A NOTE ON THE DISCRETE FOURIER RESTRICTION PROBLEM

Transcription

1 A NOTE ON THE DISCRETE FOURIER RESTRICTION PROBLEM XUDONG LAI AND YONG DING arxv: v1 [mathap] 4 Oct 017 Abstract In ths aer we establsh a general dscrete Fourer restrcton theorem As an alcaton we make some rogress on the dscrete Fourer restrcton assocated wth KdV equaton 1 Introducton Recently the Fourer restrcton roblem has been wdely studed for examle see [10] [11] [1] [5] [3] In ths aer we nvestgate the dscrete Fourer restrcton roblems Let us frst see the dscrete Fourer restrcton assocated wth KdV equatons More recsely we are gong to seek the best constant A N satsfyng 11 ˆfn n 3 A N f L T where f s a erodc functon on T ˆf s the Fourer transform of f on T e ˆfξ = T e πx ξ fxdx N s a suffcent large nteger and = 1 For any ε > 0 Bourgan [] showed that A 6N N ε Later Hu and L [7] roved that A N ε N 1 8 +ε for 14 Bourgan [] and Hu and L [7] conjectured that { C for < 8 1 A N C ε N 1 8 +ε for 8 Clearly = 8 s the crtcal number In ths aer we wll make a slght rogress of ths conjecture We wll show that A N ε N 1 8 +ε for 1 It s easy to see that the study of A N s equvalent to the erodc Strchartz nequalty assocated wth KdV equaton: 13 a n e πxn+tn3 K L N a n xt T 010 Mathematcs Subject Classfcaton 4B0511L07 Key words and hrases Dscrete Fourer restrcton exonental sums The work s suorted by NSFC No No No SRFDP No the Fundamental Research Funds for the Central Unverstes No014KJJCA10 and the Chna Scholarsh Councl No

2 XUDONG LAI AND YONG DING In fact we have A N KN by usng the dual method Later whle consderng the Cauchy roblem of the ffth-order KdV-tye equatons Hu and L [8] studed the followng Strchartz nequalty 14 a n e πxn+tnk K L N a n xt T where k s a ostve nteger and k They [8] roved that K 6N N ε f k s odd and K N ε N 1 0 = 1 k+1 +ε for 0 where { k k + 6 f k s odd k 1 k + 4 f k s even In 13 and 14 the dscrete Fourer restrcton roblems are studed n two dmensons when the Fourer transform s ndeed restrcted to the curve n n 3 and n n k It s natural to consder a smlar roblem for hgher dmensons when the Fourer transform s restrcted to the general curve n k 1 n k d where k 1 k d are ostve ntegers Let K dn be the best constant n the followng nequalty 15 a n e πα 1n k 1 + +α d n k d K L dn a n Our man result n the resent aer s as follows Theorem 11 Let a n be a comlex number for all n N Let d > 1 and k 1 k d be ostve ntegers wth 1 k 1 < < k d = k Set K = d =1 k Let K dn be defned n 15 Suose kk + 1 Then for any ε > 0 we have 16 K dn ε N 1 1 K +ε where the mlct constant deends on k 1 k d ε but does not deend on N Remark 1 In [9] TD Wooley adated the effcent congruencng method to rove that 16 holds for kk + 1 And whenever > kk + 1 one may take ε = 0 n 16 Remark 13 In Secton 3 we wll show the bound n 16 s shar u to a constant ε One may conjecture 16 holds for all K Notce f k = = 1 d then K = dd + 1 Thus 16 s vald for K n ths case By usng Theorem 11 one could make some rogresses on the revous results Alyng Theorem 11 wth d = k 1 = 1 k = 3 and d = k 1 = 1 k = khere k we may obtan the followng corollares Corollary 14 Let K N be defned n 13 Suose 1 Then for any ε > 0 we get K N ε N ε

