+ = G,l,k+ j= k G,j,k+ G,j,k+ = G,l,k+ j= Here L stands for localization. Also let so that Then.9) = Ṽ x) = V x) w x)) j k j l j k j l Ṽ x l x k+ ) = V x l x k+ )G,l,k+ Ṽ x l x k+ ) = V x l x k+ )G,l,k+ W k) R B,l,k+W k+) ) α k+) )x k ; x k) Ṽ x l x k+ ) L,l,k+ + ) α k+)...)dx k+ R Ṽ x l x k+ ) L,l,k+ + ) α k+)...)dx k+ Separate "the k-body part" and "the -body part": k.) Bk+),many αk+) = B,many,l,k+ α k+), where.) B,many,l,k+ α k+) and.) so that B k+) k k l=, α k+) k l= B,l,k+ α k+), k k l= l= def = k Ṽx l x k+ )L,l,k+ R G,j,k+ G,j,k+, G,j,k+ G,j,k+. Ṽx l x k+ )L,l,k+)α k+) ) dx k+ R Ṽx l x k+ ) Ṽx l x k+ ))α k+) ) dx k+ W k) B,l,k+W k+) ) = k+) k+) B,many + B

15 THE KLAIERMA-MACHEDO COJECTURE FOR β, ) 5 The operator B,many will give rise to the k-body interaction part and the interaction part in the Duhamel-Born series below. Finally, introduce the operator Ṽ k) αk) = Ak) Ak) )αk) + Ek) Ek) )αk) which will give rise to the potential part in the Duhamel-Born series below. From.7),.) α k) t k) = U k) t k )α k), i tk tk i i tk U k) t k t k+ ) U k) t k t k+ ) F P k, + P P k, + KIP k, + IP k,, B k) U k) t k t k+ )Ṽ k) αk) t k+) dt k+ B k+),many αk+) t k+ ) dt k+ k+) B α k+) t k+ ) dt k+ will give rise to Here, U k) t k ) = e it k xj it k e x j, F P k, stands for the free part of α k) with coupling level, P P k, stands for the potential part of α k) with coupling level, KIP k, stands for the k-body interaction part of α k) with coupling level, and IP k, stands for the -body interaction part of α k) with coupling level. We will use this notation for the rest of the paper. Remark. In the case β =, B,l,k+ is where 8πa shows up. In fact as shown in []. G,l,k+ V,l,k+ 8πa δx l x k+ ). P T. We prove Theorem. as a warm up to the proof of Theorem.. Here "warm up" means that we do not need to iterate.) many times to get a good enough decay in time for the interaction part and do not need to use the Littlewood-Paley theory or the X,b spaces. To prove Theorem., we prove that hierarchy.) converges to hierarchy.7) which written in the integral form is.) γ k) t k ) = U k) t k )γ k) ic tk U k) t k t k+ ) Tr k+ [ δxj x k+ ), γ k+) t k+ ) ] dt k+. It has been proven in [, 8,,,,, 45,, ] that, provided that the energy bound.6) holds, the st term and the last term on the right handside of.) do converge to the right hand side of.) weak*-ly in L T L. In particular, it is proved that, as trace class operators tk U k) t k t k+ ) B,j,k+ α k+) t k+ ) dt k+ lim ) tk Ṽ x)dx U k) t k t k+ )B j,k+ γ k+) t k+ ) dt k+ weak*

16 6 XUWE CHE AD JUSTI HOLMER where lim Ṽ x)dx is exactly the c defined in.8). So we only need to prove the following two estimates: P P k,.) L as T L KIP k,.) L as. T L where tk P P k, t k ) = i KIP k, t k ) = i tk U k) t k t k+ )Ṽ k) αk) t k+) dt k+, U k) t k t k+ ) B k+),many αk+) t k+ ) dt k+. Before delving into the proof, we remark that condition.9) implies that ) S sup k+) α k+) + S S k) α k) C k+ t L x k,x k In fact, consider the second term for k = : S S S S α ) = L x,x Cauchy-Schwarz in dx, ) ψ x, x ) S S G ) x ) ) = Tr S S S ) α ) Tr S ) α ) C by condition.9) with k =. L x k,x k ) SS ψ x, x ) ψ G ) x S S x, x ) dx ) G k) dx x ) dx ).. Estimate for the Potential Term. Recall where and Ṽ k) αk) = Ak) A k) αk) = Ak) )αk) + Ek) i,j,l k i,j,l distinct E k) αk) = j,l k j l S S ψ x, x ) G k) x ) Ek) )αk), xl G,l,i xl G,l,j α k) G,l,i G,,l,j xl G,j,l G,j,l xl α k). dx

