Discrete Fourier Restriction Associated with Schrödinger Equations

Transcription

1 Georgia Southern University Digital Southern Mathematical Sciences Faculty Publications Mathematical Sciences, Department of 2014 Discrete Fourier Restriction Associate with Schröinger Equations Yi Hu Georgia Southern University, Xiaochun Li University of Illinois at Urbana-Champaign Follow this an aitional works at: Part of the Mathematics Commons Recommene Citation Hu, Yi, Xiaochun Li "Discrete Fourier Restriction Associate with Schröinger Equations." Revista Matemática Iberoamericana, 30 (4): oi: /RMI/815 source: This article is brought to you for free an open access by the Mathematical Sciences, Department of at Digital Southern. It has been accepte for inclusion in Mathematical Sciences Faculty Publications by an authorize aministrator of Digital Southern. For more information, please contact

2 DISCRETE FOURIER RESTRICTION ASSOCIATED WITH SCHRÖDINGER EQUATIONS arxiv: v1 [math.ca] 17 Aug 2011 YI HU AND XIAOCHUN LI Abstract. In this paper, we present a ifferent proof on the iscrete Fourier restriction. The proof recovers Bourgain s level set result on Strichartz estimates associate with Schröinger equations on torus. Some sharp estimates on L 2(+2) norm of certain exponential sums in higher imensional cases are establishe. As an application, we show that some iscrete multilinear maximal functions are boune on L 2 (Z). 1. Introuction In this paper, we consier iscrete Fourier restriction problems associate with Schröinger equations. More precisely, for any given N N, let S,N stan for the set { } (n 1,,n ) Z : n j N, 1 j. For p > 1, let A p,,n represent the best constant satisfying (1.1) f(n, n 2 ) 2 A p,,n f 2 p, n S,N where n = (n 1,,n ) S,N, n = n n2, f is any Lp -function on T +1, f stans for Fourier transform of perioic function f on T +1, an p = p/(p 1). A harmonic analysis metho was introuce by Bourgain [1] to obtain (1.2) A p,,n CN 2(+2) p +ε for p > 2(+4). It was conjecture by Bourgain in [1] that {C p N 2(+2) +ε p (1.3) A p,,n for p 2(+2) C p for 2 p < 2(+2) The unerstaning of this conjecture is still incomplete. For instance, the esire upper bouns for A 5,1,N, A 3,2,N or A2(+2),,N for 3 are not yet obtaine. The most crucial estimate establishe by Bougain in [1] is certain (sharp) level set estimate. In this paper we provie a ifferent proof of the level set estimate.. These problems arise from the stuy of perioic nonlinear Schröinger equations: { x u+i t u+u u p 2 = 0 (1.4). u(x,0) = u 0 (x) This work was partially supporte by an NSF grant DMS

3 2 YI HU AND XIAOCHUN LI Here x = (x 1,,x ) T, an u(x,t) is a function of +1 variables which is perioic in space. The corresponing Strichartz estimate is the inequality seeking for the best constant K p,,n satisfying ( ) 1/2 (1.5) a n e 2πi(n x+ n 2t) K p,,n a n 2, n S,N L p (T +1 n ) where {a n } is a sequence of complex numbers. The restriction estimate (1.1) is essentially the Strichartz estimate because (1.6) K p,,n A p,,n follows easily by uality. The Duhamel s principle allows us to represent the ifferential equation as an integral equation u(x,t) = e it u 0 (x)+i t 0 e i(t τ) ( u(x,τ) p 2 u(x,τ) ) τ. Applying Picar s iteration an the Strichartz estimate (1.5), Bourgain in [1] obtaine local (global) well-poseness of the Schröinger equations (1.4). Hence, the iscrete restriction problems are crucial to stuy the ispersive equations on torus. Moreover, they are closely relate to Vinograov mean value conjecture on exponential sums, which is very interesting an important in aitive number theory. Let us introuce Vinograov s mean value in orer to see more clearly the connection between aitive number theory an iscrete Fourier restriction. For any given polynomial P(x,α 1,,α ) = k α jx j for α 1,,α k T, the mean value J k (N,b) is efine by 2b N J k (N,b) = e 2πiP(n,α 1,,α k ) α 1 α k. T k n=1 The Vinograov mean value conjecture aske the following question. For positive integers k an b, is it true that (1.7) J k (N,b) C k,b,ε (N b+ε +N 2b k(k+1) 2 +ε )? Vinograov invente a metho (now calle Vinograov metho) to establish some partial results on the mean value conjecture, an then utilize these partial results for exponential sums to gain new pointwise estimates, which can not be one via Weyl s classical squaring metho. One of main points in Vinograov s metho is that pointwise estimates of the exponential sums follow from the suitable upper boun of the mean value. Despite many brilliant mathematicians evote consierable time an energy to this conjecture, only k = 2 case is completely settle, an the conjecture is also answere affirmatively for cubic polynomials provie b > 8 ue to Hua s work. In terms of the language of iscrete restriction, Vinograov s mean value conjecture can be rephrase as a statement asking whether the following inequality is true: N (1.8) f(n,,n k ) 2 CN 1 k(k+1) p +ε f 2 p n=1