3 DISCRETE FOURIER RESTRICTION 3 where the mlct constant s ndeendent of N If > 4 one may take ε = 0 Corollary 15 Let K N be defned n 14 Suose kk + 1 Then for any ε > 0 we have K N ε N 1 k+1 1 +ε where the mlct constant s ndeendent of N If > kk + 1 one may take ε = 0 By settng d = k k = for = 1 k a n = 1 for n = 1 N a n = 0 for n = 0 1 N n Theorem 11 one obtan N 17 e πα 1n+ +α k n k dα ε N kk+1+ε T k for kk + 1 whch s the Vnogradov s mean value theorem roved by Bourgan Demeter and Guth [4] recently 15 can be regarded as a weghted verson of 17 and 15 s aarently harder than 17 Notce that the curve t k 1 t k t k d may be degenerate for examle the curve t t 3 has zero curvature at ont 0 0 It seems to be dffculty to use the method develoed n [3] and [4] to rove 16 for K snce what they deal wth are hyersurface wth nonzero Gaussan curvature or nondegenerate curve The roof of Theorem 11 s based on a key lemma from [4] Bourgan et al [4] used ths lemma to rove 17 Throughout ths aer the letter C stands for a ostve constant and C a denotes a constant deendng on a A ε B means A C ε B for some constant C ε A B means that A B and B A For a set E R d we denote Lebesgue measure of E by E Proof of Theorem 11 Before gvng the roof of Theorem 11 we frst ntroduce some lemmas Lemma 1 see Theorem 41 n [4] For each 1 n N let t n be a ont n n 1 N n N ] Suose B R s a ball n R d wth center c B and radus R Defne w BR x = 1 + x c B 00 R Then for each R N d each ball B R n R d each a n C each and ε > 0 we have 1 N a n e πx 1t n+ +x d t d n wbr xdx B R 1 ε N ε + N 1 1 dd+1 +ε N a n where the mlct constant does not deend on N R and a n

4 4 XUDONG LAI AND YONG DING Lemma Suose a n C and Then for any ε > 0 we get a n e πx 1n+x n + +x d n d dx ε N ε + N 1 dd+1 1 +ε where the mlct constant s ndeendent of N and a n Proof We frst notce that the functon a n e πx 1n+x n + +x d n d a n s erodc wth erod 1 n the varables x 1 x d By usng Mnkowsk s nequalty makng a change of varables and the above erodc fact one may get a n e πx 1n+x n + +x d n d dx a N a n e πx 1n+x n + +x d n d dx N a n e πx 1n+x n + +x d n d dx Hence to rove t suffces to show that N a n e πx 1n+x n + +x d n d dx has the desred bound Alyng Lemma 1 wth R = dn d t n = n N and B R = B0 R whch s centred at 0 we may obtan 3 N N d a n e πx 1 n N + +x d n N d w BR xdx ε N ε + N 1 1 dd+1 +ε N a n Snce w BR x 1 on B0 R and [0 N d ] d B0 R the left sde of 3 s bgger than N d [0N d ] d N a n e πx 1 n N + +x d n N d dx

5 DISCRETE FOURIER RESTRICTION 5 By makng a change of varables x 1 = Nα 1 x d = N d α d the above ntegral equals to 4 N d + dd+1 A N N a n e πα 1n+ +α d n d dα where A N = [0 N d 1 ] [0 N d ] [0 1] Notce that the functon K N α = N a n e πα 1n+α n + +α d n d s erodc wth erod 1 n the varables α 1 α d Snce A N has N dd 1 number of unt cubes by the erodc fact of K N α t follows that 4 s equal to N a n e πα 1n+ +α d n d dα whch comletes the roof Now we begn wth the roof of Theorem 11 We frst show that the roof can be reduced to the case k = kk + 1 that s 5 a n e πα 1n k 1 + +α d n k d L k ε N 1 1 K k +ε Suose 5 s true Utlzng Cauchy-Schwarz nequalty we get a n e πα 1n k 1 + +α d n k d N 1 L a n a n By usng the Resz-Thorn nterolaton theorem see for examle [6] to nterolate 5 and the above L estmate one could easly get the requred bound of L estmate for kk + 1 n Theorem 11 Therefore t remans to show 5 Consder ostve ntegers k 1 k d wth 1 k 1 < < k d = k and denote by {l 1 l s } the comlement set of {k 1 k d } n {1 k} Set K = d k n Then we may see 6 s =1 l = 1 kk + 1 K Note that k = kk + 1 s an even nteger therefore we may set k = u By usng the smle fact 1 0 eπxy dy = δx here δ s a Drac measure at 0

6 6 XUDONG LAI AND YONG DING we have 7 Λ := = = a n e πα 1n k 1 + +α d n k d u dα a n e πα 1n k 1 + +α d n k d n 1 n u N m 1 m u N u δ n k 1 m k 1 =1 m N a n1 a nu a m1 a mu u δ =1 n k d a m e πα 1m k 1 + +α d m k d u dα m k d Thus 7 equals to the number of ntegral soluton of the system of equatons u n k 1 m k 1 = 0 =1 8 u n k d m k d = 0 =1 n N m N = 1 u wth each soluton counted wth weght a n1 a nu a m1 a mu For each soluton n 1 n u m 1 m u of 8 there exst ntegers h j j = 1 k such that n 1 n u m 1 m u s an ntegral soluton of the followng system of equatons u n m = h 1 =1 u n m = h 9 =1 u n k mk = h k =1 n N m N = 1 u where h j = 0 f j = k for some = 1 d By the last condton of 9 t s easy to see that h j un j for j = 1 k On the other hand for each ntegral soluton n 1 n u m 1 m u of 9 wth h j un j for j = 1 k and h j = 0 f j = k for some 1 d n 1 n u m 1 m u s also an ntegral soluton of 8 Now we defne Λh = a n e πα 1n+α n + +α k n k u e π α 1h 1 α k h k dα T k