17 THE KLAIERMA-MACHEDO COJECTURE FOR β, ) 7 Let us define.4) A k),i,j,l αk) = x l G,l,i xl G,l,j G,l,i G,l,j α k),.5) E k),j,l αk) = x l G,j,l xl α k) G,,j,l then to prove estimate.), it suffi ces to prove the following estimates tk L U k) t k t k+ )A k),i,j,l αk) t k+) dt k+ C T +, T L x,x tk L U k) t k t k+ )E k),j,l αk) t k+) dt k+ C T +. T L x,x In fact, assume the above estimates for the moment, we have P P k, L T L = P P k, L T L x,x C T k + + C T k + as, for β, ], where we used the facts that L T L = L T L and there are k summands x,x in A k) while there are k summands in E k). So we finish the estimate for the potential part in the proof of Theorem. with the following two lemmas. Lemma.. We have the estimate: tk L U k) t k t k+ )A k),i,j,l αk) t k+ ) dt k+ T L x,x C T + xi xj xl α k) t k+ ) L t L x,x. In particular, if one assumes the energy bound.6), it reads tk L U k) t k t k+ )A k),i,j,l αk) t k+) dt k+ C T +. T L x,x Lemma.. tk U k) t k t k+ )E k),j,l αk) t k+ ) dt k+ L T L x,x C T + xj xl α k) t k+ ) L t L x,x In particular, if one assumes the energy bound.6), it reads tk L U k) t k t k+ )E k),j,l αk) t k+) dt k+ C T +. T L x,x Proof of Lemma.. Define v, x) = β ω ) β x ) G x),

18 8 XUWE CHE AD JUSTI HOLMER then So v, x) = β ω ) β x ) tk tk G x) β β x ṽ, x). U k) t k t k+ )A k),i,j,l αk) t k+ ) dt k+ L T L x,x U k) t k t k+ ) [ ṽ, x l x i )ṽ, x l x j )α k) t k+ ) ] dt k+ L T L x,x Insert a smooth cut-off θt) with θt) = for t [ T, T ] and θt) = for t [ T, T ] c into the above, tk θt k) U k) t k t k+ ) [ ṽ, x l x i )ṽ, x l x j )θt k+ )α k) t k+ ) ] L dt k+ t L x,x Since L t L C x,x k) X, we have + C θt k) By Lemma 5., tk U k) t k t k+ ) [ ṽ, x l x i )ṽ, x l x j )θt k+ )α k) t k+ ) ] X dt k+. k) + C ṽ, x l x i )ṽ, x l x j )θt k+ )α k) t k+ ) X k) +. Use the first inequality of 5.8) in Corollary 5., C ṽ, L + θtk+ ) xi xj xl α k) t k+ ) L t L x,x C T ṽ, L + xi xj xl α k) t k+ ) L t L x,x where That is tk ṽ, L + = C +. U k) t k t k+ )A k),i,j,l αk) t k+ ) dt k+ L T L x,x C T + xi xj xl α k) t k+ ) L t L x,x as claimed. Proof of Lemma.. As in the proof of Lemma., we replace v, x) = β ω ) β x ) β β x = ṽ, x) G x)

19 THE KLAIERMA-MACHEDO COJECTURE FOR β, ) 9 with ṽ, x) = β β x. Then, we have tk L U k) t k t k+ )E k),j,l αk) t k+ ) dt k+ T L x,x tk U k) t k t k+ ) [ ṽ, x) xl α k) t k+ ) )] L dt k+ T L x,x ṽ, x) θt k+ ) xl α k) t k+ ) ) k) X Use the second inequality of 5.6) in Corollary 5.6, C ṽ, L + θtk+ ) xj xl α k) t k+ ) L t L x,x + C T + xj xl α k) t k+ ) L t L x,x. So we have finished the proof of Lemma.... Estimate for the k-body Interaction Part. Recall B k+),many αk+) = k l= B,many,l,k+ α k+). To prove estimate.), we prove the estimate: tk.6) U k) t k t k+ ) B L +,many,l,k+ αk+) t k+ ) dt k+ T L x,x C T C k+ +, where B +,many,l,k+ is half of B,many,l,k+. Assume estimate.6), then KIP k, L T L = KIP k, L T L x,x kc T C k+ + as. The rest of this section is the proof of estimate.6). We first give the following lemma. Lemma.. One can decompose B +,many,l,k+ αk+) t k+ ), defined in.), as the sum of at most 8 k terms of the form B +,many,l,k+,σ αk+) = k R t k+ ) Ṽ x l x k+ ) β w β x σ x k+ ))A σ α k+) x k, x k+ ; x k, x k+ ) dx k+. Here, x σ is some x j or x j but not x l and A σ is a product of [ β w β x j x k+ )) ], [ β w β x j x k+ )) ] or with x j not equal to x l or x σ. Proof. Recall, B +,many,l,k+ αk+) = k Ṽ x l x k+ )L,l,k+ R α k+) x k, x k+ ; x k, x k+ ) dx k+