4 DISCRETE FOURIER RESTRICTION ASSOCIATED WITH SCHRÖDINGER EQUATIONS 3 for p k(k+1). Of course, (1.8) is apparently harer. In fact, (1.8) implies the conjecture. But the conjecture only yiels some partial results for (1.8). It will be very interesting if the equivalence of (1.7) an (1.8) coul be establishe. Despite the overwhelming ifficulty of (1.8), we pose a relatively simple question here. Let k 3 be a positive integer. Suppose p 2(k +1). Is it true that N (1.9) f(n,n k ) 2 CN 1 2(k+1) p +ε f 2 p? n=1 This question is essentially about the Strichartz estimates associate with higher orer ispersive equations. Bourgain s proof on (1.2) is base on three ingreients: Weyl s sum estimates, Hary-Littlewoo circle metho, an Tomas-Stein s restriction theorem. It is ifficult to employ Bourgain s metho for (1.9). Hence we are force to seek a metho, which can be ajuste to hanle the higher orer polynomials like ax + bx k. This is our main motivation. In this paper, we present a ifferent proof of (1.2). This paper is our first paper on the iscrete restriction. In the subsequent papers, we will moify this metho to obtain an affirmative answer to (1.9) for p large enough an then provie applications on the corresponing nonlinear ispersive equations. Our first theorem is about weighte restriction estimates, which eal with the large p cases of (1.1). Moreover, there is no ε require in the upper boun that we obtain. Theorem 1.1. For any σ > 0, any N, an any p > 4(+2), there exists a constant C inepenent of N such that σ n 2 (1.10) N f(n, n 2 2 ) 2 CN 2(+2) p f 2 p, for all f L p (T +1 ). n Z e Theorem 1.1 yiels (1.2) for large p immeiately. The proof of Theorem 1.1 presente in Section 2 is very straightforwar. The tool we use is Hary-Littlewoo circle metho. The ecay factor e σ n 2 /N 2 makes it possible to calculate L p norm of the kernel restricte to major arcs or minor arcs. For small p cases, we nee a new level set estimate, which implies Bourgain s level set estimate (see Corollary 1.1). Its proof relies on a ecomposition of the kernel, which is a sum of a L function an a function with boune Fourier transform (see Proposition 3.1). Theorem 1.2. Suppose that F is a perioic function on T +1 given by (1.11) F(x,t) = n S,N a n e 2πin x e 2πi n 2t, where {a n } is a sequence with n a n 2 = 1 an (x,t) T T. For any λ > 0, let { } E λ = (x,t) T +1 : F(x,t) > λ. Then for any positive number Q satisfying Q N, (1.12) λ 2 E λ 2 C 1 Q /2 E λ 2 + C 2N ε Q E λ

5 4 YI HU AND XIAOCHUN LI hols for all λ. Here C 1 an C 2 are constants inepenent of N an Q. Applying Theorem 1.2, we can easily obtain the following corollaries, which were prove by Bourgain in [1] in a ifferent way. The etails will appear in Section 3. Corollary 1.1. If λ CN /4 for some suitably large constant C, then the level set efine in Theorem 1.2 satisfies Corollary 1.2. E λ C 1 N ε λ 2(+2). (1.13) K p,,n C ε N 2 +2 p +ε if p > 2(+4) Remark 1.1. Corollary 1.2 clealy yiels (1.2) because K p,,n A p,,n. Moreover, the tiny positive number ε in (1.13) can be remove. Clearly from Theorem 1.1, we see immeiately that the ε is superfluous for larger p. For 2(+4) < p 4(+2), Bourgain in [1] succeee in removing the ε via a elicate interpolation argument. At the moment we were writing this paper, a new paper [4] pose by Bougain shows that the lower boun of p can be improve to by a multi-linear restriction theory. be 2(+3) Moreover, Theorem 1.2 implies the following recurrence relation on K p,,n in the sense of inequality. Corollary 1.3. For p > 2, we have (1.14) K p p,,n CN K p 2 p 2,,N +CN p 2 2+ε. Here C is inepenent of N. These three corollaries will be prove in Section 3. Carrying on the iea use in the proof of Theorem 1.2, we can get the following theorem. Theorem 1.3. Let N 1,,N N an S N1,,N be efine by (1.15) S N1,,N (x,t) = e 2πin x e 2πi n 2t. where S(N 1,,N ) is given by n S(N 1,,N ) (1.16) S(N 1,,N ) = {n = (n 1,,n ) Z : n j N j for all j {1,,}}. For any ε > 0, there exists a constant C inepenent of N such that (1.17) S N1,,N 2(+2) C(N 1 N ) 2(+2) max{n 1,,N } Observe that if N 1 = = N = N, (1.17) implies that (1.18) e 2πin x e 2πi n 2t N 2 +ε, n S,N 2(+2)/ that is, (1.19) ( ) 1/2 a n e 2πin x e 2πi n 2t N ε a n 2 n S,N n 2(+2)/ +2 +ε.