7 DISCRETE FOURIER RESTRICTION 7 By usng orthogonalty the above term s equal to n 1 n u N m 1 m u N a n1 a nu a m1 a mu u u δ n m h 1 δ n k m k h k =1 whch counts the number of ntegral soluton of 9 wth each soluton counted wth weght a n1 a nu a m1 a mu Combnng above arguments we conclude that Λ = h l1 un l 1 =1 h ls un ls Λh where h n the sum also satsfes h j = 0 f j = k for some = 1 d Obvously Λh Λ0 Hence we obtan Λ h l1 un l 1 h ls un ls Λ0 u s N l1+ +ls Λ0 N 1 kk+1 K N kε a n k where n the last nequalty we use 6 and aly Lemma wth = kk + 1 Hence we establsh 5 whch comletes the roof of Theorem 11 3 Sharness of Theorem 11 In ths secton we show that N 1 1 K s the best uer bound for K dn when K Therefore Theorem 11 s shar u to a factor of N ε Prooston 31 Let K dn be defned n 15 Suose s an even nteger Then there exst constants C 1 C such that { K dn max C 1 C N 1 1 K } Proof Set = u Let 1 k 1 < k < < k d and K = k k d Defne ΛN u = e πα 1n k 1 + +α d n k d u dα

8 8 XUDONG LAI AND YONG DING By usng orthogonalty ΛN u counts the number of ntegral soluton of the followng system of equatons u n k 1 m k 1 = 0 =1 31 u n k d m k d = 0 =1 n N m N = 1 u Notce that the system of equatons 31 has N + 1 u number of trval solutons In fact for each n 1 n u wth n N = 1 u one may choose m 1 m u = n 1 n u Hence we have 3 ΛN u CN Defne the set Ω N as { Ω N = α : α 1 } 8dN k = 1 d Then we have Ω N N K If α Ω N and n N then e πα 1n k 1 + +α d n k d Re Now we conclude that 33 ΛN u Ω N e πα 1n k 1 + +α d n k d e πα 1n k 1 + +α d n k d cosπα 1 n k α d n k d CN u dα CN Ω N CN K Recall K dn s the best constant for the followng nequalty a n e πα 1n k 1 + +α d n k d K L dn a n Choosng a n = 1 for all n N then we have K dn N 1 ΛN Combnng the estmates 3 and 33 we may get K dn max {C 1 C N 1 1 K } whch comletes the roof Acknowledgement The authors would lke to exress ther dee grattude to the referee for hs/her very careful readng and valuable suggestons

9 DISCRETE FOURIER RESTRICTION 9 References 1 J Bennett A Carbery and T Tao On the multlnear restrcton and Kakeya conjectures Acta Math no J Bourgan Fourer transform restrcton henomena for certan lattce subsets and alcatons to nonlnear evoluton equatons II: The KdV-equaton Geom Funct Anal No J Bourgan and C Demeter The roof of the l decoulng conjecture Ann of Math no J Bourgan C Demeter and L Guth Proof of the man conjecture n Vnogradov s mean value theorem for degrees hgher than three Ann of Math no J Bourgan and L Guth Bounds on oscllatory ntegral oerators based on multlnear estmates Geom Funct Anal 1011 no L Grafakos Classc Fourer Analyss Graduate Texts n Mathematcs Vol 49 Thrd edton Srnger New York Y Hu and X L Dscrete Fourer restrcton assocated wth KdV equatons Anal PDE no Y Hu and X L Local well-osedness of erodc ffth-order KdV-tye equatons J Geom Anal no T Wooley Dscrete Fourer restrcton va Effcent Congruencng Int Math Res Notces 017 no E M Sten Harmonc analyss: real-varable methods orthogonalty and oscllatory ntegrals Prnceton Unv Press Prnceton NJ T Tao Some recent rogress on the restrcton conjecture Fourer analyss and convexty Al Numer Harmon Anal Brkhäuser Boston Boston MA 004 Xudong LaCorresondng author: Insttute for Advanced Study n Mathematcs Harbn Insttute of Technology Harbn Peole s Reublc of Chna E-mal address: xudongla@malbnueducn Yong Dng: School of Mathematcal Scences Bejng Normal Unversty Bejng Peole s Reublc of Chna E-mal address: dngy@bnueducn