20 XUWE CHE AD JUSTI HOLMER otice that, L,l,k+ + = G,l,k+ jk j l G,j,k+ G,j,k+ G,j,k+ = β w β x j x k+ )) Thus, ) taken as a binomial expansion, L,l,k+ is a sum of k classes where each class has k, j =,..., k, terms inside, that is: j = L,l,k jk j l β w β x j x k+ )) + + [ β w β x l x k+ )) ] jk j l jk β w β x j x k+ )) [ β w β x j x k+ )) ] [ β w β x j x k+ )) ] ). Thus L,l,k+ can be written as a sum of at most 8 k terms which individually looks like β w β x σ x k+ ))A σ where x σ is some x j or x j but not x l and A σ is a product of [ β w β x j x k+ )) ], [ β w β x j x k+ )) ] or with x j not equal to x l or x σ. Inserting this into.), we have the claimed decomposition. With Lemma., we have the following estimate. Lemma.4. Let B +,many,l,k+,σ be defined as in Lemma., we have tk U k) t k t k+ ) B L +,many,l,k+,σ αk+) t k+ ) dt k+ T L x,x C T C k+ + xk+ k+ α k+) t k+ ). L t L x,x Consequently, tk U k) t k t k+ ) B L +,many,l,k+ αk+) t k+ ) dt k+ T L x,x C T C k+ + xk+ k+ α k+) t k+ ) L t L x,x because, by Lemma., B +,many,l,k+ αk+) = σ B +,loc,l,k+,σ,

21 THE KLAIERMA-MACHEDO COJECTURE FOR β, ) where the sum has at most 8 k terms inside. In particular, if one assumes the energy bound.6), it reads tk C T C k+ +, which is exactly estimate.6). Proof. Recall U k) t k t k+ ) B +,many,l,k+ αk+) t k+ ) dt k+ L T L x,x B +,many,l,k+,σ αk+) t k+ ) = k Ṽ x l x k+ ) β w β x σ x k+ ))A σ α k+) x k, x k+ ; x k, x k+ )dx k+. R There is no need to write out the variables in A σ. In fact, A σ is a harmless factor because β w β x j x k+ )) is in L uniformly in if β. As in the proof of Lemmas. and., we insert a smooth cut-off θt), tk θt k) U k) t k t k+ ) B L +,many,l,k+,σ αk+) t k+ ) dt k+ T L x,x tk U k) t k t k+ ) B +,many,l,k+,σ θtk+ )α k+) t k+ ) ) dt k+, L t L x,x and proceed to C B +,many,l,k+,σ θtk+ )α k+) t k+ ) ) X k) + = C Ṽ x l x k+ ) β w β x σ x k+ )) R A σ θt k+ )α k+) x k, x k+ ; x k, x k+ )dx k+. k) X + The third inequality of 5.) of Lemma 5. gives C Ṽ β w β )) L + L + A σ L xk+ k+ θt k+ )α k+) x k, x k+ ; x k, x k+ ) L t L x,x where Ṽ L + β w β )) L + A σ L C k+ +.