6 DISCRETE FOURIER RESTRICTION ASSOCIATED WITH SCHRÖDINGER EQUATIONS 5 provie a n = 1 for all n. If the conitions a n = 1 for all n coul be remove, then the Bourgain conjecture woul be solve for all p s not less than the critical inex 2(+2)/. Theorem 1.3 has a irect application to some multi-linear maximal functions, relate to maximal ergoic theorem, for instance, to pointwise convergence of the non-conventional bi-linear average N N 1 f 1 (T n )f 2 (T n2 ), n=1 where T is a measure preserving transformation on a probability space (X, A, µ). This application will appear in Section Large p Cases In this section we provie a proof of Theorem 1.1. All we nee to employ is the Hary- Littlewoo circle metho. Observethat forlarge p, A p,,n CN 2(+2) p by noticing n S,N f(n, n 2 ) 2 e σ e σ n 2 n S,N follows immeiately N f(n, n 2 2 ) 2 e σ 2 e σ n N f(n, n 2 2 ) 2. n Z Thus Theorem 1.1 yiels the esire upper bounsof A p,,n for large p cases. Here the ecay factor e σ n 2 N 2 will make our calculation much easier. The key iea is to ecompose the circle into arcs (calle major arcs an minor arcs) an then estimate L p norm of the corresponing kernel over each arcs. First we present some technical lemmas. In orer to introuce the major arcs, we shoul state Dirichlet principle. Lemma 2.1. (Dirichlet Principle) For any given N N an any t (0,1], there exist a,q N, 1 q N, 1 a q, (a,q) = 1, such that t a q 1 Nq. This principle can be prove by utilizing the pigeonhole principle or by the Farey issection of orer N. For any integer q, efine P q by P q = {a Z : 1 a q,(a,q) = 1}, an for any a P q, set the interval J a/q by J a/q = ( a q 1 Nq, a q + 1 Nq ). If q < N/10, the interval J a/q is calle a major arc, otherwise, a minor arc. Clearly we can partition (0,1] into a union of major arcs an minor arcs, that is, (0,1] = J a/q = M 1 M 2. 1 q N,a P q Here M 1 is the collection of all major arcs an M 2 is the union of all minor arcs. Lemma 2.2. Let 1 A enote the inicator function of a measurable set A. Then (2.20) J M 1 1 J J M 2 1 J

7 6 YI HU AND XIAOCHUN LI Proof. It is easy to see that all major arcs are isjoint. Thus it suffices to prove that 80. J M 2 1 J In fact, for any given minor arc J a0 /q 0, let Q enote the collection of all rational numbers a/q s such that each J a/q is a minor arc an there is a common point of J a0 /q 0 an all J a/q s. We shoul prove that the carinality of Q is less than 40. Notice that for any a/q Q, a 0 a q 0 q < Nq 0 Nq. This implies that a 0 q aq 0 < 2. Since a 0 q aq 0 Z, we conclue that either a 0 q aq 0 = 1 or a 0 q aq 0 = 1 if a/q a 0 /q 0. Hence if a/q a 0 /q 0, a/q Q must satisfy the iophantine equation a 0 x q 0 y = 1 or a 0 x q 0 y = 1 with x N. The general solution of the iophantine equation is x = x 0 +q 0 k an y = y 0 +a 0 k for all k Z an any given particular solution (x 0,y 0 ). Then kq 0 2N. By q 0 N/10, we have k 20. Thus the number of solutions of either iophantine equation is no more than 40. This completes the proof. Remark 2.1. Lemma 2.2 is about the finite overlapping property of minor arcs. The reason why we use this lemma is that we try to only calculate L p norm of the kernel restricte to each arc. Of course, this is not necessarily neee. An alternative way, which is very classic, is to obtain L norm for the kernel restricte to the union of minor arcs, an then to fin L p norm of the kernel on each major arc. Let K σ be a kernel efine by (2.21) K σ (x,t) = n Z e σ n 2 N 2 e 2πi n 2t e 2πin x. We set K a/q to be (2.22) K a/q (x,t) = K σ (x,t)1 Ja/q (t). The following lemma gives an upper boun for L p norm of K a/q. Lemma 2.3. For any integer 1 q N, any integer a P q an any p > 2(+1), (2.23) CN +2 p Ka/q p q 2 p Proof. For any given t J a/q, let β = t a q an write n = kq+l. Here l Z q = {(l 1,,l ) : l j Z q }. Then we have K σ (x,t) = k Z l Z q Interchanging the sums, we represent the kernel as K σ (x,t) = e 2πi l 2 a q l Z q e σ kq+l 2 N 2 e 2πi(kq+l) x e 2πi kq+l 2 ( a q +β). k Z e kq+l 2(. σ N 2 2πiβ) e 2πi(kq+l) x.

8 DISCRETE FOURIER RESTRICTION ASSOCIATED WITH SCHRÖDINGER EQUATIONS 7 Applying Poisson summation formula to the inner sum, we have σ N 2 2πiβ) e 2πi(kq+l) x = ( ) π q σ e 2πil k 2πiβ k Z N 2 k Z e kq+l 2( Henceforth, the kernel can be written as ( ) π (2.24) K σ (x,t) = q σ 2πiβ N 2 k Z e π2 x k q 2 σ N 2 2πiβ l Z q 2 x k q 2 q e π σ N 2 2πiβ e 2πi l 2 a qe 2πil k q. From the well-known result on the upper boun of the Gauss sum, it follows that e 2πi l 2 a qe 2πil k q (2q)/2. l Z q Thus by inserting the absolute value, the kernel can be majorize by K σ (x,t) q 2 (2π) /2 ( ) σ 2 +4π N 2 β k Z e π2 x k q 2 σ N 2 σ 2 N 4 +4π2 β 2. Integrating K σ p on each arc J a/q, we obtain that Ka/q p p = Notice that for β 1 Nq This yiels that β 1 Nq β 1 Nq T (2π) p/2 q p 2 q p 2 an q N, σ ( )p σ 2 +4π N 2 β (2π) p/2 ( )p σ 2 +4π N 2 β q 2 N 2 1 σ 2 +4π N 2 β C σ. 2 4 e k Z π2 x k q 2 σ N 2 σ 2 For p > 2(+1), we estimate L p norm of K a/q by Ka/q p p β 1 Nq q p 2 (2π) p/2 0 N 4 +4π2 β 2 C σ. ( )p σ 2 N +4π 2 β k Z e e k Z e k Z π2 x k q 2 σ N 2 σ 2 N 4 +4π2 β 2 π2 x k q 2 σ N 2 σ 2 N 4 +4π2 β 2 π2 x k q 2 σ N 2 σ 2 p p N 4 +4π2 β 2 x xβ x β, β.

9 8 YI HU AND XIAOCHUN LI which can be boune by β 1 Nq Therefore, we finish our proof. Lemma 2.4. For p > 2(+2), C(2π) p 2 N q p ( )p σ π N 2 β (2.25) K σ p C p,σ N +2 p. Proof. By Lemma 2.2 an Lemma 2.3, we have that N K σ p p C which yiels Lemma 2.4. q=1 a P q Ka/q p p C N q=1 β CNp 2. q p 2 Np 2 CN p 2, a P q q p 2 We now return to the proof of Theorem 1.1. Inee, observe that σ n 2 N f(n, n 2 2 ) 2 = K σ f,f. n Z e Applying Höler s inequality an then Hausorff-Young s inequality on convolution, we get K σ f,f K σ p/2 f 2 p. Since p > 4(+2), we employ Lemma 2.4 to conclue Theorem Level Set Estimates In this section, we provie a proof of Theorem 1.2. Theorem 1.2 can be utilize for hanling small p cases. First, we state an arithmetic result. Lemma 3.1. For any integer Q 1 an any integer n 0, an any ε > 0, a q n C ε (n,q)q 1+ε. Q q<2q a P q e 2πi Here (n,q) enotes the number of ivisors of n less than Q an C ε is a constant inepenent of Q,n. Lemma 3.1 can be prove by observing that the arithmetic function efine by f(q) = a P q e 2πia q n is multiplicative, an then utilize the prime factorization for q to conclue the lemma. The etails can be foun in [1]. We now state a proposition crucial to our proof. Proposition 3.1. For any given positive number Q with N Q N 2, the kernel K σ given by (2.21) can be ecompose into K 1,Q +K 2,Q such that (3.26) K 1,Q C 1 Q 2.

10 an DISCRETE FOURIER RESTRICTION ASSOCIATED WITH SCHRÖDINGER EQUATIONS 9 (3.27) K 2,Q C 2N ε Q. Here the constants C 1,C 2 are inepenent of Q an N. Proof. We can assume that Q is an integer, since otherwise we can take the integer part of Q. For a stanar bump function ϕ supporte on [1/200,1/100], we set (3.28) Φ(t) = ( ) t a/q 1/q 2. ϕ Q q<2qa P q Clearly Φ is supporte on [0,1]. We can exten Φ to other intervals perioically to obtain a perioic function on T. For this perioic function generate by Φ, we still use Φ to enote it. Then it is easy to see that F R ϕ(0) (3.29) Φ(0) = q 2 = q Qa P q q Q φ(q) q 2 F Rϕ(0) is a constant inepenent of Q. Here φ is Euler s totient function, an F R enotes Fourier transform of a function on R. Also we have a (3.30) Φ(k) = q k F R ϕ(k/q 2 ). We efine that 1 q 2e 2πi q Qa P q K 1,Q (x,t) = 1 Φ(0) K σ(x,t)φ(t), an K 2,Q = K σ K 1,Q. We prove (3.27) first. In fact, write Φ as its Fourier series to get K 2,Q (x,t) = 1 Φ(k)e 2πikt K σ (x,t). Φ(0) Thus its Fourier coefficient is K 2,Q (n,n +1 ) = e σ n 2 /N 2 Φ(k)1 {n+1 = n Φ(0) 2 +k}(k). k 0 Here n Z an n +1 Z. This implies that K 2,Q (n,n +1 ) = 0 if n +1 = n 2, an if n +1 n 2, K 2,Q (n,n +1 ) = e σ n 2 /N 2 Φ(n +1 n 2 ). Φ(0) k 0 Applying (3.30) an Lemma 3.1, we estimate K 2,Q (n,n +1 ) by K 2,Q (n,n +1 ) CNε Q, since N Q N 2. Henceforth we obtain (3.27). We now prove (3.26). Observe that [ a q + 1, a 200q 2 q + 1 ] s are pairwise isjoint. Thus 100q 2 we can fix q Q an a P q an try to obtain the upper boun of K 1,Q restricte to