22 XUWE CHE AD JUSTI HOLMER Thus tk U k) t k t k+ ) B L +,many,l,k+,σ αk+) t k+ ) dt k+ T L x,x C T C k+ + xk+ k+ α k+) t k+ ). L t L x,x which is good enough to conclude the proof of Lemma P T. We will use Littlewood-Paley theory to prove Theorem.. Let P M i be the projection onto frequencies M and PM i the analogous projections onto frequencies M, acting on functions of x i R the ith coordinate). We take M to be a dyadic frequency range l. Similarly, we define P M i i and PM, which act on the variable x i. Let k 4.) P k) M = i= P i MP i M. As observed in earlier work [, 9, ], to establish Theorem., it suffi ces to prove the following theorem. 6 Theorem 4.. Under the assumptions of Theorem., there exists a C independent of k, M, ) such that for each M there exists depending on M) such that for, there holds 4.) P k) M Rk) B,j,k+ γ k+) t) L T L x,x C k. In fact, passing to the weak limit γ k) γk) as, we obtain P k) M Rk) B j,k+ γ k+) L T L x,x C k Since it holds uniformly in M, we can send M and, by the monotone convergence theorem, we obtain R k) B j,k+ γ k+) L T L C k x,x which is exactly the Klainerman-Machedon space-time bound.). This completes the proof Theorem., assuming Theorem 4.. The rest of this section is devoted to proving Theorem 4.. We will first establish estimate 4.) for a suffi ciently small T which depends on the controlling constant in condition.9) and is independent of k, and M, then a bootstrap argument together with condition.9) give estimate 4.) for every finite time at the price of a larger constant C. The first step of the proof of Theorem 4. is to iterate.) p times and get to the formula α k) t k) = F P k,p t k ) + P P k,p t k ) + KIP k,p t k ) + IP k,p t k ), 6 To be precise, this formulation with frequency localization is from []. The formulations in [, 9] do not have the Littlewood-Paley projector inside.

23 THE KLAIERMA-MACHEDO COJECTURE FOR β, ) then estimate each term, that is, prove the following estimates: 4.) 4.4) 4.5) 4.6) P k ) M Rk ) B,,k F P k,p L C k, T L x,x P k ) M Rk ) B,,k P P k,p L C k, T L x,x P k ) M Rk ) B,,k KIP k,p L C k, T L x,x P k ) M Rk ) B,,k IP k,p L C k. T L x,x for all k and for some C and a suffi ciently small T determined by the controlling constant in condition.9) and independent of k, and M. Here, we iterate.) because it is diffi cult to show 4.6) unless p = ln, a fact first observed by Chen and Pavlovic [], who proved.) for β, /4), and then used in the β, /7] work [9] by X.C and in the β, /) work [] by X.C and J.H. As proven in [9, ], once p is set to be ln, one can prove estimates 4.) and 4.6) for all β, ). The obstacle in achieving higher β lies solely in proving 4.4) and 4.5). Hence, in the rest of this section, we prove estimates 4.4) and 4.5) only and refer the readers to [9, ] for the proof of estimates 4.) and 4.6). To make formulas shorter, for q, we introduce the following notation: J k,q) t k,q)f k+q) ) = U k) t k t k+ ) ) k+) B U k+q ) t k+q t k+q ) ) k+q) B f k+q), where t k,q means t k+,..., t k+q ). When q =, the above product is degenerate and we let J k,) t k,)f k) ) = f k). ow plug the k + ) version of.) into the last term only of.) to obtain where the free part is with α k) t k) = F P k, t k ) + P P k, t k ) + KIP k, t k ) + IP k, t k ) F P k, t k ) = U k) t k )α k), + i) tk = i) q q= J k,q) t k+q t k 4.7) f k,q) F P U k) t k t k+ ) t k,q)f k,q) F P k+) B U k+) t k+ )α k+), dt k+ )dt k,q = U k+q) t k+q )α k+q),

24 4 XUWE CHE AD JUSTI HOLMER the potential part is with = i = P P k, t k ) tk + i) tk U k) t k t k+ )Ṽ k) αk) t k+) dt k+ i) q+ q= 4.8) f k,q) P P = tk+ U k) t k t k+ ) J k,q) t k+q t k tk+q the k-body interaction part is with = i = KIP k, t k ) tk + i) tk U k) t k t k+ ) i) q+ q= tk+ 4.9) f k,q) KIP = tk+q and the interaction part is IP k, t k ) = i) tk = i) + tk B k+) t k,q)f k,q) U k+) t k+ t k+ )Ṽ k+) α k+) t k+ ) dt k, P P )dt k,q U k+q) t k+q t k+q+ )Ṽ k+q) B k+),many αk+) t k+ ) dt k+ U k) t k t k+ ) J k,q) t k+q t k tk+ tk+ α k+q) t k+q+ ) dt k+q+ k+) B U k+) k+) t k+ t k+ ) B,many αk+) t k+ ) dt k, t k,q)f k,q) KIP )dt k,q U k+q) t k+q t k+q+ ) U k) t k t k+ ) k+q+) B,many αk+q+) k+) B U k+) t k+ t k+ ) J k,) t k,)α k+) t k+ ))dt k, ow we iterate this process p ) more times to obtain t k+q+ ) dt k+q+ α k) t k) = F P k,p t k ) + P P k,p t k ) + KIP k,p t k ) + IP k,p t k ) k+) B α k+) t k+ ) dt k, where the free part is 4.) F P k,p t k ) = p i) q q= J k,q) t k+q t k t k,q)f k,q) F P )dt k,q.