11 10 YI HU AND XIAOCHUN LI [ a q q, a 2 q q ]. Let β = t a 2 q. Hence we have β 1/q2 for t [ a q q, a 2 q + 1 As we i in the previous section, by Poisson summation formula, we have ( ) π K σ (x,t) = q π2 x k q 2 σ σ e 2πi l 2 a qe 2πil k q. N 2πiβ 2 Hence for β 1/q 2, we estimate which is boune by K σ (x,t) CN q /2 C k Z e ( ( ) q /2 σ ) 2 4 N +β 2 2 e π2 N 2 σ k q x 2 k Z This implies (3.26). Therefore we complete the proof. N 2 2πiβ l Z q k Z e π 2 k q x 2 ( σ N 2 ) 2 +β 2 C σ q /2 C σ Q /2. σ N 2, 100q 2 ]. We now start to prove Theorem 1.2. For the function F an the level set E λ given in Theorem 1.2, we efine f to be f(x,t) = F(x,t) F(x,t) 1 E λ (x,t). Clearly λ E λ F(x,t)f(x,t)xt. T +1 By the efinition of F, we get λ E λ a n f(n, n 2 ). n S,N Utilizing Cauchy-Schwarz s inequality, we have λ 2 E λ 2 e σ n n S,N f(n, n 2 ) 2. The right han sie is boune by e σ n 2 N f(n, n 2 2 ) 2 = e σ K σ f,f. For any Q with N Q N 2, we employ Proposition 3.1 to ecompose the kernel K σ. Then we have λ 2 E λ 2 C σ K 1,Q f,f +C σ K 2,Q f,f From (3.26) an (3.27), we then obtain λ 2 E λ 2 C 1 Q /2 f C 2N ε Q f 2 2 C 1 Q /2 E λ 2 + C 2N ε Q E λ. The case Q N 2 is trivial since the level set E λ is empty if λ > CN /2. Therefore, we finish the proof of Theorem 1.2.

12 DISCRETE FOURIER RESTRICTION ASSOCIATED WITH SCHRÖDINGER EQUATIONS 11 We now start to prove Corollary 1.1 by using Theorem 1.2. We shoul take Q /2 = 1 2C 1 λ 2, where C 1 is the constant state in (1.12). Since Q N, we nee to restrict λ > 2C 1 N /4. Then E λ CN ε λ 2(+2)/ follows immeiately from (1.12). This completes the proof of Corollary 1.1. To prove Corollary 1.2, write which equals to F p p = C p 0 λ p 1 E λ λ, CN /4 C p λ p 1 E λ λ+c p λ p 1 E λ λ. 0 CN /4 Utilizing the trivial estimate E λ Cλ 2 for the first term an employing Corollary 1.1 for the secon term, we then obtain, for p > 2(+4), F p p CN p 2 (+2)+ε as esire. Therefore the proof of Corollary 1.2 is complete. We now prove Corollary 1.3. Multiply (1.12) by λ p 3 to get, for N Q, (3.31) λ p 1 E λ C 1 Q /2 λ p 3 E λ + C 2N ε Integrating (3.31) in λ from 0 to CN /2, we obtain that Q λp 3. (3.32) F p p C 1Q /2 F p 2 p 2 +C N p 2 +ε 2. Q Taking Q = N 2, we then have (3.33) F p p C 1N K p 2 p 2,,N +C 2N p 2 2+ε. This finishes the proof of Corollary Proof of Theorem 1.3 In this section, we prove Theorem 1.3 by carrying the similar iea shown in Section 3. We introuce a level set G λ for any λ > 0 by setting, { } (4.34) G λ = (x,t) T T : S N1,,N (x,t) > λ. As we i in Section 3, let f = 1 Gλ S N1,,N / S N1,,N an we then have (4.35) λ G λ f(n,n 2 ) = f N1,,N,S N1,,N, n S(N 1,,N ) where f N1,,N is a rectangular Fourier partial sum efine by (4.36) f N1,,N (x,t) = f(n,n +1 )e 2πn x e 2πin+1t. n S(N 1,,N ) n +1 max{n 1,,N } 2

13 12 YI HU AND XIAOCHUN LI Here unlike what we i in Section 3, we o not use Cauchy-Schwarz inequality for the right han sie of (4.35). We actually nee to get a ecomposition of S N1,,N. Before we state this ecomposition, we shoul inclue a famous result on Weyl s sum. Lemma 4.1. Suppose t is a real number satisfying t a q 1 q 2. Here a an q are relatively prime integers. Then N { (4.37) e 2πi(tn2 +xn) N Cmax q, N logq, } qlogq. n=1 The proof can be one by Weyl s squaring metho. See [5] or [8] for etails. Lemma 4.2. For any real number Q with max{n 1,,N } Q max{n 1,,N } 2, the function S N1,,N efine in (1.15) can be written as a sum of S 1,Q an S 2,Q, where S 1,Q satisfies (4.38) S 1,Q CQ /2 (logq) /2 an S 2,Q satisfies (4.39) Ŝ2,Q Cmax{N 1,,N } ε. Q Here the constant C is inepenent of N 1,,N an Q. Proof. Let Φ be the function efine in (3.28). We then obtain (4.40) S N1,,N = S 1,Q +S 2,Q, where S 1,Q is given by (4.41) S 1,Q (x,t) = 1 Φ(0) S N 1,,N (x,t)φ(t) an S 2,Q is (4.42) S 2,Q = S N1,,N S 1,Q. (4.38) follows immeiately from (4.37). Notice that S 2,Q (x,t) = 1 Φ(k)e 2πikt S N1,,N (x,t). Φ(0) k 0 (4.39) follows by using Lemma 3.1, as we i in the proof of (3.27). Hence we finish the proof. We now return to the proof of Theorem 1.3. From (4.35) an Lemma 4.2, the level set G λ satisfies (4.43) λ G λ f N1,,N,S 1,Q + f N1,,N,S 2,Q, which can be boune by (4.44) C Q/2 (logq) /2 f N1,,N 1 + n S(N 1,,N ) n +1 max{n 1,,N } 2 Ŝ2,Q(n,n +1 ) f(n,n +1 ).

14 DISCRETE FOURIER RESTRICTION ASSOCIATED WITH SCHRÖDINGER EQUATIONS 13 Thus from the fact that L 1 norm of Dirichlet kernel D N is comparable to logn, (4.39), an Cauchy-Schwarz inequality, we have (4.45) λ G λ CQ /2 (logq) 2 G λ + C(N 1 N ) 1/2 max{n 1,,N } 1+ε G λ 1/2. Q For λ Cmax{N 1,,N } 2 +ε, take Q to beanumbersatisfying Q /2 max{n 1,,N } ε = λ an then Lemma 4.2 yiels (4.46) G λ CN 1 N max{n 1,,N } 2+ε. Notice that λ 2(+2) (4.47) S N1,,N 2 (N 1 N ) 1/2. Thus for λ < Cmax{N 1,,N } 2 +ε, we have (4.48) G λ CN 1 N λ 2 CN 1 N max{n 1,,N } 2+ε. λ 2(+2) Henceforth (4.46) hols for all λ > 0. We now estimate L 2(+2) 2(+2) (4.49) S N1,,N 2(+2) C 2 N 1 N 1 λ 2(+2) 1 G λ λ+c norm of S N1,,N by 1 0 λ 2(+2) 1 G λ λ. Since (4.46) hols for all λ > 0, the first term in the right han sie of (4.49) can be boune by CN 1 N max{n 1,,N } 2+ε. The secon term is clearly boune by C because G λ is a set with finite measure. Putting both estimates together, we get 2(+2) (4.50) S N1,,N 2(+2) CN 1 N max{n 1,,N } 2+ε, as esire. Therefore, we complete the proof. 5. Estimates of multi-linear maximal functions In this section, we shoul provie an application of Theorem 1.3. Definition 5.1. Let N an K {1,,}. A subset S of N is calle K-amissible if for every element (n 1,,n ) S, there exist n i1,,n ik such that i 1 < i 2 < < i K an i 1,,i K {1,,}; max{n 1,,n } Cmin{n i1,,n ik }. Here the constant C is inepenent of (n 1,,n ). Theorem 5.1. Let,M 1,,M N, K {1,,}, an A M1,,M be a multi-linear operator efine by setting A M1,,M (f 1,,f +1 )(n) to be (5.51) 1 M 1 M M 1 m 1 =1 M m =1 Here n Z. Suppose T is a maximal function given by f 1 (n m 1 ) f (n m )f +1 ( n (m m 2 )). (5.52) T (f 1,,f +1 )(n) = sup (M 1,,M ) S K A M1,,M (f 1,,f +1 )(n).

15 14 YI HU AND XIAOCHUN LI Here S K is any K-amissible subset of N. Then if K satisfies (5.53) K > 2 +4, then we have +1 (5.54) T (f 1,,f +1 ) L 2 (Z) C f j L 2 (Z). Here L 2 (Z) stans for L 2 norm associate with counting measure on Z, an C is inepenent of f j s but may epen on K an. 2 Remark 5.1. Notice that for = 1,2,3, +4 < 1. Thus the conition (5.53) is superfluous in Theorem 5.1 for = 1,2,3. Thus for = 1,2,3, the set S K in Theorem 5.1 can be replace by N because N is 1-amissible accoring to Definition 5.1. It is very possible that, for 4, the conition (5.53) on K is reunant too. A elicate analysis involving the circle metho shoul be utilize in orer to remove (5.53) for the 4 cases. We woul not iscuss this in this paper. Remark 5.2. It is natural to ask whether the following inequality hols. +1 (5.55) T (f 1,,f +1 ) 2 C f j L L +1(Z) 2 (Z)? This seems to be ifficult but also be interesting. So far we are only able to establish the bouneness of T from L 2 L 2 to L p for p > 2/(+1) by an interpolation argument an Theorem 5.1. To prove Theorem 5.1, we first introuce a simple multi-linear estimate. Lemma 5.1. Let M N an F 1,,F M+1 be perioic functions on T. Let T(F 1,,F M+1 ) be a multilinear operator given by (5.56) T(F 1, F M+1 )(x 1,,x M ) = F 1 (x 1 ) F M (x M )F M+1 (x 1 + +x M ), for (x 1,,x M ) T M. If 1 p 2M M+1, (5.57) T(F 1, F M+1 ) L p (T M ) M+1 F j L 2 (T). Proof. We only nee to prove the case when p = 2M M+1, since other cases follow easily by Höler equality. By a change of variables, we get (5.58) T(F 1, F M+1 ) L p (T M ) F i F j p, j i j {1,,M+1} for any i {1,,M + 1}. Now set α 1,,α M+1 Q M+1 by α 1 = (0, 1 p,, 1 p ), α 2 = ( 1 p,0, 1 p,, 1 p ),, α M+1 = ( 1 p,, 1 p ( 1 (5.59) 2,, 1 ) 2 2M,0). Clearly for p = M+1, we have = 1 M +1 (α 1 + +α M+1 ). Thus ( 1 2,, 1 2) is in the convex hull generate by α1,,α M+1. (5.57) follows immeiately by interpolation.