25 The potential part is 4.) P P k,p t k ) = THE KLAIERMA-MACHEDO COJECTURE FOR β, ) 5 p i) q+ q= The k-body interaction part is p 4.) KIP k,p t k ) = i) q+ q= The interaction part is 4.) IP k,p t k ) = i) p+ J k,q) t k+q t k J k,q) t k+q t k J k+p+) t k+p t k t k,q)f k,q) P P )dt k,q. t k,q)f k,q) KIP )dt k,q. t k,p+ )α k+p+) t k+p+ ))dt k,p+. We then apply the Klainerman-Machedon board game to the free part, potential part, k-body interaction part, and interaction part. Lemma 4. Klainerman-Machedon board game). [47]One can express J k,q) t k,q)f k+q) )dt k,q, t k+q t k as a sum of at most 4 q terms of the form J k,q) t k,q, µ m )f k+q) )dt k,q, or in other words, J k,q) t k+q t k D t k,q)f k+q) )dt k,q = m D J k,q) t k,q, µ m )f k+q) )dt k,q. Here D [, t k ] q, µ m are a set of maps from {k +,..., k + q} to {k,..., k + q } satisfying µ m k + ) = k and µ m l) < l for all l, and J k,q) t k,q, µ m )f k+q) ) = U k) t k t k+ ) B,k,k+ U k+) t k+ t k+ ) B,µm k+),k+ U k+q ) t k+q t k+q ) B,µm k+q),k+qf k+q) ). 4.. Estimate for the k-body Interaction Part. To make formulas shorter, let us write since P k) M k R k) M k = P k) M k R k), and R k) are usually bundled together Step I. Applying Lemma 4. to 4.), we get R k ) M k B,,k KIP k,p L T L x,x p M k B,,k J k,q) t k,q, µ m )f k,q) q= m Rk ) D KIP )dt k,q L T L x,x

26 6 XUWE CHE AD JUSTI HOLMER where f k,q) KIP is given by 4.9) and the sum m has at most 4q terms inside. By Minkowski s integral inequality, = Rk ) T M k B,,k dt k [,T ] q+ D R k ) D R k ) Cauchy-Schwarz in the t k integration, T By Lemma 5., C ε T [,T ] q M k M k J k,q) t k,q, µ m )f k,q) KIP )dt k,q L T L x,x M k B,,k J k,q) t k,q, µ m )f k,q) KIP )dt k,q M k B,,k U k) t k t k+ ) B,k,k+... L dt k L x,x x,x dt k dt k,q. ) R k ) M k B,,k U k) t k t k+ ) B,k,k+... dt k L x,x Mk M k ) ε [,T ] q Iterate the previous steps q ) times, C ε T ) q R k) M k+q M k M k R k+q ) M k+q B,µm k+q),k+q = C ε T ) q M k+q M k M k R k+q ) B,µm M k+q k+q),k+q M B,k,k+ k U k+) t k+ t k+ ) [ M k M k Mk+q M k M k+ M k+q ) ] f k,q) L KIP T L x,x [ ) ε M k M k+q f k,q) KIP ) L T L x,x ] ) ε dt k,q L x,x dt k,q where the sum is over all M k,..., M k+q dyadic such that M k+q M k M k. Hence R k ) M k B,,k KIP k,p L T L x,x p C ε T ) q q= M k+q M k M k R k+q ) B,µm M k+q k+q),k+q [ M k M k+q f k,q) KIP ) L T L x,x ) ε ]

27 THE KLAIERMA-MACHEDO COJECTURE FOR β, ) 7 We then insert a smooth cut-off θt) with θt) = for t [ T, T ] and θt) = for t [ T, T ] c into the above estimate to get R k ) M k B,,k KIP k,p L T L x,x with 4.4) f k,q) KIP = tk+q p C ε T ) q q= M k+q M k M k R k+q ) B,µm M k+q k+q),k+q U k+q) t k+q t k+q+ )θt k+q+ ) [ M k M k+q ) ε ) k,q) L θt k+q ) f KIP T L x,x B k+q+),many αk+q+) t k+q+ ) dt k+q+, where the sum is over all M k,..., M k+q dyadic such that M k+q M k M k Step II. With Lemma 5., the X b space version of Lemma 5., we turn Step I into R k ) M k B,,k KIP k,p L T L x,x ] p C ε T ) q+ q= R k+q) M k+q M k+q M k+q M k M k θt k+q ) f k,q) KIP ) X k+q) + ]. [ M k M k+q ) ε Use Lemma 5. gives us p C ε T ) q+ q= R k+q) M k+q M k+q M k+q M k M k θt k+q+ ) [ M k M k+q ) ε ) k+q+) X B,many αk+q+) t k+q+ ) Carry out the sum in M k M k+q with the help of Lemma 4.: p [ ) ε M k+q ) C ε T ) q+ M log k M k + q) q M q= k+q q! M k+q M k ) ] R k+q) k+q+) X M k+q θt k+q+ ) B,many αk+q+) t k+q+ ). Take a T j/4 from the front to apply Lemma 4.4: { p C ε T ) C ε T 4 ) q q= R k+q) M k+q θt k+q+ ) M k+q M k [ ) M ε k M ε k+q k+q) + k+q) + ]. ) ] } k+q+) X B,many αk+q+) t k+q+ ). k+q) +

28 8 XUWE CHE AD JUSTI HOLMER where the sum is over dyadic M k+q such that M k+q M k. Lemma 4. [, Lemma.]). M k M k M k+j M k+j log M k+j M k + j) j, j! where the sum is in M k M k+j over dyads, such that M k M k M k+j M k+j. Lemma 4.4 [, Lemma.]). For each α > possibly large) and each ɛ > arbitrarily small), there exists t > independent of M) suffi ciently small such that j, M, we have t j α log M + j) j j! M ɛ 4... Step III. Recall the ending result of Step II, R k ) M k B,,k KIP k,p L T L x,x { p C ε T ) C ε T 4 ) q q= R k+q) M k+q θt k+q+ ) M k+q M k [ ) M ε k M ε k+q ) k+q+) X B,many αk+q+) t k+q+ ) k+q) + ] }. Use Corollary 4.5, { p C ε T ) C ε T 4 ) q q= M k+q M k [ ) M ε k M ε k+q C k+q+ + min β, M k+q ) ] } θtk+q+ )S S k+q+) α k+q+), L tk+q+ L xl x because there are k + q) terms inside k+q+) B,many. Rearranging terms { p C k C ε T ) CT 4 ) q θtk+q+ )S S k+q+) α k+q+) q= M ε k + minm +ε k+q M k+q M k β, M ε k+q) }, L tk+q+ L xl x

29 THE KLAIERMA-MACHEDO COJECTURE FOR β, ) 9 We carry out the sum in M k+q by dividing into M k+q β for which minm +ε k+q β, Mk+q ε ) = Mj ɛ ) and M k+q β for which minm +ε k+q β, Mk+q ε ) = M +ε k+q β ). This yields 4.5) minm +ε k+q β, Mk+q) ε...) +...) M k+q M k β M k+q M k M k+q M k,m k+q β Mk+q ɛ + β M k+q M k ɛ. M +ε k+q M k+q β Remark. The above is exactly what we meant by writing "gains one derivative via Littlewood-Paley" in.. So we have reached R k ) M k B,,k KIP k,p L T L x,x { p C k C ε T ) CT 4 ) q θtk+q+ )S S k+q+) α k+q+) q= M ε k +ε }. Via Condition.9) the energy estimate), it becomes L tk+q+ L x L x β p C k C ε T ) CT 4 ) q C k+q+ q= C k C ε T ) CT 4 ) q C q+ q= M ε k +ε M ε k +ε. We can then choose a T independent of M k, k, p and such that the infinite series converges. We then have R k ) M k B,,k KIP k,p L T L x,x C k M ε k +ε for some C larger than C. Therefore, on the one hand, there is a C independent of M k, k, p, and s.t. given a M k, there is M k ) which makes R k ) M k B,,k KIP k,p L C k, for all, T L x,x on the other hand, R k ) M k B,,k KIP k,p L T L x,x as which matches Theorem. as well. Whence we have finished the proof of estimate 4.5).