16 DISCRETE FOURIER RESTRICTION ASSOCIATED WITH SCHRÖDINGER EQUATIONS 15 To finish the proof of Theorem 5.1, we nee the following proposition. Proposition 5.1. Let N, K {1,,}, M K+1,,M N. Let A M,MK+1,,M be efine by setting A M,MK+1,,M (f 1,,f +1 )(n) to be (5.60) M 1 K M j ( M K f j (n m j ) f j (n m j ) f +1 n (m 2 M 1 + +m 2 )) K+1 M m j =1 j=k+1m j =1 Suppose that M Cmax{M K+1,,M }. Then we have (5.61) A M,MK+1,,M (f 1,,f +1 ) L 2 (Z) C(M K+1 M ) +4 2(+2) M (+4)K+2 Proof. By uality, it is sufficient to prove that for any f +2 L 2 (Z), (5.62) A M,MK+1,,M (n)f +2 (n) C(M K+1 M ) +4 2(+2) M (+4)K+2 n Now efine F j for any j {1,,+2} by 2(+2) +ε +1 2(+2) +ε +2 f j L 2 (Z). f j L 2 (Z). (5.63) F j (x) = n f j (n)e 2πinx. Then the left han sie of (5.62) can be represente by (5.64) 1 M K M K+1 M T +1 Here S(x 1,,x +1 ) is given by (5.65) M M S(x 1,,x +1 ) = m 1 =1 Utilizing Theorem 1.3, we have S 2(+2) Then Höler inequality yiels that +1 F j (x j )F +2 (x 1 + +x +1 )S(x 1,,x +1 )x 1 x +1. M K+1 m K =1m K+1 =1 M m =1 C(M K+1 M ) (5.64) C T(F 1,,F +2 ) 2(+2) +4 e 2πi(m 1x 1 + +m x ) e 2πi(m m2 )x +1. 2(+2) M K 2(+2) ε. (M K+1 M ) +4 2(+2) M K 2(+2) + +2 K+ε. Since 2(+2) +4 2(+1) +2, we can apply Lemma 5.1 to obtain (5.66) (5.64) C(M K+1 M ) +4 2(+2) M (+4)K+2 2(+2) +ε +2 F j L 2 (T).

17 16 YI HU AND XIAOCHUN LI We now prove Theorem 5.1. Since S K is K-amissible, without loss of generality, we assume that M 1 = = M K = M an M Cmax{M K+1,,M }. Moreover, we may also assume that M is yaic. Henceforth we only nee to consier T (f 1,,f +1 ) given by (5.67) T (f 1,,f +1 ) = sup M,M K+1,,M AM,MK+1,,M (f 1,,f +1 ). Clearly we have T (f 1,,f +1 ) M,M K+1,,M A M,MK+1,,M (f 1,,f +1 ) 1/2 2. Taking L 2 norm for both sies, we then get (5.68) T (f 1,,f +1 ) L 2 (Z) A M,MK+1,,M (f 1,,f +1 ) 2 L 2 (Z) M,M K+1,,M 1/2. Employing Proposition 5.1, we estimate T (f 1,,f +1 ) L 2 (Z) by (5.69) +4 (+2) M (+4)K+2 +ε (+2) which is boune by M,M K+1,,M C(M K+1 M ) C +1 f j L 2 (Z), 1/2 +1 f j L 2 (Z), since K > 2 (+4)K 2 +4 implies (+2) > 0. This completes the proof of Theorem 5.1. A similar argument yiels Theorem 5.2. We omit its proof. Theorem 5.2. Let N, N N, an A N be a multi-linear operator efine by setting A N (f 1,,f +1 )(n) to be (5.70) 1 N N m 1 =1 N m =1 Here n Z. Suppose T be a maximal function given by f 1 (n m 1 ) f (n m )f +1 ( n (m m 2 )). (5.71) T (f 1,,f +1 )(n) = sup A N (f 1,,f +1 )(n). N N Then we have +1 (5.72) T (f 1,,f +1 ) L 2 (Z) C f j L 2 (Z). Also we are able to obtain L 2 estimate for the corresponing bilinear Hilbert transform. Theorem 5.3. Let K be a function on Z satisfying (5.73) K(n) C n