30 XUWE CHE AD JUSTI HOLMER Corollary ) R k+q) M Bk+q+) k+q,many αk+q+) k+q) X + ) { M C k+q S S k+q+) α k+q+) k+q + L t L xx Proof. Recall.), which gives the expansion k+q 4.7) Bk+q),many = l= B,many,l,k+q+ β + where B,many,l,k+q+ is defined by.) and itself decomposed in Lemma. into a sum of at most 8 k+q terms of the form 4.8) β k+q) = k Ṽ x l x k+q+ ) β w β x σ x k+q+ )) R Here, x σ {x,..., x k+q+, x,..., x k+q+ }\{x l} and A σ = Z j Z j A σ α k+q+) x k+q, x k+q+ ; x k+q, x k+q+ ) dx k+q+. j k+q j l, j σ where Z j is either or β w β x j x k+q+ )), and likewise Z j is either or β w β x j x k+q+ )). Since there are k + q) terms in 4.7) and 8 k+q terms of the type β k+q) in 4.8), we multiply by a factor C k+q. For each individual term β k+q), the derivatives xj for j k + q, j l, j σ can either land on Z j Z j or α k+q+), giving k+q terms. Each possibility is accommodated by a suitable variant of Proposition 5.. Of course, we actually need to modify 5.5) so that it has a k + q + )-component density as opposed to 4) and multiple factors of the type f x x 4 ) in 5.5), but these modifications are straightforward and amount to bookkeeping. The remaining coordinates act as passive variables and are placed in L on the inside of the estimates, and otherwise do not play any role. 4.. Estimate for the Potential Part. Repeating Steps I and II in the treatment of the k-body interaction part, we have R k ) M k B,,k P P k,p L T L x,x { p [ ) C ε T ) C ε T 4 ) q M ε k M ε k+q Recall q= R k+q) M k+q Ṽ k) αk) = Ak) M k+q M k θt k+q+ )Ṽ k+q) Ak) )αk) + Ek) ) α k+q) X t k+q+ ) k+q) + Ek) )αk). ] }.

31 THE KLAIERMA-MACHEDO COJECTURE FOR β, ) From here on out, we will call A k) the two-body error term. E k) )αk) Ak) )αk) the three-body potential term and Ek) By Step III in the estimate of the k-body interaction term, it suffi ces to prove the following two corollaries. Corollary 4.6. Recall as defined in.4). Then R k+q) Corollary 4.7. Recall A k+q),i,j,l αk+q) = x l G,l,i xl G,l,j G,l,i G,l,j α k+q), M k+q A,i,j,l α k+q) k+q) X + as defined in.5), we have ) R k+q) M k+q E k) X,j,l αk) t k+ ) 4.9) { β S k+q) α k+q) L t L xx Mk+q + E k+q),j,l α k+q) = x l G,j,l xl α k+q) G,j,l k+q) + ) M S S k+q) α L t L + S k) k+q β 4 α L x k+q x t L x k+q k+q x k+q where, for convenience, we have assumed that β >. and Then one merely needs to estimate the following two sums: + min ) M +ε k+q β, M +ε k+q, M k+q M k β 4 + min M +ε k+q β, M ε M k+q M k k+q). β 4 In fact, separate the above sums at M k+q β and M k+q β, then use the same method as in estimate 4.5), we get to + min ) M +ε k+q β, M +ε k+q + β+ε M k+q M k β 4 + min ) M +ε k+q β, M ε k+q β 4 + ε M k+q M k which is enough to conclude the estimates of the potential part for β, ). Remark. We remark that the estimate for the three-body interaction term is the only place in this paper which requires β <.

32 XUWE CHE AD JUSTI HOLMER Proof of Corollary 4.6. Since xµ and µ move directly onto α k+q), for µ {,..., k + q}\{i, j, l}, it suffi ces to use the obvious extension of Proposition 5. where {,, } is replaced by {l, i, j}, α ) is replaced by α k+q), and X ) where ote that and A ijl = β U x l x i )U x l x j ) Ux) = Ux) = w )x) β w x) w w x)) + β w x) is replaced by Xk+q). ote that + + Ux) x, Ux) x, Ux) x 4 uniformly in. Hence U, U, and U all belong to L p for p > Proof of Corollary 4.7. ote that xl G,j,l G,j,l = β U x j x l ) ) uniformly in ). where w x) Ux) = β w x) We then appeal to the straightforward generalization of Proposition 5.7 to k + q)-level density, noting that Ux) x, Ux) x, and Ux) x 4, uniformly in, so C U < and independent of. Define the norm 7 α k) X k) b 5. C S E = τ + ξ k ξ k b ˆα k) τ, ξ k, ξ k) dτ dξ k dξ k We will use the case b = + of the following lemma. X k) b ) / Lemma 5. [, Lemma 4.]). Let < b < and θt) be a smooth cutoff. Then t 5.) θt) U k) t s)β k) s) ds β k) k) X b Lemma 5. [, Lemma 4.4]). For each ε >, there is a C ε independent of M k, j, k, and such that R k) M B,j,k+ k U k+) t)f k+) ) ε Mk R k+) L t L C ε Ṽ x,x L M M k+ f k+) L k+ x,x M k+ M k where the sum on the right is in M k+, over dyads such that M k+ M k. 7 To be precise, this X b should be written as X,b in the usual notation for the X s,b spaces. Since we are not using the s in X s,b, we write it as X b.

33 THE KLAIERMA-MACHEDO COJECTURE FOR β, ) Lemma 5. [, Lemma 4.5]). For each ε >, there is a C ε independent of M k, j, k, and such that ) ε R k) M B,j,k+ k α k+) Mk R k+) L t L C ε x,x M M k+ α k+) X. k) k+ + M k+ M k where the sum on the right is in M k+, over dyads such that M k+ M k. The D endpoint Strichartz estimate directly yields the following multiparticle estimate: β k) X k) + β k) L t L x L c where c stands for the remaining spatial coordinates x,..., x k, x,..., x k ). However, when β k) = V x x )γ k), this estimate does not allow us to put V in L since the L x norm comes before the L x norm. In order to effectively put the L x norm after the L x norm, we need to translate coordinates before applying the Strichartz estimate. This maneuver was introduced in our earlier paper [, Lemma 4.6]. We restate the relevant estimate in the following lemma. Since we will need to deal with Fourier transforms in only selected coordinates, we introduce the following notation: F denotes the Fourier transform in t, F j denotes the Fourier transform in x j, and F j denotes Fourier transform in x j. Fourier transforms in multiple coordinates will be denoted as combined subscripts for example, F = F F denotes the Fourier transform in t and x. Lemma 5.4 D endpoint Strichartz in transformed coordinates). Let Then 5.) β k) X k) + T fx, x ) = fx + x, x ) T fx, x ) = fx, x + x ) F T β k) )t, x, ξ ) L t L ξ L x L c F T β k) )t, ξ, x ) L t L ξ L x L c where in each case c stands for complementary coordinates, specifically coordinates x,..., x k, x,..., x k ). Lemma 5.5 Hölder and Sobolev). If 5.) β k) t, x, x ) = V x x )γ k) t, x, x ) then 5.4) F T β k) )t, x, ξ ) L t L ξ L x L c V 6 L 5 + x γ k) L t L xx V L + x γ k) L t L xx V L + γ k) L t L xx

34 4 XUWE CHE AD JUSTI HOLMER 5.5) F T β k) )t, ξ, x ) L t L ξ L x L c Proof. Consider 5.4). By 5.), and hence, applying F, we obtain Applying Hölder, V 6 L 5 + x γ k) L t L xx V L + x γ k) L t L xx V L + γ k) L t L xx T β k) )t, x, x ) = V x )T γ k) )t, x, x ) F T β k) )t, x, ξ ) = V x )F T γ k) )t, x, ξ ) By Sobolev, F T β k) )t, x, ξ ) L x L c V 6 L 5 + F T γ k) )t, x, ξ ) L x L c F T β k) )t, x, ξ ) L x L c Applying the L ξ norm and Plancherel in x, F T β k) )t, x, ξ ) L ξ L x L c Reviewing the definition of T, we see that V L x F T γ k) )t, x, ξ ) L x L c = V L F x T γ k) )t, x, ξ ) L x L c = V L x T γ k) )t, x, x ) L xx = V L x γ k) t, x, x ) L xx. The other cases are similar. Using frequency localization, we can share derivatives between two coordinates, as in the following corollary. Corollary 5.6. If γ k) is symmetric and then 5.6) β k) X k) + β k) t, x, x ) = V x x )γ k) t, x, x ) V 6 L 5 + x 4 x 4 γ k) L t L xx V L + xi γ k) L t L with i =, xx V L + γ k) L t L xx Proof. We need only to prove the first inequality of 5.6). The other two are directly from Lemma 5.5 and the fact that γ k) is symmetric i.e. x γ k) = x γ k). Split γ k) according to the relative magnitude of the ξ and ξ frequencies: γ k) = P ξ ξ γ k) + P ξ ξ γ k)