18 DISCRETE FOURIER RESTRICTION ASSOCIATED WITH SCHRÖDINGER EQUATIONS 17 for n 0. Let T(f 1,f 2 ) be efine by (5.74) T(f 1,f 2 )(n) = m 0K(m)f 1 (n m)f 2 (n m 2 ), for Schwartz functions f 1,f 2 : R C. Then we have (5.75) T(f 1,f 2 ) L 2 (Z) C f 1 L 2 (Z) f 2 L 2 (Z). Proof. For any yaic number M 1, efine T M (f 1,f 2 ) by (5.76) T M (f 1,f 2 )(n) = 1 f 1 (n m)f 2 (n m 2 ). M Apply Proposition 5.1 to get m M (5.77) T M (f 1,f 2 ) L 2 (Z) M 1/2+ε f 1 L 2 (Z) f 2 L 2 (Z). (5.75) follows from (5.77). Remark 5.3. If the kernel K in Theorem 5.3 has some cancellation conition, then T(f 1,f 2 ) coul be a boune operator from L 2 L 2 to L 1. This problem is still open an seems to be challenging. 6. Estimate for K p,,n when p is even In this section, we give a proposition on K p,,n when p is even. The iea is not new, an it is utilize often in the fiel of number theory. For the sake of self-containeness, we inclue it here. By using it an an arithmetic argument, one can get sharp estimates, up to a factor of N ε, for K 6,1,N, K 4,2,N, etc. See [1] for etails. Proposition 6.1. If p > 0 is an even integer, then we have (6.78) K p p,,n sup e 2πεm F T T (Fp/2 (, +iε))(l,m). (l,m) S,pN/2 {1,,pN 2 /2} Here F T T is Fourier transform of functions on T T, ε is any positive number, an F is given by (6.79) F(x,z) = n Z e 2πiz n 2 +2πix n. Proof. Let k = p/2. A irect calculation yiels (6.80) 2k a n e 2πi(n x+ n 2t) xt = a n1 a nk a m1 a mk. T +1 n S,N (n 1,,n k,m 1,,m k ) S,N,k Here S,N,k is given by k S,N,k = (n 1,,n k,m 1,,m k ) S p,n : n j = k m j, k n j 2 = k m j 2

19 18 YI HU AND XIAOCHUN LI For any l S,kN an any positive integer m kn 2, we set k S k (l,m) = (n 1,,n k ) S,N k : n j = l, We now can estimate (6.80) by (6.81) kn 2 l S,kN m=1 (n 1,,n k ) S k (l,m) k n j 2 = m. a n1 a nk Utilizing Cauchy-Schwarz inequality an the fact that {S k (l,m)} forms a partition of S k,n, we ominate (6.81) by ( ) k (6.82) max l S,kN,1 m kn 2 S k(l,m) a n 2, where S k (l,m) enotes the carinality of S k (l,m). Employing the elementary fact 1 0 e2πinθ θ = 0 if n 0 an 1 0 e2πinθ θ = 1 if n = 0, for any l S,kN an any positive integer m kn 2, we can estimate S k (l,m) by 1 (6.83) e 2πit( k n j 2 m) t e 2πi k x n j e 2πix l x, T which equals to (6.84) (n 1,,n k ) S k,n (n 1,,n k ) S k,n 0 e 2πεm 1 0 n e 2πi(t+iε) k n j 2 e 2πimt t e 2πi k x n j e 2πix l x, T for any real number ε. This term can also be written as (6.85) e 2πεm k e 2πix n e 2πix l e 2πimt xt. T T n S,N e 2πi(t+iε) n 2 Notice that we may replace S,N by Z in (6.83), (6.84) an (6.85) to make upper bouns larger. Thus, by the efinition of F in (6.79), we ominate S k (l,m) by (6.86) e 2πεm (F(x,t+iε)) k e 2πix l e 2πimt xt. T T This finishes the proof of Proposition 6.1. Acknowlegement. The first author wishes to thank his avisor Xiaochun Li, for his valuable an insightful suggestions an enthusiastic guiance. 2

20 DISCRETE FOURIER RESTRICTION ASSOCIATED WITH SCHRÖDINGER EQUATIONS 19 References [1] J.Bourgain, Fourier transform restriction phenomena for certain lattice subsets an applications to nonlinear evolution equations. Part I: Schröinger equations, GAFA, Vol. 3, No. 2, 1993, [2] J.Bourgain, Fourier transform restriction phenomena for certain lattice subsets an applications to nonlinear evolution equations. Part II: The KDV-equations, GAFA, Vol. 3, No. 3, 1993, [3] J. Bourgain, Pointwise ergoic theorems for arithmetic sets, Inst. Hautes Etues Sci. Publ. Math. 69 (1989), [4] J. Bourgain, Moment inequalities for trigonometric polynomials with spectrum in curve hypersurfaces, arxiv: v1. [5] L. K. Hua, Aitive theory of prime numbers, translations of math. monographs, Vol. 13, AMS, [6] A. Ionescu an S. Wainger, L p bouneness of iscrete singular raon transforms, JAMS, Vol. 19, No. 2 (2007), [7] A. Magyar, E. Stein, an S. Wainger, Discrete analogues in harmonic analysis: spherical averages. Ann. of Math. (2) 155 (2002), no. 1, [8] H. L. Montgomery, Ten lectures on the interface between analytic number theory an harmonic analysis, CBMS, No. 84, AMS, [9] I. M. Vinograov, The metho of trigonometrical sums in the theory of numbers, Intersci. Publishers, ING., New York, Yi Hu, Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL, 61801, USA aress: yihu1@illinois.eu Xiaochun Li, Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL, 61801, USA aress: xcli@math.uiuc.eu